From a73961416b54b2a95b13af7afdf7f7ed8c83e634 Mon Sep 17 00:00:00 2001 From: Bharat Ramanathan Date: Wed, 11 Sep 2024 22:21:35 +0530 Subject: [PATCH 1/3] feat: add initial dspy solution solver --- try_dspy.ipynb | 384 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 384 insertions(+) create mode 100644 try_dspy.ipynb diff --git a/try_dspy.ipynb b/try_dspy.ipynb new file mode 100644 index 0000000..d25c4f6 --- /dev/null +++ b/try_dspy.ipynb @@ -0,0 +1,384 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "import pathlib\n", + "import dspy\n", + "from dspy import Example\n", + "from dspy.evaluate.evaluate import Evaluate\n", + "from dspy.evaluate.metrics import answer_exact_match\n", + "from functools import partial\n", + "from dspy import Signature, InputField, OutputField\n", + "from pydantic import BaseModel, Field\n", + "from datetime import datetime\n", + "from dspy.teleprompt import MIPROv2\n", + "from dspy.teleprompt.random_search import BootstrapFewShotWithRandomSearch\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "class Problem(BaseModel):\n", + " problem_dir: pathlib.Path = Field(\n", + " ..., description=\"The path to the problem directory\"\n", + " )\n", + " problem_name: str = Field(..., description=\"The name of the problem\")\n", + " problem_description: str = Field(..., description=\"The description of the problem\")\n", + " sample_input: str = Field(..., description=\"The sample input of the problem\")\n", + " sample_output: str = Field(..., description=\"The sample output of the problem\")\n", + " problem_input: pathlib.Path = Field(..., description=\"The path to the input file\")\n", + " problem_output: pathlib.Path = Field(..., description=\"The path to the output file\")\n", + " solution: str = Field(..., description=\"The solution to the problem\")\n", + "\n", + " @property\n", + " def as_xml(self) -> str:\n", + " return f\"\"\"\n", + "\n", + "\n", + "{self.problem_description}\n", + "\n", + "\n", + "\n", + "{self.sample_input}\n", + "\n", + "\n", + "{self.sample_output}\n", + "\n", + "\n", + "\n", + "\"\"\"\n", + "\n", + "\n", + "def load_problem(problem_name: str, problem_dir: pathlib.Path) -> Problem:\n", + " problem_input = problem_dir / f\"{problem_name}.in\"\n", + " problem_output = problem_dir / f\"{problem_name}.out\"\n", + " sample_input = problem_dir / f\"{problem_name}_sample_input.txt\"\n", + " sample_output = problem_dir / f\"{problem_name}_sample_output.txt\"\n", + " problem_description = problem_dir / f\"{problem_name}.md\"\n", + " solution = problem_dir / f\"{problem_name}_sol.md\"\n", + " return Problem(\n", + " problem_dir=problem_dir,\n", + " problem_name=problem_name,\n", + " problem_description=problem_description.read_text(),\n", + " sample_input=sample_input.read_text(),\n", + " sample_output=sample_output.read_text(),\n", + " problem_input=problem_input,\n", + " problem_output=problem_output,\n", + " solution=solution.read_text(),\n", + " )" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "problems = list(pathlib.Path(\"data/dataset/\").rglob(\"*.in\"))\n", + "eval_problems = list(filter(lambda x: \"2022/final\" in str(x), problems))\n", + "test_problems = list(filter(lambda x: \"2023/practice\" in str(x), problems))\n", + "train_problems = list(filter(lambda x: x not in eval_problems + test_problems, problems))\n", + "\n", + "\n", + "def load_problem_from_path(problem_path: pathlib.Path):\n", + " problem_name = problem_path.stem\n", + " problem_dir = problem_path.parent\n", + " problem = load_problem(problem_name, problem_dir)\n", + " return problem\n", + "\n", + "\n", + "\n", + " \n", + "train_problems = list(map(load_problem_from_path, train_problems))\n", + "eval_problems = list(map(load_problem_from_path, eval_problems))\n", + "test_problems = list(map(load_problem_from_path, test_problems))\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "(len(train_problems), len(eval_problems), len(test_problems))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "trainset = [Example(\n", + " problem_statement=problem.problem_description,\n", + " sample_input=problem.sample_input,\n", + " sample_output=problem.sample_output,\n", + " solution=problem.solution).with_inputs(\"problem_statement\", \"sample_input\", \"sample_output\") for problem in train_problems]\n", + "\n", + "devset = [Example(\n", + " problem_statement=problem.problem_description,\n", + " sample_input=problem.sample_input,\n", + " sample_output=problem.sample_output,\n", + " solution=problem.solution).with_inputs(\"problem_statement\", \"sample_input\", \"sample_output\") for problem in eval_problems]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "\n", + "lm = dspy.OpenAI(\n", + " model=\"gpt-4o-mini\", # Note: didn't find much a difference btwn mini & full gpt-4o\n", + " max_tokens=4000,\n", + " temperature=0.1,\n", + " )\n", + "\n", + "dspy.settings.configure(lm=lm)\n", + "dspy.configure(experimental=True)\n", + "\n", + "\n", + "def validate_solution(example, prediction, trace=None,frac=0.8):\n", + " \n", + " result = answer_exact_match(\n", + " example=Example(answer=example.solution),\n", + " pred=Example(answer=prediction.solution),\n", + " trace=trace, \n", + " frac=frac)\n", + " return result\n", + " \n", + "\n", + "# Setup evaluation function\n", + "evaluate = Evaluate(\n", + " devset=devset,\n", + " num_threads=6, # Note: Set this to 1 for debugging purposes \n", + " display_progress=True,\n", + " display_table=5,\n", + " metric=validate_solution\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "class GenerateSolution(Signature):\n", + " \"\"\"You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + " When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + " Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + " Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + " Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + " Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + " Address time and space complexity, and provide pseudocode if appropriate. \n", + " Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + " Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + " \"\"\"\n", + "\n", + " problem_statement: str = InputField(format=str)\n", + " sample_input: str = InputField(format=str, desc=\"The sample input provided with the problem statement.\")\n", + " sample_output: str = InputField(format=str, desc=\"The sample output provided with the problem statement.\")\n", + " solution: str = OutputField(\n", + " format=str, desc=\"a solution explanation for how we should go about solving this problem.\"\n", + " )" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "dspy.ChainOfThought(GenerateSolution)(\n", + " problem_statement=trainset[0].problem_statement, \n", + " sample_input=trainset[0].sample_input, \n", + " sample_output=trainset[0].sample_output\n", + " ).solution\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "# prediction = SimpleGenerateSolution()(problem_statement=trainset[0].problem_statement, \n", + "# sample_input=trainset[0].sample_input, \n", + "# sample_output=trainset[0].sample_output)\n", + "\n", + "# print(prediction.solution)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "simple_program = SimpleGenerateSolution()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "evaluate(program=simple_program, devset=devset)\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "\n", + "\n", + "def optimize_with_mipro(program, prompt_model, task_model, metric, trainset):\n", + " teleprompter = MIPROv2(\n", + " prompt_model=prompt_model,\n", + " task_model=task_model,\n", + " metric=metric,\n", + " num_candidates=5,\n", + " init_temperature=0.5,\n", + " verbose=False,\n", + " log_dir=\"./logs\",\n", + " )\n", + "\n", + " optimized_program = teleprompter.compile(\n", + " program.deepcopy(),\n", + " trainset=trainset,\n", + " eval_kwargs=dict(num_threads=16),\n", + " max_bootstrapped_demos=0, # 0-shot optimization\n", + " max_labeled_demos=0,\n", + " num_batches=20,\n", + " minibatch=False, # turning this off bc we have a small trainset already\n", + " seed=9\n", + " )\n", + " now = datetime.now()\n", + " date_time = now.strftime(\"%Y%m%d_%H%M%S\")\n", + "\n", + " optimized_program.save(f\"mipro_optimized_{date_time}\")\n", + "\n", + " return optimized_program\n", + "\n", + "\n", + "def optimize_with_bootstrap_fewshot(program, task_model, teacher_model, metric, trainset):\n", + " rs_optimizer = BootstrapFewShotWithRandomSearch(\n", + " metric=metric,\n", + " num_threads=8,\n", + " num_candidate_programs=5,\n", + " max_labeled_demos=0,\n", + " max_bootstrapped_demos=2,\n", + " max_errors=10000,\n", + " teacher_settings=dict(lm=teacher_model)\n", + " )\n", + " \n", + " optimized_program = rs_optimizer.compile(\n", + " program,\n", + " trainset=trainset,\n", + " )\n", + "\n", + " now = datetime.now()\n", + " date_time = now.strftime(\"%Y%m%d_%H%M%S\")\n", + "\n", + " optimized_program.save(f\"fewshot_optimized_{date_time}\")\n", + "\n", + "\n", + " return optimized_program" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "optimized_program = optimize_with_bootstrap_fewshot(simple_program, lm, lm, validate_solution, trainset)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "evaluate(program=optimized_program, devset=devset)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "mipro_optimized_program = optimize_with_mipro(simple_program, lm, lm, validate_solution, trainset)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "evaluate(program=mipro_optimized_program, devset=devset)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.9" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} From 913e7fa57569944949828df5cafd6cd17493d331 Mon Sep 17 00:00:00 2001 From: Thomas Capelle Date: Wed, 11 Sep 2024 21:35:24 +0200 Subject: [PATCH 2/3] small fixes and add Mistral (not working) --- try_dspy.ipynb | 8946 +++++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 8905 insertions(+), 41 deletions(-) diff --git a/try_dspy.ipynb b/try_dspy.ipynb index d25c4f6..ebac4b8 100644 --- a/try_dspy.ipynb +++ b/try_dspy.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "code", - "execution_count": null, + "execution_count": 1, "metadata": {}, "outputs": [], "source": [ @@ -21,11 +21,10 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ - "\n", "class Problem(BaseModel):\n", " problem_dir: pathlib.Path = Field(\n", " ..., description=\"The path to the problem directory\"\n", @@ -78,11 +77,19 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 12, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Error loading problem zero_crossings_ch1: [Errno 2] No such file or directory: 'dataset/2022/round3/zero_crossings_ch1.md'\n" + ] + } + ], "source": [ - "problems = list(pathlib.Path(\"data/dataset/\").rglob(\"*.in\"))\n", + "problems = list(pathlib.Path(\"dataset/\").rglob(\"*.in\"))\n", "eval_problems = list(filter(lambda x: \"2022/final\" in str(x), problems))\n", "test_problems = list(filter(lambda x: \"2023/practice\" in str(x), problems))\n", "train_problems = list(filter(lambda x: x not in eval_problems + test_problems, problems))\n", @@ -91,29 +98,44 @@ "def load_problem_from_path(problem_path: pathlib.Path):\n", " problem_name = problem_path.stem\n", " problem_dir = problem_path.parent\n", - " problem = load_problem(problem_name, problem_dir)\n", + " try:\n", + " problem = load_problem(problem_name, problem_dir)\n", + " except Exception as e:\n", + " print(f\"Error loading problem {problem_name}: {e}\")\n", + " return None\n", " return problem\n", "\n", "\n", "\n", - " \n", - "train_problems = list(map(load_problem_from_path, train_problems))\n", - "eval_problems = list(map(load_problem_from_path, eval_problems))\n", - "test_problems = list(map(load_problem_from_path, test_problems))\n" + "# in 2022 there are some problems that are missing the sample input and output\n", + "train_problems = list(l for l in map(load_problem_from_path, train_problems) if l is not None)\n", + "eval_problems = list(l for l in map(load_problem_from_path, eval_problems) if l is not None)\n", + "test_problems = list(l for l in map(load_problem_from_path, test_problems) if l is not None)\n" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 13, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "(44, 6, 5)" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "(len(train_problems), len(eval_problems), len(test_problems))" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 14, "metadata": {}, "outputs": [], "source": [ @@ -132,17 +154,30 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 15, "metadata": {}, "outputs": [], "source": [ + "import weave\n", + "weave.init(\"dspy_hackercup\")\n", "\n", + "MODEL = \"mistral-large-latest\"\n", + "BASE_URL = \"http://195.242.25.198:8000/v1\"\n", + "API_KEY = \"dummy_key\"\n", "\n", "lm = dspy.OpenAI(\n", - " model=\"gpt-4o-mini\", # Note: didn't find much a difference btwn mini & full gpt-4o\n", - " max_tokens=4000,\n", - " temperature=0.1,\n", - " )\n", + " model=MODEL,\n", + " api_base=BASE_URL,\n", + " api_key=API_KEY,\n", + " max_tokens=4000,\n", + " temperature=0.1,\n", + ")\n", + "\n", + "# lm = dspy.OpenAI(\n", + "# model=\"gpt-4o-mini\", # Note: didn't find much a difference btwn mini & full gpt-4o\n", + "# max_tokens=4000,\n", + "# temperature=0.1,\n", + "# )\n", "\n", "dspy.settings.configure(lm=lm)\n", "dspy.configure(experimental=True)\n", @@ -170,7 +205,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 16, "metadata": {}, "outputs": [], "source": [ @@ -198,9 +233,20 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 17, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "'1. **Input Parsing**: Read the number of test cases. For each test case, read the number of tries and their respective structures (nodes and edges).\\n\\n2. **DFS for String Extraction**:\\n\\n- For each trie, initialize a set to store unique strings.\\n\\n- Perform a DFS from the root node, appending characters to form strings as you traverse down to each node. Add each formed string to the set.\\n\\n3. **Calculate Unique Strings for Triplets**:\\n\\n- For each combination of three tries (using combinations from itertools), compute the union of their string sets.\\n\\n- Count the number of unique strings in this union and add it to a running total for the test case.\\n\\n4. **Output the Result**: After processing all triplets for a test case, format the output as specified.\\n\\nPseudocode:\\n\\n```\\n\\nfor each test_case in T:\\n\\nread N\\n\\ntries = []\\n\\nfor i from 1 to N:\\n\\nread M_i\\n\\ntrie = {}\\n\\nfor j from 2 to M_i:\\n\\nread P_{i,j}, C_{i,j}\\n\\nadd edge P_{i,j} -> j with label C_{i,j} to trie\\n\\ntries.append(trie)\\n\\nunique_strings = []\\n\\nfor trie in tries:\\n\\nstrings_set = set()\\n\\ndfs(trie.root, \"\", strings_set)\\n\\nunique_strings.append(strings_set)\\n\\ntotal_count = 0\\n\\nfor (i, j, k) in combinations(range(N), 3):\\n\\ncombined_set = unique_strings[i] | unique_strings[j] | unique_strings[k]\\n\\ntotal_count += len(combined_set)\\n\\nprint(\"Case #{}: {}\".format(test_case + 1, total_count))\\n\\n```\\n\\nThis approach ensures that we efficiently gather and combine strings from the tries while adhering to the constraints provided. The use of sets allows for quick union operations, making the solution scalable within the given limits.'" + ] + }, + "execution_count": 17, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "dspy.ChainOfThought(GenerateSolution)(\n", " problem_statement=trainset[0].problem_statement, \n", @@ -211,7 +257,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 18, "metadata": {}, "outputs": [], "source": [ @@ -232,7 +278,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 19, "metadata": {}, "outputs": [], "source": [ @@ -245,7 +291,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 20, "metadata": {}, "outputs": [], "source": [ @@ -254,21 +300,173 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 21, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 6 (0.0): 100%|██████████| 6/6 [00:08<00:00, 1.38s/it]\n" + ] + }, + { + "data": { + "text/html": [ + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "
 problem_statementsample_inputsample_outputexample_solutionpred_solutionvalidate_solution
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", + "5 3\n", + "6 3\n", + "6 2\n", + "243447 42273\n", + "Case #1: 1\n", + "Case #2: 2\n", + "Case #3: 4\n", + "Case #4: 4\n", + "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...To solve the problem, we will follow these steps: 1. For each test case, read \\(N\\) and \\(K\\). 2. Calculate the number of distinct bracelets...False
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", + "Case #2: 28\n", + "Case #3: 42\n", + "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...To implement the solution, we will follow these steps: 1. Read the number of test cases \\(T\\). 2. For each test case: - Read the...False
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", + "Case #2: 2 1 6 4\n", + "Case #3: 1 1 2\n", + "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...1. **Graph Representation**: Use an adjacency list to represent the tree. Each node will have a list of tuples containing the connected node and the...False
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", + "3 1 4\n", + "0 0\n", + "2 2\n", + "4 0\n", + "5 1 4\n", + "0 0\n", + "1 2\n", + "0 3\n", + "4 3\n", + "3 1\n", + "Case #1: 0.1169663730642699\n", + "Case #2: 0.1353445414060363\n", + "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...To solve this problem, we can follow these steps: 1. **Input Parsing**: Read the number of test cases \\(T\\). For each test case, read \\(N\\),...False
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", + "Case #2: 40 50 38 45 13\n", + "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...To solve this problem, we will follow these steps: 1. **Calculate the Centroid**: For each test case, compute the centroid of the tree coordinates: \\[...False
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num_batches=20,\n", @@ -325,39 +522,8706 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 23, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Going to sample between 1 and 2 traces per predictor.\n", + "Will attempt to bootstrap 5 candidate sets.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [06:47<00:00, 9.26s/it] \n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "New best sscore: 0.0 for seed -3\n", + "Scores so far: [0.0]\n", + "Best score: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2737.84it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0]\n", + "Best score: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 537.60it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2751.22it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 1923.25it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 1999.19it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 444.09it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2338.94it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 2258.04it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2755.17it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 2107.47it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2674.63it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 44/44 [00:00<00:00, 1814.57it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 4006.98it/s]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Score: 0.0 for set: [0]\n", + "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", + "Best score: 0.0\n", + "Average of max per entry across top 1 scores: 0.0\n", + "Average of max per entry across top 2 scores: 0.0\n", + "Average of max per entry across top 3 scores: 0.0\n", + "Average of max per entry across top 5 scores: 0.0\n", + "Average of max per entry across top 8 scores: 0.0\n", + "Average of max per entry across top 9999 scores: 0.0\n", + "8 candidate programs found.\n", + "[('generate_code', Predict(StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")))]\n" + ] + } + ], "source": [ "optimized_program = optimize_with_bootstrap_fewshot(simple_program, lm, lm, validate_solution, trainset)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 24, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0 / 6 (0.0): 100%|██████████| 6/6 [00:00<00:00, 3886.61it/s]\n" + ] + }, + { + "data": { + "text/html": [ + "\n", + "\n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + " \n", + "
 problem_statementsample_inputsample_outputexample_solutionpred_solutionvalidate_solution
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", + "5 3\n", + "6 3\n", + "6 2\n", + "243447 42273\n", + "Case #1: 1\n", + "Case #2: 2\n", + "Case #3: 4\n", + "Case #4: 4\n", + "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...To solve the problem, we will follow these steps: 1. For each test case, read \\(N\\) and \\(K\\). 2. Calculate the number of distinct bracelets...False
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", + "Case #2: 28\n", + "Case #3: 42\n", + "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...To implement the solution, we will follow these steps: 1. Read the number of test cases \\(T\\). 2. For each test case: - Read the...False
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", + "Case #2: 2 1 6 4\n", + "Case #3: 1 1 2\n", + "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...1. **Graph Representation**: Use an adjacency list to represent the tree. Each node will have a list of tuples containing the connected node and the...False
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", + "3 1 4\n", + "0 0\n", + "2 2\n", + "4 0\n", + "5 1 4\n", + "0 0\n", + "1 2\n", + "0 3\n", + "4 3\n", + "3 1\n", + "Case #1: 0.1169663730642699\n", + "Case #2: 0.1353445414060363\n", + "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...To solve this problem, we can follow these steps: 1. **Input Parsing**: Read the number of test cases \\(T\\). For each test case, read \\(N\\),...False
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", + "Case #2: 40 50 38 45 13\n", + "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...To solve this problem, we will follow these steps: 1. **Calculate the Centroid**: For each test case, compute the centroid of the tree coordinates: \\[...False
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If the projected costs exceed your budget or expectations, you may consider:\n", + "\n", + "\u001b[93m- Reducing the number of trials (`num_batches`), the size of the trainset, or the number of LM calls in your program.\u001b[0m\n", + "\u001b[93m- Using a cheaper task model to optimize the prompt.\u001b[0m\n", + "To proceed with the execution of this program, please confirm by typing \u001b[94m'y'\u001b[0m for yes or \u001b[94m'n'\u001b[0m for no.\n", + "\n", + "If you would like to bypass this confirmation step in future executions, set the \u001b[93m`requires_permission_to_run`\u001b[0m flag to \u001b[93m`False` when calling compile.\u001b[0m\n", + "\n", + "\u001b[93mAwaiting your input...\u001b[0m\n", + "\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:root:OpenAI Response Token Usage: 21435\n", + "DEBUG:root:OpenAI Response Token Usage: 16411\n", + "DEBUG:root:OpenAI Response Token Usage: 17230\n", + "DEBUG:root:OpenAI Response Token Usage: 18132\n", + "DEBUG:root:OpenAI Response Token Usage: 6955\n", + "DEBUG:root:OpenAI Response Token Usage: 1978\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "b: 10\n", + "b: 20\n", + "b: 30\n", + "b: 40\n", + "summary: Prediction(\n", + " summary='The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.'\n", + ")\n", + "DATA SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + " 0%| | 0/44 [00:00 rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\nPlease provide the output field SUMMARY OF PROGRAM ABOVE. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field SUMMARY OF PROGRAM ABOVE.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.connection:close.started\n", + "DEBUG:httpcore.connection:close.complete\n", + "DEBUG:httpcore.connection:connect_tcp.started host='api.openai.com' port=443 local_address=None timeout=5.0 socket_options=None\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Bootstrapped 0 full traces after 44 examples in round 0.\n", + "Using a randomly generated configuration for our grounded proposer.\n", + "Selected tip: description\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:httpcore.connection:connect_tcp.complete return_value=\n", + "DEBUG:httpcore.connection:start_tls.started ssl_context= server_hostname='api.openai.com' timeout=5.0\n", + "DEBUG:httpcore.connection:start_tls.complete return_value=\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:41 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'828'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149995123'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'1ms'), (b'x-request-id', b'req_5b634f8bead26cf055648b1bdea5f3cd'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d51bfbd11b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:41 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '828', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149995123', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '1ms', 'x-request-id': 'req_5b634f8bead26cf055648b1bdea5f3cd', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d51bfbd11b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_5b634f8bead26cf055648b1bdea5f3cd\n", + "DEBUG:root:OpenAI Response Token Usage: 810\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'Below is some pseudo-code for a pipeline that solves tasks with calls to language models. Please describe the purpose of one of the specified module in this pipeline.\\n\\n---\\n\\nFollow the following format.\\n\\nPROGRAM CODE: Pseudocode for a language model program designed to solve a particular task.\\n\\nEXAMPLE OF PROGRAM IN USE: An example of the program in use.\\n\\nSUMMARY OF PROGRAM ABOVE: Summary of the task the program is designed to solve, and how it goes about solving it.\\n\\nMODULE: The module in the program that we want to describe.\\n\\nMODULE DESCRIPTION: Description of the module\\'s role in the broader program.\\n\\n---\\n\\nPROGRAM CODE:\\nStringSignature(problem_statement, sample_input, sample_output -> rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\n\\nEXAMPLE OF PROGRAM IN USE: \\n\\nSUMMARY OF PROGRAM ABOVE: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\\n\\nMODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\nPlease provide the output field MODULE DESCRIPTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field MODULE DESCRIPTION.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:42 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'998'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994916'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_0fe73b12eb11c15b676ef7f60f5de277'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d592e9911b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:42 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '998', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994916', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_0fe73b12eb11c15b676ef7f60f5de277', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d592e9911b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_0fe73b12eb11c15b676ef7f60f5de277\n", + "DEBUG:root:OpenAI Response Token Usage: 963\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\\n\\n---\\n\\nFollow the following format.\\n\\nDATASET SUMMARY: A description of the dataset that we are using.\\n\\nPROGRAM CODE: Language model program designed to solve a particular task.\\n\\nPROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\\n\\nMODULE: The module to create an instruction for.\\n\\nTASK DEMO(S): Example inputs/outputs of our module.\\n\\nBASIC INSTRUCTION: Basic instruction.\\n\\nTIP: A suggestion for how to go about generating the new instruction.\\n\\nPROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\\n\\n---\\n\\nDATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\\n\\nPROGRAM CODE:\\nStringSignature(problem_statement, sample_input, sample_output -> rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\n\\nPROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\\n\\nMODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\nTASK DEMO(S): \\n\\nBASIC INSTRUCTION:\\nYou are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\\n\\nTIP: Make sure your instruction is very informative and descriptive.\\n\\nPlease provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "task_demos \n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:45 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'2357'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994467'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_b536aa6a0398153ead07d7a0fb23fa1c'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d622f6b11b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:45 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '2357', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994467', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_b536aa6a0398153ead07d7a0fb23fa1c', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d622f6b11b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_b536aa6a0398153ead07d7a0fb23fa1c\n", + "DEBUG:root:OpenAI Response Token Usage: 1399\n", + "DEBUG:root:OpenAI Response Token Usage: 810\n", + "DEBUG:root:OpenAI Response Token Usage: 963\n", + "DEBUG:root:OpenAI Response Token Usage: 1399\n", + "DEBUG:root:OpenAI Response Token Usage: 810\n", + "DEBUG:root:OpenAI Response Token Usage: 963\n", + "DEBUG:root:OpenAI Response Token Usage: 1399\n", + "DEBUG:root:OpenAI Response Token Usage: 810\n", + "DEBUG:root:OpenAI Response Token Usage: 963\n", + "DEBUG:root:OpenAI Response Token Usage: 1399\n", + "DEBUG:root:OpenAI Response Token Usage: 810\n", + "DEBUG:root:OpenAI Response Token Usage: 963\n", + "DEBUG:root:OpenAI Response Token Usage: 1399\n", + "[I 2024-09-11 19:37:45,326] A new study created in memory with name: no-name-b5aec049-cdbd-44b3-a4de-7e318de19e3e\n", + "INFO:root:Starting trial num: 0\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with B2, with key differences in bold.*\\n\\nGiven a positive integer \\\\(P\\\\), please find an array of at most \\\\(100\\\\) positive integers which have a sum of \\\\(41\\\\) and a product of \\\\(P\\\\), or output \\\\(-1\\\\) if no such array exists.\\n\\nIf multiple such arrays exist, **you may output any one of them**.\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 965\\\\)\\n\\\\(1 \\\\leq P \\\\leq 10^9\\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is one line containing a single integer \\\\(P\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\\\(N\\\\), the size of your array, followed by the array itself as \\\\(N\\\\) more space-separated positive integers.\\n\\n# Sample Explanation\\nIn the first sample, we must find an array with product \\\\(2023\\\\), and sum \\\\(41\\\\). One possible answer is \\\\([7, 17, 17]\\\\).\\n\\n\\n\\nSample Input:\\n7\\n2023\\n114\\n41\\n175\\n434\\n666\\n1872\\n\\n\\nSample Output:\\nCase #1: 3 7 17 17\\nCase #2: 2 3 38\\nCase #3: 1 41\\nCase #4: 3 1 5 35\\nCase #5: 4 1 2 7 31\\nCase #6: -1\\nCase #7: 5 1 2 6 6 26\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_This problem shares some similarities with A1, with key differences in bold._\\n\\n_Atari 2600? More like Atari 2600 BCE!_\\n\\nThe classic board game _Go_ is a two-player game played on an \\\\(R \\\\times C\\\\) board. One player places white stones while the other places black stones. On a player\\'s turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \\\\(5 \\\\times 5\\\\) game looks like this:\\n\\n{{PHOTO_ID:142163805651048|WIDTH:180}}\\n\\nAn orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space.\\n\\nYou\\'re playing as Black and it\\'s your turn. Given a valid board (i.e. no groups have \\\\(0\\\\) adjacent empty spaces), **what’s the maximum number of white stones you can capture** with a single black stone?\\n\\nHere are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured:\\n\\n{{PHOTO_ID:1327337367897977|WIDTH:300}}\\n\\nHere, Black can capture a group of \\\\(3\\\\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it\\'s placed:\\n\\n{{PHOTO_ID:311797128262237|WIDTH:240}}\\n\\nBlack can even capture multiple groups at once. Here, Black captures a group of \\\\(2\\\\) stones and a group of \\\\(3\\\\) stones:\\n\\n{{PHOTO_ID:975143063563862|WIDTH:400}}\\n\\nThe Go board is represented as a character array \\\\(A\\\\) where \\\\(A_{i, j}\\\\) is one of:\\n* `B` for a black stone\\n* `W` for a white stone\\n* `.` for an empty space\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 150\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\textbf{3{,}000}\\\\)\\n\\\\(A_{i, j} \\\\in \\\\{\\\\)\\'`.`\\', \\'`B`\\', \\'`W`\\'\\\\(\\\\}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) characters \\\\(A_{i, 1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the maximum number of white stones you can capture on your turn.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, Black can capture 3 white stones by playing in the bottom-right corner.\\n\\nIn the second case, there are no white stones that can be captured.\\n\\nIn the third case, Black can capture both white groups at once, for a total of 6 + 3 = 9 white stones.\\n\\nIn the fourth case, there are 6 different white stones that can be captured, but Black can capture at most 4 of them at once (by playing in the center of the board).\\n\\n\\n\\nSample Input:\\n4\\n4 4\\nW...\\nB.BB\\n.BWW\\n.BW.\\n5 5\\nW...W\\n.W.W.\\nBBWBB\\n.W.W.\\nW...W\\n5 5\\nB..B.\\nWBBWB\\nW.WWB\\nWWBB.\\nWWB..\\n5 5\\n.....\\nWB.BW\\nBW.WB\\nBWBWB\\n.B.B.\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 0\\nCase #3: 9\\nCase #4: 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nPatrick\\'s big sister Sam is visiting for Halloween, and Sandy wants to impress the two Stars by dressing as an astronaut. Unfortunately her regular outfit already resembles an astronaut, so she\\'ll have to go as a computer scientist instead. Sandy knows a lot about [double stars](https://en.wikipedia.org/wiki/Double_star) in astronomy, but will now have to learn about double stars in graph theory.\\n\\nThere is an unrooted tree with \\\\(N\\\\) nodes, and an edge between nodes \\\\(i\\\\) and \\\\(P_i\\\\) (for each \\\\(i \\\\ge 2\\\\)). A *double star* is a tree centered at some edge \\\\(u \\\\leftrightarrow v\\\\), where \\\\(u\\\\) and \\\\(v\\\\) each have \\\\(x\\\\) chains connected to them, each chain consisting of \\\\(y\\\\) nodes (\\\\(x, y \\\\ge 1\\\\)). Alternatively, you can think of a double star as two identical star graphs (each with \\\\(x\\\\) chains of length \\\\(y\\\\)) connected by their centers. Thus, a distinct double star is specified by a tuple \\\\((u, v, x, y)\\\\), where \\\\(u \\\\lt v\\\\). \\n\\nPlease help Sandy count the number of distinct tuples \\\\((u, v, x, y)\\\\) for which the tree contains the double star described by that tuple as a subgraph.\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 110\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(8{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case has two lines. The first contains the integer \\\\(N\\\\), and the second contains the \\\\(N-1\\\\) integers \\\\(P_2\\\\) through \\\\(P_N\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of distinct double star tuples \\\\((u, v, x, y)\\\\), where \\\\(u \\\\lt v\\\\).\\n\\n# Sample Explanation\\nThe first sample tree is depicted below. There are \\\\(5\\\\) distinct double star tuples: \\\\((1, 2, 1, 1)\\\\), \\\\((1, 2, 1, 2)\\\\), \\\\((1, 2, 2, 1)\\\\), \\\\((1, 4, 1, 1)\\\\) and \\\\((2, 6, 1, 1)\\\\).\\n\\n{{PHOTO_ID:1573931960080150|WIDTH:350}}\\n\\nThe second sample tree is depicted below. There are \\\\(5\\\\) distinct double star tuples: \\\\((1, 2, 1, 1)\\\\), \\\\((1, 2, 1, 2)\\\\), \\\\((1, 8, 1, 1)\\\\), \\\\((1, 8, 2, 1)\\\\) and \\\\((2, 7, 1, 1)\\\\).\\n\\n{{PHOTO_ID:1286569645378322|WIDTH:500}}\\n\\nIn the third sample tree, no double stars exist because arm lengths of \\\\(0\\\\) are not allowed \\\\((y \\\\ge 1 )\\\\).\\n\\n\\nSample Input:\\n3\\n8\\n1 1 1 2 2 6 4\\n10\\n1 8 8 1 1 2 1 7 7\\n2\\n1\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 5\\nCase #3: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**The only differences between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/D1) is that here, \\\\(A_i \\\\in \\\\{1, 2\\\\}\\\\), and you may only swap adjacent elements of \\\\(A_i\\\\) with a single operation.**\\n\\nAs a Metal Platforms employee, you place a high value on your work-life balance. Boss Rob has assigned you \\\\(N\\\\) tasks, the \\\\(i\\\\)th of which takes \\\\(A_i\\\\) minutes to finish, where \\\\(A_i\\\\) is either **\\\\(\\\\mathbf{1}\\\\) or \\\\(\\\\mathbf{2}\\\\)**.\\n\\nYou may reorder your tasks, where each *operation* lets you **swap any two adjacent elements** of \\\\(A\\\\) (\\\\(A_i\\\\) and \\\\(A_j\\\\) for some \\\\(i\\\\) and \\\\(j\\\\) such that \\\\(|i - j| = 1\\\\)).\\n\\nTo reflect how often task requirements change in the real world, there will be \\\\(M\\\\) updates made to the task completion times, with the \\\\(i\\\\)th update setting \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n\\nAfter completing the \\\\(i\\\\)th update, you would like to know if it\\'s possible to balance the time spent at work versus at home, namely if you hope to finish the first \\\\(Z_i\\\\) tasks at work and the rest at home. Specifically, let \\\\(Q_i\\\\) be the minimum number of swap operations which must be theoretically performed so that \\\\(A_1 + ... + A_{Z_i} = A_{Z_i + 1} + ... + A_{N}\\\\), with \\\\(Q_i = -1\\\\) if it\\'s impossible. Note that it\\'s possible for \\\\(Q_i\\\\) to be \\\\(0\\\\), if the subarrays already have equal sums.\\n\\nTo reduce the size of the output, please compute the sum of \\\\(Q_1\\\\), ..., \\\\(Q_M\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 95\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le M \\\\le 1{,}000{,}000\\\\)\\n\\\\(A_i, Y_i \\\\in \\\\{1, 2\\\\}\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Z_i \\\\le N-1\\\\)\\n\\nThe sum of \\\\(N+M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line containing \\\\(N\\\\) digits, \\\\(A_1\\\\), \\\\(A_2\\\\), ..., \\\\(A_N\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(X_i\\\\), \\\\(Y_i\\\\), and \\\\(Z_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output `\"Case #i: \"` followed by a single integer, \\\\(Q_1 + ... + Q_M\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:2337609416394611|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 2]\\\\). The first update yields \\\\(A=[1, 1]\\\\) with \\\\(Z_1 = 1\\\\). The first and second tasks are already equal, so \\\\(Q_1 = 0\\\\) as no swaps are necessary. The second update yields \\\\(A=[2, 1]\\\\), and after the third update we still have \\\\([2, 1]\\\\). In neither case can we equally split the tasks, so \\\\(Q_2 = Q_3 = -1\\\\). The final answer is \\\\(0 + (-1) + (-1) = -2\\\\).\\n\\nThe second case is depicted below:\\n\\n{{PHOTO_ID:402660235386017|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 1, 1, 2]\\\\). The first update yields \\\\(A=[2, 1, 1, 2]\\\\) with \\\\(Z_1 = 2\\\\). The two subarrays both sum to \\\\(3\\\\) already, so \\\\(Q_1 = 0\\\\). The second update yields \\\\(A=[2, 2, 1, 2]\\\\). It\\'s impossible to split the tasks equally, so \\\\(Q_2 = -1\\\\). The third update yields \\\\(A=[2, 2, 1, 1]\\\\) with \\\\(Z_3 = 2\\\\). We can swap the second and third tasks so that both halves of the array sum to \\\\(3\\\\), so \\\\(Q_3 = 1\\\\). The final answer is \\\\(0 + (-1) + 1 = 0\\\\).\\n\\nIn the third case, we start with \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2]\\\\). With the updates, we get:\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_1 = 4\\\\) and \\\\(Q_1 = 4\\\\): move the left two \\\\(2\\\\)s two spaces left\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_2 = 3\\\\) and \\\\(Q_2 = 12\\\\): move the left three \\\\(2\\\\)s four spaces left\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_3 = 5\\\\) and \\\\(Q_3 = 0\\\\)\\n- \\\\(A = [2, 1, 1, 1, 2, 2, 2, 2], Z_4 = 4\\\\) and \\\\(Q_4 = -1\\\\)\\n- \\\\(A = [2, 1, 2, 1, 2, 2, 2, 2], Z_5 = 4\\\\) and \\\\(Q_5 = 1\\\\): swap \\\\(A_4\\\\) and \\\\(A_5\\\\)\\n\\nThe final answer is \\\\(4 + 12 + 0 + (-1) + 1 = 16\\\\).\\n\\n\\nSample Input:\\n3\\n2 3\\n1 2\\n2 1 1\\n1 2 1\\n1 2 1\\n4 3\\n1 1 1 2\\n1 2 2\\n2 2 2\\n4 1 2\\n8 5\\n1 1 1 1 2 2 2 2\\n5 2 4\\n7 2 3\\n6 2 5\\n1 2 4\\n3 2 4\\n\\n\\nSample Output:\\nCase #1: -2\\nCase #2: 0\\nCase #3: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with B1, with key differences in bold.*\\n\\nGiven a positive integer \\\\(P\\\\), please find an array of at most \\\\(100\\\\) positive integers which have a sum of \\\\(41\\\\) and a product of \\\\(P\\\\), or output \\\\(-1\\\\) if no such array exists.\\n\\nIf multiple such arrays exist, **print one with the fewest number of elements**. If there are multiple with the fewest number of elements, you may print any one of them.\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 960\\\\)\\n\\\\(1 \\\\leq P \\\\leq 10^9\\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is one line containing a single integer \\\\(P\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\\\(N\\\\), the size of your array, followed by the array itself as \\\\(N\\\\) more space-separated positive integers.\\n\\n# Sample Explanation\\nIn the first sample, we must find an array with product \\\\(2023\\\\), and sum \\\\(41\\\\). One possible answer is \\\\([7, 17, 17]\\\\).\\n\\n\\n\\nSample Input:\\n7\\n2023\\n114\\n41\\n175\\n434\\n666\\n1872\\n\\n\\nSample Output:\\nCase #1: 3 7 17 17\\nCase #2: 2 3 38\\nCase #3: 1 41\\nCase #4: 3 1 5 35\\nCase #5: 4 1 2 7 31\\nCase #6: -1\\nCase #7: 4 2 4 9 26\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with C1, with key differences in bold.*\\n\\nA friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it.\\n\\nThe puzzle consists of \\\\(N\\\\) buttons in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). The initial state of button \\\\(i\\\\) is white if \\\\(S_i = 1\\\\), or black if \\\\(S_i = 0\\\\). Pressing a button \\\\(k\\\\) toggles the state of itself as well as every \\\\(k\\\\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black.\\n\\nLife is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \\\\(Q\\\\) buttons sequentially, the \\\\(i\\\\)th being button \\\\(B_i\\\\). \\n\\nAfter your brother **pushes each button**, you\\'d like to add to your answer the minimum number of button presses required to turn all the buttons black.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 75\\\\)\\n\\\\(1 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 4{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) and \\\\(Q\\\\) over all cases will be at most \\\\(9{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing a bitstring \\\\(S\\\\) of length \\\\(N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains a single integer \\\\(B_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of the number of button presses needed to turn all buttons black **after each of the \\\\(Q\\\\) presses.**\\n\\n**Sample Explanation**\\n\\nIn the first case, after your brother presses the first button, the state of the puzzle is \\\\(0101\\\\). The best strategy is to press the second button, turning all lights off.\\n\\nIn the second case, the puzzle starts as \\\\(0001\\\\), and after each button press becomes \\\\(0100\\\\), \\\\(0110\\\\), \\\\(0011\\\\), and \\\\(0010\\\\) respectively. The answers after each query are \\\\(2\\\\), \\\\(3\\\\), \\\\(2\\\\), and \\\\(1\\\\) respectively, so we output their sum of \\\\(8\\\\).\\n\\n\\nSample Input:\\n5\\n4\\n1010\\n1\\n1\\n4\\n0001\\n4\\n2\\n3\\n2\\n4\\n7\\n0101101\\n8\\n1\\n3\\n2\\n6\\n7\\n4\\n2\\n5\\n7\\n0101100\\n1\\n7\\n7\\n1111111\\n1\\n1\\n\\n\\nSample Output:\\nCase #1: 1\\nCase #2: 8\\nCase #3: 36\\nCase #4: 4\\nCase #5: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*Bejeweled™* is a classic puzzle game where the player tries to match three-in-a-row in a 2D grid by swapping adjacent tile pairs. You may remember Hacker Cup\\'s [spinoff](https://www.facebook.com/codingcompetitions/hacker-cup/2022/final-round/problems/C) by the name of *Isblinged™*. The all new sequel, *Isblinged 2*, is also played on a grid of \\\\(R\\\\) rows by \\\\(C\\\\) columns of tiles where the tile at \\\\((i, j)\\\\) is of an integer type \\\\(G_{i,j}\\\\).\\n\\nAt any time, the *score* of the grid is the number of subarrays of \\\\(3\\\\) equal tiles in either a single row or column (i.e. either a \\\\(3 \\\\times 1\\\\) or \\\\(1 \\\\times 3\\\\) submatrix). Note that different subarrays can overlap, and will each count toward the score. The score of the initial grid is guaranteed to be \\\\(0\\\\).\\n\\nYou will make exactly \\\\(2\\\\) moves, where each involves swapping a pair of adjacent tiles (either in the same row or column). What is the maximum score that can be achieved after the \\\\(2\\\\) moves?\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(1 \\\\le R, C \\\\le 1{,}000\\\\)\\n\\\\(R*C \\\\ge 2\\\\)\\n\\\\(1 \\\\le G_{i,j} \\\\le 1{,}000{,}000\\\\)\\n\\nThe sum of \\\\(R * C\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) space-separated integers \\\\(G_{i,1..C}\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing `\"Case #i: \"` followed by a single integer, the maximum score.\\n\\n# Sample Explanation\\n\\nIn the first case, one possible optimal ordered pair of swaps is depicted below:\\n\\n{{PHOTO_ID:1050535959425825|WIDTH:750}}\\n\\nThe total score is \\\\(5\\\\) as there are \\\\(3\\\\) equal subarrays in the the first row and \\\\(2\\\\) in the second.\\n\\n\\n\\nSample Input:\\n3\\n2 5\\n1 1 2 1 1\\n2 2 1 3 2\\n1 4\\n1 1 2 1\\n5 1\\n1\\n2\\n1\\n2\\n1\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 1\\nCase #3: 1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_This problem shares some similarities with A2, with key differences in bold._\\n\\n_Atari 2600? More like Atari 2600 BCE!_\\n\\nThe classic board game _Go_ is a two-player game played on an \\\\(R \\\\times C\\\\) board. One player places white stones while the other places black stones. On a player\\'s turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \\\\(5 \\\\times 5\\\\) game looks like this:\\n\\n{{PHOTO_ID:1491241961665655|WIDTH:180}}\\n\\nAn orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space.\\n\\nYou\\'re playing as Black and it\\'s your turn. Given a valid board (i.e. no groups have \\\\(0\\\\) adjacent empty spaces), **is it possible to capture any white stones on this turn** with a single black stone?\\n\\nHere are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured:\\n\\n{{PHOTO_ID:632037365762295|WIDTH:300}}\\n\\nHere, Black can capture a group of \\\\(3\\\\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it\\'s placed:\\n\\n{{PHOTO_ID:850220283519385|WIDTH:240}}\\n\\nBlack can even capture multiple groups at once. Here, Black captures a group of \\\\(2\\\\) stones and a group of \\\\(3\\\\) stones:\\n\\n{{PHOTO_ID:804957498302389|WIDTH:400}}\\n\\nThe Go board is represented as a character array \\\\(A\\\\) where \\\\(A_{i, j}\\\\) is one of:\\n* `B` for a black stone\\n* `W` for a white stone\\n* `.` for an empty space\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 160\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\textbf{19}\\\\)\\n\\\\(A_{i, j} \\\\in \\\\{\\\\)\\'`.`\\', \\'`B`\\', \\'`W`\\'\\\\(\\\\}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) characters \\\\(A_{i, 1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by \"`YES`\" if you can capture any white stones, or \"`NO`\" if you cannot.\\n\\n\\n# Sample Explanation\\n\\nThe first sample case is the first capture example shown above. Black can capture the lone white stone.\\n\\nIn the second case, Black can capture a group of \\\\(5\\\\) white stones.\\n\\nThe third case is the same as the second case, except that the group of \\\\(5\\\\) white stones has two empty adjacent spaces, so it cannot be captured in one term. The other white stones can also not be captured on this turn.\\n\\nIn the fourth case, White could capture a black stone if it was their turn, but Black cannot capture any white stones.\\n\\nThe fifth case is the second capture example shown above. Black can capture \\\\(3\\\\) white stones by playing in the upper-left corner.\\n\\n\\nSample Input:\\n5\\n4 4\\n.B..\\nBW..\\n.B..\\n....\\n4 5\\nWWWB.\\nWB...\\nWB.WB\\nB.W..\\n4 5\\nWWWB.\\nWB...\\nWB.WB\\n..W..\\n3 5\\n..W..\\n.WBWW\\n.B..W\\n3 3\\n.WB\\nWWB\\nBB.\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: YES\\nCase #3: NO\\nCase #4: NO\\nCase #5: YES\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYour friend baked \\\\(N\\\\) batches of cookies, the first with chocolate chips, and the rest with raisins. Batch \\\\(i\\\\) has \\\\(C_i\\\\) cookies all having the same weight \\\\(W_i\\\\). Cookies in different batches may have the same weight. Initially, all cookies are scattered on the table, and you hope to find a hefty one to eat with your trusty balance scale.\\n\\nFirst, you select a single cookie uniformly at random from the table and place it on the left side of the scale. Then, \\\\(K\\\\) times, you will select a cookie uniformly at random from the table, place it on the empty side of the scale, and throw away the lighter of the two cookies on the scale (or the cookie on the left side if they both weigh the same).\\n\\n{{PHOTO_ID:502174485028685|WIDTH:700}}\\n\\nDetermine the probability that after throwing away \\\\(K\\\\) cookies, the remaining cookie on the scale is a yummy chocolate chip cookie from batch \\\\(1\\\\).\\n\\nIf we write this probability as a quotient of integers \\\\(p/q\\\\) in lowest terms, then you should output this quotient modulo \\\\(1{,}000{,}000{,}007\\\\) — in other words, output the unique integer \\\\(x\\\\) such that \\\\(0 \\\\le x \\\\lt 1{,}000{,}000{,}007\\\\) and \\\\(p = (x * q) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 1{,}000\\\\)\\n\\\\(2 \\\\le N \\\\le 3{,}000\\\\)\\n\\\\(1 \\\\le K \\\\lt \\\\sum_{i=1}^{N} C_i\\\\)\\n\\\\(1 \\\\le C_i \\\\le 3{,}000\\\\)\\n\\\\(1 \\\\le W_i \\\\le 10^{9}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two-space separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(C_i\\\\) and \\\\(W_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by the unique integer \\\\(x\\\\) such that \\\\(p = (x * q) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first case, there are \\\\(5\\\\) batches with \\\\(1\\\\) cookie each. Since we only weigh \\\\(K=1\\\\) time, the only way for the last cookie to be chocolate chip is if it\\'s one of the two placed on the scale. There\\'s a \\\\(\\\\frac{1}{5}\\\\) chance of picking it for the initial cookie, and a \\\\(\\\\frac{4}{5}\\\\cdot\\\\frac{1}{4}\\\\) chance of picking it for the second cookie. In either case, because it\\'s the heaviest, it will remain on the scale. Adding these independent events\\' probabilities, we get a \\\\(\\\\frac{1}{5} + \\\\frac{4}{5}\\\\cdot\\\\frac{1}{4} = \\\\frac{2}{5}\\\\) chance that the last cookie on the scale is chocolate chip. Here, we print \\\\(800{,}000{,}006\\\\) since \\\\(2 = (800{,}000{,}006 * 5) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\nThe second case is the same, except we must weigh a second time. There\\'s a \\\\(\\\\frac{2}{5}\\\\) chance of picking chocolate chip the first or second times, and a \\\\(\\\\frac{3}{5}\\\\cdot\\\\frac{1}{3}\\\\) chance of picking it the third time. The answer is \\\\(\\\\frac{2}{5} + \\\\frac{3}{5}\\\\cdot\\\\frac{1}{3} = \\\\frac{3}{5}\\\\). We print \\\\(200{,}000{,}002\\\\) since \\\\(3 = (200{,}000{,}002 * 5) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\nIn the third case, there are \\\\(2\\\\) batches of \\\\(10\\\\) cookies each, where batch \\\\(2\\\\) cookies are heavier. Since we must weigh and throw out \\\\(10\\\\) cookies, we must have weighed a cookie from batch \\\\(2\\\\) at least once, after which a batch \\\\(1\\\\) cookie can never remain on the scale. Therefore the answer is \\\\(0\\\\).\\n\\nIn the fourth case, there is a \\\\(\\\\frac{5}{24}\\\\) chance that the final cookie on the scale is chocolate chip.\\n\\n\\nSample Input:\\n6\\n5 1\\n1 3000\\n1 2000\\n1 1000\\n1 2000\\n1 1000\\n5 2\\n1 3000\\n1 2000\\n1 1000\\n1 2000\\n1 1000\\n2 10\\n10 1\\n10 2\\n5 2\\n2 50\\n1 40\\n1 50\\n1 60\\n3 50\\n4 2993\\n3000 999999999\\n2995 1000000000\\n1552 888888888\\n1336 999999999\\n3 1\\n1 10\\n2 9\\n1 11\\n\\n\\nSample Output:\\nCase #1: 800000006\\nCase #2: 200000002\\nCase #3: 0\\nCase #4: 208333335\\nCase #5: 590307096\\nCase #6: 333333336\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This is the easier version of [Chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/A2). This version involves a string, but chapter 2 involves an array of integers that can change with updates.**\\n\\nA string is *perfectly balanced* if its length is even, and the first half of the string can be shuffled to make the string a palindrome. For example, \"`abab`\" and \"`dood`\" are perfectly balanced, but \"`racecar`\" and \"`doodad`\" are not.\\n\\nA string is *almost perfectly balanced* if you can delete exactly one character from it to make it perfectly balanced. Some examples are \"`classical`\", \"`intelligent`\", and \"`www`\".\\n\\n{{PHOTO_ID:1262846811180772|WIDTH:700}}\\n\\nYou are given a larger template string \\\\(S\\\\), and \\\\(Q\\\\) substrings, the \\\\(i\\\\)th of which is \\\\(S_{L_i..R_i}\\\\). For how many of these \\\\(Q\\\\) queries is the substring almost perfectly balanced?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 90\\\\)\\n\\\\(1 \\\\le |S| \\\\le 10^6\\\\)\\n\\\\(S_i \\\\in\\\\) {\\'`a`\\', ..., \\'`z`\\'}\\n\\\\(1 \\\\le Q \\\\le 10^6\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le |S|\\\\)\\n\\nThe sum of \\\\(|S|\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single template string \\\\(S\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(L_i\\\\) and \\\\(R_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"`, followed by a single integer: the number of queries which are an almost perfectly balanced substring.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, the template string is \"`singingbanana`\".\\n\\nFor the first query, the substring is \"`banan`\". You can delete the `\"b\"` to get \"`anan`\" then reorder the first half of the string to get `\"naan\"`, which is a palindrome. Thus, \"`banan`\" is an almost perfectly balanced substring.\\n\\nFor the second query, the substring is \"`anana`\". You can delete the second \"`a`\" to get \"`anna`\", which is a palindrome. Thus, \"`anana`\" is an almost perfectly balanced substring.\\n\\nFor the third query, the substring is \"`ban`\". You cannot delete any character to get an almost perfectly balanced substring.\\n\\nFor the fourth query, the substring is \"`nan`\". You can delete the \"`a`\" to get \"`nn`\", which is a palindrome. Thus, \"`nan`\" is an almost perfectly balanced substring.\\n\\nFor \"`singing`\", you can create \"`gniing`\".\\n\\nIn the second test case, the first, second, and third queries are almost perfectly balanced substrings, but the fourth is not.\\n\\nIn the third test case, the first, second, third, and fourth queries are almost perfectly balanced substrings, but the fifth and sixth are not.\\n\\n\\nSample Input:\\n3\\nsingingbanana\\n5\\n8 12\\n9 13\\n8 10\\n10 12\\n1 7\\nprepareintelligentopinion\\n4\\n1 7\\n8 18\\n19 25\\n12 13\\nphpservers\\n6\\n1 3\\n4 10\\n1 3\\n2 2\\n3 5\\n1 10\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 3\\nCase #3: 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This problem has large input, so we recommend pre-downloading the compressed zip file.**\\n\\nMeta Getaways is a travel agency that deals with \\\\(N\\\\) airports numbered \\\\(1...N\\\\), and \\\\(M\\\\) flight paths. Flight path \\\\(i\\\\) connects airports \\\\(A_i\\\\) and \\\\(B_i\\\\) in both directions, with two direct flights operating every morning (one in each direction), and another two every evening (also one in each direction). Each of these four direct flights can carry up to \\\\(C_i\\\\) **t**ourists.\\n\\n{{PHOTO_ID:356437426702208|WIDTH:400}}\\n\\nThe first sample case is depicted above, with morning and evening flights in red and blue.\\n\\nPeak travel season is here, and will last \\\\(Q\\\\) days. For each day \\\\(i\\\\), determine \\\\(F_i\\\\), the maximum number of tourists who could possibly fly from airport \\\\(X_i\\\\) to \\\\(Y_i\\\\). Each tourist may either fly directly or take one morning and one evening flight which share an intermediate airport.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 70 \\\\)\\n\\\\(1 \\\\leq N, M, Q \\\\leq 200{,}000\\\\)\\n\\\\(1 \\\\leq C_i \\\\leq 10^9 \\\\)\\n\\\\(1 \\\\leq A_i, B_i \\\\leq N; A_i \\\\ne B_i \\\\)\\n\\\\(1 \\\\leq X_i, Y_i \\\\leq N; X_i \\\\ne Y_i\\\\)\\nAll unordered pairs \\\\((A_i, B_i)\\\\) within a given test case are distinct.\\n\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(5{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(Q\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), and \\\\(C_i\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing \"`Case #i:` \" followed by \\\\(Q\\\\) space-separated integers \\\\(F_1, ..., F_Q\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first case:\\n\\n- On day \\\\(1\\\\), we must send as many tourists from airport \\\\(1\\\\) to airport \\\\(2\\\\). We can fly \\\\(10\\\\) tourists direct in the morning and \\\\(10\\\\) more at night. Only \\\\(5\\\\) tourists can be flown from \\\\(1 \\\\to 3\\\\) in the morning and \\\\(3 \\\\to 2\\\\) in the evening (despite the evening flight capacity being \\\\(15\\\\)). Therefore, \\\\(F_1 = 10 \\\\cdot 2 + 5 = 25\\\\).\\n- On day \\\\(2\\\\), we can fly \\\\(5\\\\) tourists direct in the morning and evening, then fly \\\\(10\\\\) tourists through airports \\\\(1 \\\\to 2 \\\\to 3\\\\). Therefore, \\\\(F_2 = 5 \\\\cdot 2 + 10 = 20\\\\).\\n- \\\\(F_3 = 15 \\\\cdot 2 + 5 + 7 = 42\\\\)\\n- \\\\(F_4 = 10 \\\\cdot 2 + 7 = 27\\\\)\\n- \\\\(F_5 = 7 \\\\cdot 2 + 10 = 24\\\\)\\n- On day \\\\(6\\\\), there are no direct flights. We can fly \\\\(10\\\\) tourists through airports \\\\(4 \\\\to 2 \\\\to 1\\\\), and \\\\(5\\\\) tourists through airports \\\\(4 \\\\to 3 \\\\to 1\\\\) for a total of \\\\(F_6 = 10 + 5 = 15\\\\) tourists.\\n\\nIn the second case:\\n\\n- \\\\(F_1 = 10 \\\\cdot 2 + 20 = 40\\\\)\\n- \\\\(F_2 = 30 \\\\cdot 2 + 10 = 70\\\\)\\n- \\\\(F_3 = 0\\\\)\\n- \\\\(F_4 = 20 \\\\cdot 2 + 10 = 50\\\\)\\n- \\\\(F_5 = 0\\\\)\\n- \\\\(F_6 = 0\\\\)\\n\\n\\n\\nSample Input:\\n3\\n4 5 6\\n1 2 10\\n1 3 5\\n2 3 15\\n2 4 10\\n3 4 7\\n1 2\\n1 3\\n2 3\\n2 4\\n3 4\\n4 1\\n4 3 6\\n1 2 10\\n2 3 20\\n3 1 30\\n1 2\\n1 3\\n1 4\\n2 3\\n2 4\\n3 4\\n4 3 6\\n1 2 20\\n2 3 10\\n3 4 30\\n1 2\\n1 3\\n1 4\\n2 3\\n2 4\\n3 4\\n\\n\\nSample Output:\\nCase #1: 25 20 42 27 24 15\\nCase #2: 40 70 0 50 0 0\\nCase #3: 40 10 0 20 10 60\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem B2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/B2) is that here, scenes are smaller and do not contain rocks.**\\n\\nBoss Rob painted a beautiful scene on a 2D canvas of \\\\(R\\\\) rows by \\\\(C\\\\) columns, containing zero or more happy little trees.\\n\\nTo make sure none of his trees are lonely, Rob would like you to add as many trees as you\\'d like (possibly \\\\(0\\\\)) to empty spaces so that each tree in the final painting has at least two tree *friends*, that is, two trees which are each adjacent to it (directly to its north, south, east, or west). If there are multiple solutions, you may print any one of them.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 85\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\mathbf{100}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, each of which contains \\\\(C\\\\) characters, either \"`.`\" (an empty space) or \"`^`\" (a tree), representing the initial painting.\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print \"`Case #i: ` \", followed by \"`Possible`\" and \\\\(R\\\\) lines of \\\\(C\\\\) characters each representing the final painting (if a solution exists), otherwise \"`Impossible`\".\\n\\n\\n# Sample Explanation\\n\\nIn the first case (depicted below), we could add two tree friends to either side of the middle tree, but they themselves would each only have one tree friend. Therefore, it\\'s impossible to get two friends for each tree in the final painting.\\n\\n{{PHOTO_ID:545859687328272|WIDTH:295}}\\n\\nIn the second case, there are no trees in the initial painting, so the condition of two friends for each tree is already satisfied.\\n\\nIn the third case, one possible solution is depicted below.\\n\\n{{PHOTO_ID:5118404751621442|WIDTH:700}}\\n\\n\\nSample Input:\\n3\\n1 3\\n.^.\\n3 1\\n.\\n.\\n.\\n4 4\\n..^.\\n..^.\\n....\\n...^\\n\\n\\nSample Output:\\nCase #1: Impossible\\nCase #2: Possible\\n.\\n.\\n.\\nCase #3: Possible\\n^^^.\\n^.^.\\n^^^^\\n..^^\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nDespite his age, Mr. Krabs still goes trick-or-treating. Free candy (free anything, really) is irresistible to him. Bikini Bottom is an \\\\(R \\\\times C\\\\) grid of houses, with house \\\\((i, j)\\\\) having \\\\(A_{i,j}\\\\) pieces of candy.\\n\\nMr. Krabs plans to travel from house \\\\((1, 1)\\\\), the top-left corner, to house \\\\((R, C)\\\\), the bottom-right corner, taking as much candy as possible along the way. When Mr. Krabs is at house \\\\((i, j)\\\\), he takes all the candy, and then either moves down to house \\\\((i + 1, j)\\\\) or right to house \\\\((i, j + 1)\\\\).\\n\\nPlankton wants to sabotage him, and has decided to complete his own candy run before Mr. Krabs even starts his trip. Plankton will go from house \\\\((1, C)\\\\), the top-right corner, to house \\\\((R, 1)\\\\), the bottom-left corner. When Plankton is at house \\\\((i, j)\\\\), he takes all its candy (so there\\'s none left for Mr. Krabs later) and either moves down to house \\\\((i + 1, j)\\\\) or left to house \\\\((i, j - 1)\\\\).\\n\\nWhat\\'s the maximum amount of candy that Mr. Krabs can collect, assuming that Plankton strives to minimize this value by completing his trip before Mr. Krabs starts?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 100\\\\)\\n\\\\(2 \\\\leq R, C \\\\leq 300\\\\)\\n\\\\(0 \\\\leq A_{i,j} \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\nThe sum of \\\\(R * C\\\\) across all test cases is at most \\\\(500{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains the \\\\(C\\\\) integers \\\\(A_{i,1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the maximum pieces of candy that Mr. Krabs can get if Plankton optimally sabotages.\\n\\n\\n# Sample Explanation\\n\\nThe following depicts sample cases 1 through 5. Plankton\\'s paths to minimize Mr. Krabs\\'s candy are greyed out, while Mr. Krabs\\'s optimal paths through the remaining candy (in spite of Plankton\\'s sabotage) are drawn in red.\\n\\n{{PHOTO_ID:216354288143884|WIDTH:650}}\\n\\n\\nSample Input:\\n5\\n3 4\\n2 5 3 6\\n1 7 2 5\\n4 5 1 3\\n2 2\\n2 2\\n2 1\\n3 3\\n3 4 5\\n5 3 1\\n7 6 4\\n3 3\\n1 1 1\\n3 3 3\\n3 2 1\\n3 3\\n1 1 3\\n3 3 1\\n1 3 2\\n\\n\\nSample Output:\\nCase #1: 12\\nCase #2: 1\\nCase #3: 11\\nCase #4: 5\\nCase #5: 6\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*…right down Santa Claus Lane!*\\n\\nSanta Claus Lane is home to \\\\(N\\\\) elves, the \\\\(i\\\\)th of whom lives \\\\(X_i\\\\) meters from the start of the lane. As Santa\\'s little helper, you\\'re tasked to assign elves to work on toys for good little girls and boys. Specifically, you must assign each elf to work on some toy so that at least \\\\(2\\\\) toys get worked on, and no elf works on a toy alone.\\n\\nAll elves working on the same toy will meet at the point which minimizes the farthest distance that any of them would need to walk. Formally, if the elves assigned to a given toy live at \\\\(X_1\\\\), \\\\(X_2\\\\), \\\\(…\\\\), then they will meet at point \\\\(P\\\\) such that \\\\(\\\\max(|X_1 - P|\\\\), \\\\(|X_2 - P|\\\\), \\\\(…\\\\)\\\\()\\\\) is as small as possible.\\n\\nFor instance, the first sample case is depicted below:\\n\\n{{PHOTO_ID:838516607729325|WIDTH:700}}\\n\\nSanta is supervising, and you reckon he could use some exercise. Among all valid assignments of elves to toys, what\\'s the farthest Santa would need to walk to visit all meeting points? Santa may start and end anywhere, but will try to walk as little as possible after seeing your assignments.\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 20\\\\)\\n\\\\(4 \\\\leq N \\\\leq 10^5 \\\\)\\n\\\\(1 \\\\leq X_i \\\\leq 10^9 \\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with one line containing the integer \\\\(N\\\\), followed by a second line containing the the \\\\(N\\\\) integers \\\\(X_1 ... X_N\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single real number, the farthest distance Santa would need to walk in meters. Your answer will be considered correct if it differs from the jury\\'s answer by at most \\\\(10^{-6}\\\\) meters or at most \\\\(0.0001\\\\)%, whichever is larger.\\n\\n# Sample Explanation\\n\\nIn the first sample case, elves living at \\\\(1 \\\\,\\\\text{m}\\\\) and \\\\(3 \\\\,\\\\text{m}\\\\) will work on a toy. Elves living at \\\\(7 \\\\,\\\\text{m}\\\\), \\\\(12 \\\\,\\\\text{m}\\\\), and \\\\(13 \\\\,\\\\text{m}\\\\) will work on another toy, and elves at \\\\(17 \\\\,\\\\text{m}\\\\) and \\\\(23 \\\\,\\\\text{m}\\\\) will work on the third toy. The toys will be made at \\\\(2 \\\\,\\\\text{m}\\\\), \\\\(10 \\\\,\\\\text{m}\\\\) and \\\\(20 \\\\,\\\\text{m}\\\\) respectively. Santa would need to walk at least \\\\(18 \\\\,\\\\text{m}\\\\) in total to visit every meeting point.\\n\\nThe second sample case is depicted below. No elf is allowed to work alone and we must make two toys. One optimal way of doing this is to have the leftmost \\\\(3\\\\) elves work on one toy at \\\\(2 \\\\,\\\\text{m}\\\\), and the rest work on the other toy at \\\\(4.5 \\\\,\\\\text{m}\\\\). This would maximize the distance Santa would have to walk among all valid elf assignments.\\n\\n{{PHOTO_ID:163905706775126|WIDTH:400}}\\n\\nIn the third case, the two toys will be made at \\\\(55 \\\\,\\\\text{m}\\\\) and \\\\(5500 \\\\,\\\\text{m}\\\\).\\n\\n\\n\\nSample Input:\\n3\\n7\\n1 17 3 13 7 23 12\\n5\\n5 4 3 2 1\\n4\\n10 100 1000 10000\\n\\n\\nSample Output:\\nCase #1: 18\\nCase #2: 2.5\\nCase #3: 5445\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nSandy\\'s store has \\\\(N\\\\) pre-owned clock parts for sale, where the \\\\(i\\\\)th part is of style \\\\(S_i\\\\). The store also has two display cases, each capable of holding at most \\\\(K\\\\) parts. To maximize the aesthetics of Sandy\\'s secondhand second hands, she\\'d like to put each of the \\\\(N\\\\) parts into one of the two cases so that neither case ends up with two different parts of the same style, and neither case has more than \\\\(K\\\\) parts total. Can you determine if this is possible?\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 90\\\\)\\n\\\\(1 \\\\leq N, K, S_i \\\\leq 100\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing \\\\(2\\\\) space-separated integers, \\\\(N\\\\) and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(S_1, ..., S_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by \"`YES`\" if it\\'s possible to arrange the \\\\(N\\\\) parts into two cases satisfying the description above, or \"`NO`\" otherwise.\\n\\n\\n# Sample Explanation\\n\\nIn the first test case, there are \\\\(3\\\\) parts of styles \\\\(1\\\\), \\\\(2\\\\), and \\\\(2\\\\), with the display cases having capacity \\\\(2\\\\). One solution, depicted below, is to put the first and third parts in one display case, and the second part in the other.\\n\\n{{PHOTO_ID:459254706243127|WIDTH:500}}\\n\\nIn the second test case, there are \\\\(5\\\\) parts of styles \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(3\\\\), \\\\(1\\\\), with the display cases having capacity \\\\(3\\\\). One solution, depicted below, is to put the first three parts in one display case, and the last two in the other.\\n\\n{{PHOTO_ID:1183048075593188|WIDTH:500}}\\n\\nIn the third test case, there are \\\\(5\\\\) parts, but the display cases can each only hold \\\\(2\\\\). Therefore, there is no solution.\\n\\nIn the fourth test case, style \\\\(1\\\\) will always be duplicated in some display case for any given arrangement. Therefore, there is no solution.\\n\\n\\n\\n\\nSample Input:\\n5\\n3 2\\n1 2 2\\n5 3\\n1 2 3 3 1\\n5 2\\n1 2 3 4 5\\n5 5\\n1 1 2 2 1\\n1 1\\n1\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: YES\\nCase #3: NO\\nCase #4: NO\\nCase #5: YES\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob like to play cards on their lunch break. Their favorite card game starts with two decks on the table containing \\\\(K_1\\\\) and \\\\(K_2\\\\) cards. Players take turns, with Alice going first. Each turn is in two parts:\\n\\n1. The player selects a deck of cards from the table. Let \\\\(k\\\\) be the number of cards in that deck. They then remove somewhere between \\\\(A_k\\\\) and \\\\(B_k\\\\) \\\\((1 \\\\le A_k \\\\le B_k \\\\le k)\\\\), inclusive, cards from this deck.\\n2. The player then puts a new deck of exactly \\\\(C_k\\\\) \\\\((0 \\\\le C_k < k)\\\\) cards on the table. Here, \\\\(C_k = 0\\\\) means no deck is added.\\n\\nThe player who takes the last card wins.\\n\\nFor each possible value of \\\\(K_1\\\\) from \\\\(1\\\\) to a given value \\\\(N\\\\), find the minimum possible value of \\\\(K_2\\\\) so that Bob wins the game if both players play optimally. If there is no such \\\\(K_2\\\\) between \\\\(1\\\\) and \\\\(N\\\\), then \\\\(K_2 = -1\\\\). Output the sum of \\\\(K_2\\\\) across every \\\\(K_1 = 1..N\\\\).\\n\\nTo reduce the input size, you will not be given \\\\(A_{1..N}\\\\), \\\\(B_{1..N}\\\\), and \\\\(C_{1..N}\\\\) directly. You must instead generate them using parameters \\\\(X_a\\\\), \\\\(Y_a\\\\), \\\\(Z_a\\\\), \\\\(X_b\\\\), \\\\(Y_b\\\\), \\\\(Z_b\\\\), \\\\(X_c\\\\), \\\\(Y_c\\\\), and \\\\(Z_c\\\\) and the algorithm below:\\n\\n\\\\(P_a[0] := 0\\\\)\\n\\\\(P_b[0] := 0\\\\)\\n\\\\(P_c[0] := 0\\\\)\\n\\\\(\\\\text{for each } i := 1..N\\\\):\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_a[i] := (P_a[i - 1] * X_a + Y_a) \\\\text{ mod } Z_a\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_b[i] := (P_b[i - 1] * X_b + Y_b) \\\\text{ mod } Z_b\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_c[i] := (P_c[i - 1] * X_c + Y_c) \\\\text{ mod } Z_c\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; A[i] := \\\\min(i, 1 + P_a[i])\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; B[i] := \\\\max(A[i], i - P_b[i])\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; C[i] := \\\\min(i - 1, P_c[i])\\\\)\\n\\nNote that for any \\\\(i\\\\), the algorithm guarantees \\\\(1 \\\\le A_i \\\\le B_i \\\\le i\\\\) and \\\\(0 \\\\le C_i < i\\\\).\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 45\\\\)\\n\\\\(1 \\\\le N \\\\le 2{,}000{,}000\\\\)\\n\\\\(1 \\\\le X_a, Y_a, X_b, Y_b, X_c, Y_c \\\\le 1{,}000{,}000{,}000\\\\)\\n\\\\(1 \\\\le Z_a, Z_b, Z_c \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(50{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, first there is a line containing a single integer \\\\(N\\\\). Then, there is a line containing integers \\\\(X_a\\\\), \\\\(Y_a\\\\), \\\\(Z_a\\\\), \\\\(X_b\\\\), \\\\(Y_b\\\\), \\\\(Z_b\\\\), \\\\(X_c\\\\), \\\\(Y_c\\\\), and \\\\(Z_c\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"` followed by a single integer, the sum of the minimum \\\\(K_2\\\\) so that Bob has a guaranteed winning strategy, for every \\\\(K_1 = 1..N\\\\).\\n\\n# Sample Explanation\\n\\nIn the first sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&1&3&2&1 \\\\\\\\ 4&1&4&0&4 \\\\end{array} \\\\)\\n\\nWhen \\\\(K_1 = 1\\\\), Bob wins when \\\\(K_2 = 1\\\\) because Alice takes the first card, and Bob takes the last card.\\n\\nWhen \\\\(K_1 = 2\\\\), Alice will win if \\\\(K_2 = 1\\\\) because she can start by taking \\\\(1\\\\) card from the first deck. Bob then takes \\\\(1\\\\) card from either deck (each of which have only \\\\(1\\\\) card left), and Alice takes the last card. But if \\\\(K_2 = 2\\\\) then Bob can always win regardless of whether Alice starts by taking \\\\(1\\\\) card or \\\\(2\\\\) cards.\\n\\nWhen \\\\(K_1 = 3\\\\), Bob can always win when \\\\(K_2 = 1\\\\). If Alice takes the single card from the second deck, Bob takes \\\\(1\\\\) card from the first deck and adds a new deck of size \\\\(2\\\\) to the table. We now have two decks of size \\\\(2\\\\), and it\\'s Alice\\'s turn. That\\'s a losing state for Alice as we saw previously.\\n\\nIf Alice takes \\\\(1\\\\) card from the first deck and adds a new deck of size \\\\(2\\\\), we now have decks of size \\\\([2, 1, 2]\\\\). Bob will pick up the pile of size \\\\(1\\\\) and again we\\'re in the same losing state for Alice. If Alice takes \\\\(2\\\\) cards from the first deck, we\\'ll have decks of size \\\\([1, 1, 2]\\\\). Bob now takes the whole deck of size \\\\(2\\\\). Alice gets the next card and Bob gets the last card. Finally, if Alice takes all \\\\(3\\\\) cards from the first deck, we\\'ll have decks of size \\\\([1, 2]\\\\). Bob can take just \\\\(1\\\\) card from the deck of size \\\\(2\\\\) to win.\\n\\nIn the second sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&1&0&2 \\\\end{array} \\\\)\\n\\nIn the third sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&2&3&1&2 \\\\end{array} \\\\)\\n\\nIn the fourth sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&3&3&2&3 \\\\\\\\ 4&1&4&0&4 \\\\\\\\ 5&5&5&4&3 \\\\\\\\ 6&1&6&0&6 \\\\\\\\ 7&6&7&4&3 \\\\\\\\ 8&1&8&0&8 \\\\end{array} \\\\)\\n\\n\\nSample Input:\\n4\\n4\\n25 30 2 45 16 4 7 26 4\\n2\\n19 48 2 38 23 2 43 31 2\\n3\\n9 17 2 23 42 2 21 43 2\\n8\\n23 5 8 14 22 2 29 28 6\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 3\\nCase #3: 5\\nCase #4: 30\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Language model program designed to solve a particular task.\n", + "\n", + "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", + "\n", + "MODULE: The module to create an instruction for.\n", + "\n", + "TASK DEMO(S): Example inputs/outputs of our module.\n", + "\n", + "BASIC INSTRUCTION: Basic instruction.\n", + "\n", + "TIP: A suggestion for how to go about generating the new instruction.\n", + "\n", + "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", + "\n", + "---\n", + "\n", + "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", + "\n", + "PROGRAM CODE:\n", + "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")\n", + "\n", + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n", + "\n", + "\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "\n", + "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + "TASK DEMO(S): \n", + "\n", + "BASIC INSTRUCTION:\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "TIP: Make sure your instruction is very informative and descriptive.\n", + "\n", + "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", + "\n", + "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", + "\n", + "\n", + "\n", + "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "task_demos \n", + "\n", + "\n", + "\n", + "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "DATASET SUMMARY: A description of the dataset that we are using.\n", + "\n", + "PROGRAM CODE: Language model program designed to solve a particular task.\n", + "\n", + "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", + "\n", + "MODULE: The module to create an instruction for.\n", + "\n", + "TASK DEMO(S): Example inputs/outputs of our module.\n", + "\n", + "BASIC INSTRUCTION: Basic instruction.\n", + "\n", + "TIP: A suggestion for how to go about generating the new instruction.\n", + "\n", + "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", + "\n", + "---\n", + "\n", + "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", + "\n", + "PROGRAM CODE:\n", + "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")\n", + "\n", + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n", + "\n", + "\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "\n", + "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + "TASK DEMO(S): \n", + "\n", + "BASIC INSTRUCTION:\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "TIP: Make sure your instruction is very informative and descriptive.\n", + "\n", + "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", + "\n", + "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", + "\n", + "\n", + "\n", + "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "task_demos \n", + "\n", + "\n", + "\n", + "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "DATASET SUMMARY: A description of the dataset that we are using.\n", + "\n", + "PROGRAM CODE: Language model program designed to solve a particular task.\n", + "\n", + "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", + "\n", + "MODULE: The module to create an instruction for.\n", + "\n", + "TASK DEMO(S): Example inputs/outputs of our module.\n", + "\n", + "BASIC INSTRUCTION: Basic instruction.\n", + "\n", + "TIP: A suggestion for how to go about generating the new instruction.\n", + "\n", + "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", + "\n", + "---\n", + "\n", + "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", + "\n", + "PROGRAM CODE:\n", + "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")\n", + "\n", + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n", + "\n", + "\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "\n", + "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + "TASK DEMO(S): \n", + "\n", + "BASIC INSTRUCTION:\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "TIP: Make sure your instruction is very informative and descriptive.\n", + "\n", + "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", + "\n", + "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", + "\n", + "\n", + "\n", + "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "task_demos \n", + "\n", + "\n", + "\n", + "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "DATASET SUMMARY: A description of the dataset that we are using.\n", + "\n", + "PROGRAM CODE: Language model program designed to solve a particular task.\n", + "\n", + "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", + "\n", + "MODULE: The module to create an instruction for.\n", + "\n", + "TASK DEMO(S): Example inputs/outputs of our module.\n", + "\n", + "BASIC INSTRUCTION: Basic instruction.\n", + "\n", + "TIP: A suggestion for how to go about generating the new instruction.\n", + "\n", + "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", + "\n", + "---\n", + "\n", + "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", + "\n", + "PROGRAM CODE:\n", + "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")\n", + "\n", + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n", + "\n", + "\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "\n", + "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + "TASK DEMO(S): \n", + "\n", + "BASIC INSTRUCTION:\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "TIP: Make sure your instruction is very informative and descriptive.\n", + "\n", + "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", + "\n", + "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", + "\n", + "\n", + "\n", + "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "task_demos \n", + "\n", + "\n", + "\n", + "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "DATASET SUMMARY: A description of the dataset that we are using.\n", + "\n", + "PROGRAM CODE: Language model program designed to solve a particular task.\n", + "\n", + "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", + "\n", + "MODULE: The module to create an instruction for.\n", + "\n", + "TASK DEMO(S): Example inputs/outputs of our module.\n", + "\n", + "BASIC INSTRUCTION: Basic instruction.\n", + "\n", + "TIP: A suggestion for how to go about generating the new instruction.\n", + "\n", + "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", + "\n", + "---\n", + "\n", + "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", + "\n", + "PROGRAM CODE:\n", + "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")\n", + "\n", + "class SimpleGenerateSolution(dspy.Module):\n", + " def __init__(self):\n", + " super().__init__()\n", + " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + " def forward(self, problem_statement, sample_input, sample_output):\n", + " solution = self.generate_code(\n", + " problem_statement=problem_statement,\n", + " sample_input=sample_input,\n", + " sample_output=sample_output\n", + " ).solution\n", + "\n", + " return dspy.Prediction(solution=solution)\n", + "\n", + "\n", + "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", + "\n", + "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", + "\n", + "TASK DEMO(S): \n", + "\n", + "BASIC INSTRUCTION:\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "TIP: Make sure your instruction is very informative and descriptive.\n", + "\n", + "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", + "\n", + "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", + "\n", + "\n", + "\n", + "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob are servers at *Nim Sum Dim Sum*, a bustling dumpling restaurant. For a staff meal, the manager has generously provided \\\\(N\\\\) plates of dumplings in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). Initially, plate \\\\(i\\\\) has \\\\(A_i\\\\) dumplings. In classic fashion, the duo has decided to play a game.\\n\\nAlice and Bob will take turns eating dumplings from the plates. On a given turn, a player must pick a plate adjacent to the last picked plate by the other player, and eat any positive number of dumplings from that plate. The first player who cannot do so on their turn loses. Alice goes first, and can choose any starting plate to eat from.\\n\\nFor example, suppose there are three plates holding \\\\(4\\\\), \\\\(1\\\\) and \\\\(2\\\\) dumplings respectively. On the first turn, Alice can eat \\\\(3\\\\) dumplings from the first plate. Bob must then eat the dumpling from the middle plate. Alice can respond by eating one dumpling from the third plate. Bob must then eat from plate \\\\(2\\\\), but since it’s empty now, he loses.\\n\\nAssuming both players play optimally, how many starting moves can Alice make so that she wins? Two starting moves are considered different if Alice eats from different plates, or eats a different number of dumplings.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 220\\\\)\\n\\\\(2 \\\\le N \\\\le 800{,}000\\\\)\\n\\\\(0 \\\\le A_i \\\\lt 2^{25}\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of cases. Each case will begin with a single integer \\\\(N\\\\) followed by the \\\\(N\\\\) integers \\\\(A_1, ..., A_N\\\\) on the next line.\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"` followed by a single integer, the number of winning starting moves Alice has.\\n\\n# Sample Explanation\\n\\nIn the first case, Alice can start by taking any number of dumplings from either the first or third plate. Bob will then have to take the solitary dumpling on the middle plate, and Alice can win by taking all the dumplings from the plate she didn\\'t start with. This gives Alice 6 different winning starting moves.\\n\\nIn the second case, Alice cannot win because she takes one dumpling, Bob takes the other, and then Alice has no move to make.\\n\\nIn the third case, Alice\\'s winning moves are to take \\\\(1\\\\) or \\\\(2\\\\) dumplings from the right-hand plate.\\n\\nIn the fourth case, Bob can always force a win.\\n\\n\\n\\nSample Input:\\n6\\n3\\n4 1 2\\n2\\n1 1\\n2\\n2 4\\n3\\n1 3 2\\n6\\n2 2 3 3 4 4\\n8\\n6 2 8 3 1 8 5 3\\n\\n\\nSample Output:\\nCase #1: 6\\nCase #2: 0\\nCase #3: 2\\nCase #4: 0\\nCase #5: 0\\nCase #6: 19\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. 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We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem B1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/B1) is that here, scenes are larger and may contain rocks.**\\n\\nBoss Rob painted a beautiful scene on a 2D canvas of \\\\(R\\\\) rows by \\\\(C\\\\) columns, containing zero or more happy little trees **and zero or more rocks.**\\n\\nTo make sure none of his trees are lonely, Rob would like you to add as many trees as you\\'d like (possibly \\\\(0\\\\)) to empty spaces so that each tree in the final painting has at least two tree *friends*, that is, two trees which are each adjacent to it (directly to its north, south, east, or west). If there are multiple solutions, you may print any one of them. \\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 80\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\mathbf{3{,}000}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, each of which contains \\\\(C\\\\) characters, either \"`.`\" (an empty space), \"`^`\" (a tree), **or \"`#`\" (a rock)**, representing the initial painting.\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print \"`Case #i:` \", followed by \"`Possible`\" and \\\\(R\\\\) lines of \\\\(C\\\\) characters each representing the final painting (if a solution exists), otherwise \"`Impossible`\".\\n\\n\\n# Sample Explanation\\n\\nIn the first case (depicted below), we could add two tree friends to either side of the middle tree, but they themselves would each only have one tree friend. Therefore, it\\'s impossible to get two friends for each tree in the final painting.\\n\\n{{PHOTO_ID:604840697695882|WIDTH:295}}\\n\\nIn the second case, there are no trees in the initial painting, so the condition of two friends for each tree is already satisfied.\\n\\nIn the third case, one possible solution is depicted below.\\n\\n{{PHOTO_ID:749016109740997|WIDTH:700}}\\n\\n\\nSample Input:\\n3\\n1 3\\n.^.\\n3 1\\n.\\n#\\n#\\n4 4\\n..^.\\n.#^#\\n....\\n...^\\n\\n\\nSample Output:\\nCase #1: Impossible\\nCase #2: Possible\\n.\\n#\\n#\\nCase #3: Possible\\n^^^.\\n^#^#\\n^^^^\\n..^^\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:53 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'7750'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149973327'), (b'x-ratelimit-reset-requests', b'10ms'), (b'x-ratelimit-reset-tokens', b'10ms'), (b'x-request-id', b'req_231e01ec2387af56a121b2fee5bf93dc'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d734dc959a7-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:53 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '7750', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149973327', 'x-ratelimit-reset-requests': '10ms', 'x-ratelimit-reset-tokens': '10ms', 'x-request-id': 'req_231e01ec2387af56a121b2fee5bf93dc', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d734dc959a7-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_231e01ec2387af56a121b2fee5bf93dc\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou, a perfect speller, have a vocabulary of \\\\(N\\\\) distinct words, \\\\(V_1, ..., V_N\\\\), each consisting of exactly \\\\(L\\\\) lowercase letters from the alphabet \\\\(\\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\).\\n\\nYour friend, a truely terrable speler, has attempted to write \\\\(Q\\\\) of these words as \\\\(W_1, ..., W_Q\\\\), each also consisting of \\\\(L\\\\) lowercase letters from the same alphabet.\\n\\nLet \\\\(S_i\\\\) be the number of words in your vocabulary that differ from \\\\(W_i\\\\) at exactly two indices. Please determine the sum \\\\(S_1 + ... + S_Q\\\\).\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 95\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 750{,}000\\\\)\\n\\\\(1 \\\\le L = |V_i| = |W_i| \\\\le 20{,}000\\\\)\\n\\\\((N+Q)*L \\\\le 15{,}000{,}000\\\\)\\n\\\\(V_{ij} \\\\in \\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\)\\n\\\\(W_{ij} \\\\in \\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\)\\nAll \\\\(V_i\\\\) in a given test case are distinct.\\n\\nThe sum of lengths of all strings across all cases is at most \\\\(18{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(V_i\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(W_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum \\\\(S_1 + ... + S_Q\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:1332823754208247|WIDTH:500}}\\n\\nThe answer is \\\\(4\\\\), since:\\n- \\\\(W_1\\\\) = \"`teammate`\" differs from \"`metamate`\" at three indices, so \\\\(S_1 = 0\\\\).\\n- \\\\(W_2\\\\) = \"`meatmate`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_2 = 1\\\\).\\n- \\\\(W_3\\\\) = \"`metatame`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_3 = 1\\\\).\\n- \\\\(W_4\\\\) = \"`mememate`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_4 = 1\\\\).\\n- \\\\(W_5\\\\) = \"`metameme`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_5 = 1\\\\).\\n\\nIn the second case, the answer is \\\\(0\\\\), since:\\n- \\\\(W_1\\\\) = \"`tata`\" differs from \\\\(V_1\\\\) = \"`meet`\" at four indices, \\\\(V_2\\\\) = \"`emma`\" at three indices, and \\\\(V_3\\\\) = \"`tate`\" at only one index, so \\\\(S_1 = 0\\\\).\\n- \\\\(W_2\\\\) = \"`maam`\" differs from \\\\(V_1\\\\) = \"`meet`\" at three indices, \\\\(V_2\\\\) = \"`emma`\" at four indices, and \\\\(V_3\\\\) = \"`tate`\" at three indices, so \\\\(S_2 = 0\\\\).\\n\\nIn the third case, the answer is \\\\(5\\\\), since:\\n- \\\\(W_1\\\\) = \"`tam`\" differs from both \\\\(V_1\\\\) = \"`mem`\" and \\\\(V_3\\\\) = \"`mat`\" at exactly two indices, so \\\\(S_1 = 2\\\\).\\n- \\\\(W_2\\\\) = \"`mat`\" differs from \\\\(V_1\\\\) = \"`mem`\" at exactly two indices, so \\\\(S_2 = 1\\\\).\\n- \\\\(W_3\\\\) = \"`tea`\" differs from both \\\\(V_1\\\\) = \"`mem`\" and \\\\(V_2\\\\) = \"met\" at exactly two indices, so \\\\(S_3 = 2\\\\).\\n\\n\\nSample Input:\\n3\\n1\\nmetamate\\n5\\nteammate\\nmeatmate\\nmetatame\\nmememate\\nmetameme\\n3\\nmeet\\nemma\\ntate\\n2\\ntata\\nmaam\\n3\\nmem\\nmet\\nmat\\n3\\ntam\\nmat\\ntea\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 0\\nCase #3: 5\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:53 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'8229'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29998'), (b'x-ratelimit-remaining-tokens', b'149992417'), (b'x-ratelimit-reset-requests', b'3ms'), (b'x-ratelimit-reset-tokens', b'3ms'), (b'x-request-id', b'req_a363ec43fb446bf88a69feaaa4775d83'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d732e4227b7-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:53 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '8229', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29998', 'x-ratelimit-remaining-tokens': '149992417', 'x-ratelimit-reset-requests': '3ms', 'x-ratelimit-reset-tokens': '3ms', 'x-request-id': 'req_a363ec43fb446bf88a69feaaa4775d83', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d732e4227b7-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_a363ec43fb446bf88a69feaaa4775d83\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this and [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-3/problems/D1/) is that here, you must find the sum of answers for \\\\(K = 1..N\\\\).**\\n\\nBoss Rob ain’t Som Tawyer, but he can paint a fence alright.\\n\\nRob\\'s fence is made of \\\\(N\\\\) wooden stakes, numbered \\\\(1\\\\) to \\\\(N\\\\) from left to right. Initially (at time \\\\(0\\\\)), the \\\\(i\\\\)th stake is of color \\\\(i\\\\). There is a fencepost before stake \\\\(1\\\\) and after stake \\\\(N\\\\), as well as after every \\\\(K\\\\)th stake starting from the left.\\n\\nRob has a simple and joyful plan to repaint his fence, consisting of \\\\(M\\\\) moments in time. At time \\\\(i\\\\), he\\'ll repaint all stakes which are color \\\\(A_i\\\\) to color \\\\(B_i\\\\). Doing so, when would be the *first time* that all pairs of stakes not separated by a fencepost have the same color? If it will never occur, consider the answer to be \\\\(-1\\\\).\\n\\nRob is still on the _fence_ about the value of \\\\(K\\\\), so please print **the sum of answers over \\\\(K = 1..N\\\\).** Sorry for the pun (we know there is a lot at _stake_ in this round _gating_ the finals). In our de\\u200b_fence_, it makes for good _post_-problem content.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 30\\\\)\\n\\\\(1 \\\\le N, M \\\\le 600{,}000\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For the \\\\(i\\\\)th test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of answers for each \\\\(K = 1..N\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe progressions of fences for the first two sample cases are depicted below (with fenceposts depicted for \\\\(K=2\\\\) and \\\\(K=3\\\\) respectively):\\n\\n{{PHOTO_ID:631752615085471|WIDTH:700}}\\n\\nIn the first case, the answers are \\\\([0, 2, 4, 3, 4]\\\\) for \\\\(K = 1..5\\\\), so we print \\\\(2+4+3+4 = 13\\\\).\\n\\nIn the second case, the answers are \\\\([0, 1, 3]\\\\) for \\\\(K = 1..3\\\\), so we print \\\\(1+3=4\\\\).\\n\\nIn the third case, the fence (with fenceposts depicted for \\\\(K=3\\\\)) progresses as follows:\\n\\n{{PHOTO_ID:413921204256250|WIDTH:700}}\\n\\nFor \\\\(K = 1..8\\\\), the answers are \\\\([0, 5, -1, 6, -1, -1, -1, -1]\\\\), so we print the sum \\\\(6\\\\).\\n\\n\\n\\nSample Input:\\n3\\n5 4\\n2 1\\n3 4\\n1 4\\n5 4\\n3 4\\n1 2\\n2 1\\n1 3\\n3 1\\n8 6\\n1 4\\n2 3\\n4 3\\n8 7\\n6 5\\n7 5\\n\\n\\nSample Output:\\nCase #1: 13\\nCase #2: 4\\nCase #3: 6\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:54 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'8533'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29997'), (b'x-ratelimit-remaining-tokens', b'149985420'), (b'x-ratelimit-reset-requests', b'5ms'), (b'x-ratelimit-reset-tokens', b'5ms'), (b'x-request-id', b'req_78f005987356c166cc8ebc6808d4f5df'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d734cff0dad-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:54 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '8533', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29997', 'x-ratelimit-remaining-tokens': '149985420', 'x-ratelimit-reset-requests': '5ms', 'x-ratelimit-reset-tokens': '5ms', 'x-request-id': 'req_78f005987356c166cc8ebc6808d4f5df', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d734cff0dad-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_78f005987356c166cc8ebc6808d4f5df\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this and [Chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-3/problems/D2/) is that here, you only need to find the answer for a single \\\\(K\\\\).**\\n\\nBoss Rob ain’t Som Tawyer, but he can paint a fence alright.\\n\\nRob\\'s fence is made of \\\\(N\\\\) wooden stakes, numbered \\\\(1\\\\) to \\\\(N\\\\) from left to right. Initially (at time \\\\(0\\\\)), the \\\\(i\\\\)th stake is of color \\\\(i\\\\). There is a fencepost before stake \\\\(1\\\\) and after stake \\\\(N\\\\), as well as after every \\\\(K\\\\)th stake starting from the left.\\n\\nRob has a simple and joyful plan to repaint his fence, consisting of \\\\(M\\\\) moments in time. At time \\\\(i\\\\), he\\'ll repaint all stakes which are color \\\\(A_i\\\\) to color \\\\(B_i\\\\). Doing so, when would be the *first time* that all pairs of stakes not separated by a fencepost have the same color? If it will never occur, consider the answer to be \\\\(-1\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 40\\\\)\\n\\\\(1 \\\\le N, M \\\\le 600{,}000\\\\)\\n\\\\(1 \\\\le K \\\\le N\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For the \\\\(i\\\\)th test case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(K\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the first time that the condition is satisfied, or \\\\(-1\\\\) if will never be.\\n\\n# Sample Explanation\\n\\nThe progressions of fences for the first two sample cases are depicted below:\\n\\n{{PHOTO_ID:1207042806525380|WIDTH:700}}\\n\\nIn the first case, all stakes between fenceposts will have the same color after time \\\\(2\\\\).\\n\\nIn the second case, all stakes between fenceposts will have the same color after time \\\\(3\\\\).\\n\\nIn the third case, the fence progresses as follows:\\n\\n{{PHOTO_ID:1173386423530703|WIDTH:700}}\\n\\nAt no point in time will all stakes between pairs of fenceposts have the same color.\\n\\n\\n\\nSample Input:\\n3\\n5 4 2\\n2 1\\n3 4\\n1 4\\n5 4\\n3 4 3\\n1 2\\n2 1\\n1 3\\n3 1\\n8 6 3\\n1 4\\n2 3\\n4 3\\n8 7\\n6 5\\n7 5\\n\\n\\nSample Output:\\nCase #1: 2\\nCase #2: 3\\nCase #3: -1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9355'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149993854'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_ebfa85fe3d6aa826b84059ba24fc1aba'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d7359215234-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9355', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149993854', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_ebfa85fe3d6aa826b84059ba24fc1aba', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d7359215234-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_ebfa85fe3d6aa826b84059ba24fc1aba\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**The only differences between this chapter and [chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/D2) is that here, \\\\(A_i \\\\in \\\\{1, 2, 3\\\\}\\\\), and you may swap any two elements of \\\\(A_i\\\\) with a single operation.**\\n\\nAs a Metal Platforms employee, you place a high value on your work-life balance. Boss Rob has assigned you \\\\(N\\\\) tasks, the \\\\(i\\\\)th of which takes \\\\(A_i\\\\) minutes to finish, where \\\\(A_i\\\\) is either **\\\\(\\\\mathbf{1, 2}\\\\), or \\\\(\\\\mathbf{3}\\\\)**.\\n\\nYou may reorder your tasks, where each *operation* lets you **swap any two elements** of \\\\(A\\\\).\\n\\nTo reflect how often task requirements change in the real world, there will be \\\\(M\\\\) updates made to the task completion times, with the \\\\(i\\\\)th update setting \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n\\nAfter completing the \\\\(i\\\\)th update, you would like to know if it\\'s possible to balance the time spent at work versus at home, namely if you hope to finish the first \\\\(Z_i\\\\) tasks at work and the rest at home. Specifically, let \\\\(Q_i\\\\) be the minimum number of swap operations which must be theoretically performed so that \\\\(A_1 + ... + A_{Z_i} = A_{Z_i + 1} + ... + A_{N}\\\\), with \\\\(Q_i = -1\\\\) if it\\'s impossible. Note that it\\'s possible for \\\\(Q_i\\\\) to be \\\\(0\\\\), if the subarrays already have equal sums.\\n\\nTo reduce the size of the output, please compute the sum of \\\\(Q_1\\\\), ..., \\\\(Q_M\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le M \\\\le 1{,}000{,}000\\\\)\\n\\\\(A_i, Y_i \\\\in \\\\{1, 2, 3\\\\}\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Z_i \\\\le N-1\\\\)\\n\\nThe sum of \\\\(N+M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line containing \\\\(N\\\\) digits, \\\\(A_1\\\\), \\\\(A_2\\\\), ..., \\\\(A_N\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(X_i\\\\), \\\\(Y_i\\\\), and \\\\(Z_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output `\"Case #i: \"` followed by a single integer, \\\\(Q_1 + ... + Q_M\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:463552492491498|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 2]\\\\). The first update yields \\\\(A=[1, 1]\\\\) with \\\\(Z_1 = 1\\\\). The first and second tasks are already equal, so \\\\(Q_1 = 0\\\\) as no swaps are necessary. The second update yields \\\\(A=[1, 3]\\\\) with \\\\(Z_2 = 1\\\\). Whether or not we swap these tasks, they can never be equal, so \\\\(Q_2 = -1\\\\). The final answer is \\\\(0 + (-1) = -1\\\\).\\n\\nThe second case is depicted below:\\n\\n{{PHOTO_ID:774910673835114|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 1, 3, 3]\\\\). The first update yields \\\\(A=[1, 1, 3, 1]\\\\). We need the sum of the first three tasks to equal the fourth. We can swap the third and fourth tasks to get \\\\(1 + 1 + 1 = 3\\\\), so \\\\(Q_1 = 1\\\\). The second update yields \\\\(A=[3, 1, 3, 1]\\\\). We need the sum of the first two tasks to equal the sum of the last two. This is already true, so \\\\(Q_2 = 0\\\\). The final answer is \\\\(1 + 0 = 1\\\\).\\n\\nIn the third case, we start with \\\\(A = [1, 2, 3, 3, 3, 3]\\\\). With the updates, we get:\\n- \\\\(A = [1, 2, 3, 3, 3, 2], Z_1 = 3, Q_1 = 1\\\\): we can swap \\\\(A_1\\\\) and \\\\(A_6\\\\)\\n- \\\\(A = [1, 2, 3, 1, 3, 2], Z_2 = 2, Q_2 = 2\\\\): we can swap both \\\\(3\\\\)s to the front\\n- \\\\(A = [1, 2, 1, 1, 3, 2], Z_3 = 2, Q_3 = 1\\\\): we can swap \\\\(A_1\\\\) and \\\\(A_5\\\\)\\n- \\\\(A = [3, 2, 1, 1, 3, 2], Z_4 = 5, Q_4 = -1\\\\): impossible\\n\\nThe final answer is \\\\(1 + 2 + 1 + (-1) = 3\\\\).\\n\\n\\nSample Input:\\n3\\n2 2\\n1 2\\n2 1 1\\n2 3 1\\n4 2\\n1 1 3 3\\n4 1 3\\n1 3 2\\n6 4\\n1 2 3 3 3 3\\n6 2 3\\n4 1 2\\n3 1 2\\n1 3 5\\n\\n\\nSample Output:\\nCase #1: -1\\nCase #2: 1\\nCase #3: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nHacker Cup contest strategy often involves a metagame, where choosing which problems to work on might just be an important decision. On a Quest to become more Pro, you encounter an oracle promising to teach you the contest meta if you play her own Meta-game.\\n\\nThe oracle presents a peg board with \\\\(2N\\\\) moving dots. The initial \\\\(y\\\\)-positions of the dots are given as two arrays \\\\(A_{1..N}\\\\) and \\\\(B_{1..N}\\\\). Each second, simultaneously, \\\\(A_1\\\\) will move to the end of \\\\(B\\\\), while \\\\(B_1\\\\) will move to the end of \\\\(A\\\\) (with all elements shifting left accordingly).\\n\\nYou can connect the dots to form a *Meta-like logo* if all of the following are true:\\n* For the first half of both arrays, each dot in \\\\(A\\\\) is below the corresponding dot in \\\\(B\\\\).\\n* For the last half of both arrays, each dot in \\\\(A\\\\) is above the corresponding dot in \\\\(B\\\\).\\n* \\\\(A\\\\) equals the reverse of \\\\(B\\\\).\\n\\nFormally:\\n* \\\\(A_i < B_i\\\\) for every \\\\(i < (N+1)/2\\\\)\\n* \\\\(A_i > B_i\\\\) for every \\\\(i > (N+1)/2\\\\)\\n* \\\\(A_i = B_{N-i+1}\\\\) for every \\\\(i = 1..N\\\\)\\n\\nNote that if \\\\(N\\\\) is odd, the arrays\\' middle elements are not subject to the first two constraints.\\n\\nThe following is a visualization of a Meta-like logo (corresponding to the first sample case), with dots in \\\\(A\\\\) shown in red and dots in \\\\(B\\\\) shown in blue.\\n\\n{{PHOTO_ID:359057163229199|WIDTH:500}}\\n\\nYou must answer the oracle: how many seconds must pass before the first time a Meta-like logo appears? If one never appears, output \\\\(-1\\\\).\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 135\\\\)\\n\\\\(2 \\\\leq N \\\\leq 2{,}000{,}000\\\\)\\n\\\\(0 \\\\leq A_i, B_i \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(9{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing integers \\\\(B_1, ..., B_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the number of seconds that must pass before a Meta-like logo appears, or \\\\(-1\\\\) if that will never happen.\\n\\n\\n# Sample Explanation\\n\\nThe first test case is shown above. \\\\(A\\\\) and \\\\(B\\\\) already form a Meta-like logo, so the answer is 0.\\n\\nThe second case is not initially a Meta-like logo, for several reasons. One reason is that it is not symmetric. Specifically, the \\\\([3, 3, 2, 3, 5 ,6]\\\\) is not the reverse of \\\\([4, 4, 6, 5, 3 ,2]\\\\). After \\\\(1\\\\) second though, this case turns into the case above and is Meta-like.\\n\\nThe third and fourth cases will never turn into a Meta-like logo, no matter how many seconds we wait.\\n\\nIn the fifth case, after 6 seconds we see the first Meta-like logo. In this case \\\\(A = [1, 1, 2, 2]\\\\) and \\\\(B = [2, 2, 1, 1]\\\\).\\n\\n\\n\\n\\nSample Input:\\n8\\n6\\n3 2 3 5 6 4\\n4 6 5 3 2 3\\n6\\n3 3 2 3 5 6\\n4 4 6 5 3 2\\n6\\n4 3 2 3 5 6\\n3 4 6 5 3 2\\n2\\n1 1\\n1 1\\n4\\n2 2 2 2\\n1 1 1 1\\n5\\n3 3 3 3 3\\n1 1 1 1 1\\n5\\n3 3 3 3 3\\n1 1 1 1 3\\n5\\n1 1 1 1 3\\n3 3 3 3 3\\n\\n\\nSample Output:\\nCase #1: 0\\nCase #2: 1\\nCase #3: -1\\nCase #4: -1\\nCase #5: 6\\nCase #6: -1\\nCase #7: 7\\nCase #8: 2\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob are playing *Tower Rush,* a two-phase game involving \\\\(N\\\\) types of blocks, numbered from \\\\(1\\\\) to \\\\(N\\\\). Blocks of type \\\\(i\\\\) have height \\\\(H_i\\\\), and no two types of blocks have the same height.\\n\\nPhase 1 of the game consists of \\\\(K \\\\ge 2\\\\) alternating turns, with Alice going first. On each player\\'s turn, they will choose a block type that has not yet been chosen. Note that if \\\\(K\\\\) is odd, this phase would end with Alice having chosen one more type of block than Bob. From this point on, players have access to infinitely many blocks of each of the types they chose.\\n\\nPhase 2 consists of an indefinite number of alternating turns, with Bob going first. Each player starts with a tower of height \\\\(0\\\\). On each player\\'s turn, they may pick a block of any type \\\\(i\\\\) available to them, and extend their tower height by \\\\(H_i\\\\). They may also choose to skip their turn, leaving their tower unchanged.\\n\\nIn a break from tradition, Alice and Bob want to work *together* to see if it\\'s possible for Alice to build a tower that\\'s exactly \\\\(D\\\\) units taller than Bob\\'s. In how many different ways can phase 1 be played out such that it will be possible for Alice to get her tower to be exactly \\\\(D\\\\) units taller than Bob\\'s in phase 2? Two ways are considered different if there exists an \\\\(i\\\\) such that different block types are chosen on turn \\\\(i\\\\) between the two ways.\\n\\nAs this value may be large, output it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 24\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(2 \\\\leq K \\\\leq \\\\min(N, 20)\\\\)\\n\\\\(1 \\\\leq D \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq H_i \\\\leq 1{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(7{,}500{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, the first line contains the integers \\\\(N\\\\), \\\\(K\\\\), and \\\\(D\\\\). Then, there is a line containing the \\\\(N\\\\) integers \\\\(H_1\\\\), ..., \\\\(H_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the number of unique ways to accomplish Bob\\'s goal, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are \\\\(54\\\\) valid choices of blocks in phase 1. One such example is \\\\(\\\\{H_2 = 4,\\\\, H_3 = 5\\\\}\\\\) for Alice and \\\\(\\\\{H_4 = 7\\\\}\\\\) for Bob. Four turns in phase 2 can go as follows:\\n* Bob adds \\\\(H_4 = 7\\\\). His tower now has height \\\\(7\\\\).\\n* Alice adds \\\\(H_3 = 5\\\\). Her tower now has height \\\\(5\\\\).\\n* Bob chooses not to add a block. His tower still has height \\\\(7\\\\).\\n* Alice adds \\\\(H_3 = 5\\\\). Her tower now has height \\\\(10\\\\).\\n\\nAlice’s tower is now exactly \\\\(3\\\\) units taller than Bob’s, so this is a valid choice.\\n\\nAn example of an invalid choice in the first test case would be \\\\(\\\\{H_1 = 2,\\\\, H_2=4\\\\}\\\\) for Alice and \\\\(\\\\{H_5 = 8\\\\}\\\\) for Bob, since no matter how they build their towers, it is impossible for the difference in the height of their towers to ever be \\\\(3\\\\).\\n\\n\\nSample Input:\\n2\\n5 3 3\\n2 4 5 7 8\\n3 2 4\\n3 6 9\\n\\n\\nSample Output:\\nCase #1: 54\\nCase #2: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/B1) is that here, coordinates may be up to \\\\(\\\\mathbf{10^9}\\\\).**\\n\\nBoss Rob just planted \\\\(N\\\\) happy little trees in his yard, which can be represented on a Cartesian plane. The \\\\(i\\\\)th tree is located at coordinates \\\\(t_i = (A_i, B_i)\\\\). Now, he\\'s looking for the best spot to build a well in order to provide water to them. He considers the *inconvenience* of a potential well location \\\\(p\\\\) to be the sum of the squared Euclidean distances to every tree:\\n\\n\\\\[\\\\sum_{i=1}^{N} \\\\Vert \\\\,p - t_i \\\\Vert ^ 2 \\\\]\\n\\nRob wants to pick a location for his well, well... well. Help him determine the inconvenience for \\\\(Q\\\\) different potential well locations, \\\\((X_1, Y_1), ..., (X_Q, Y_Q)\\\\). To reduce output size, please print the sum of inconveniences for all potential well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 50\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le A_i, B_i, X_i, Y_i \\\\le \\\\mathbf{10^9}\\\\)\\nAll \\\\((A_i, B_i)\\\\) are distinct within a given test case.\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe total sum of \\\\(N\\\\) and \\\\(Q\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\). Then there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"`, followed by a single integer, the sum of inconveniences for all \\\\(Q\\\\) well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first two sample cases are depicted below:\\n\\n{{PHOTO_ID:3620154144878669|WIDTH:700}}\\n\\nIn the first case, the total inconvenience is \\\\(18 + 34 = 52\\\\):\\n\\n- For the well at \\\\((2, 5)\\\\), the inconvenience is the sum of the squared Euclidean distance to both trees, which is \\\\(3^2 + 3^2 = 18\\\\).\\n- For the well at \\\\((6, 6)\\\\), the inconvenience is \\\\(32 + 2 = 34\\\\).\\n\\nIn the second case, the total inconvenience is \\\\(47 + 31 + 53 = 131\\\\):\\n\\n- For the well at \\\\((3, 1)\\\\), the inconvenience is \\\\(4 + 5 + 13 + 25 = 47\\\\).\\n- For the well at \\\\((5, 2)\\\\), the inconvenience is \\\\(17 + 2 + 2 + 10 = 31\\\\).\\n- For the well at \\\\((6, 5)\\\\), the inconvenience is \\\\(41 + 8 + 4 + 0 = 53\\\\).\\n\\n\\nSample Input:\\n4\\n2\\n2 2\\n5 5\\n2\\n2 5\\n6 6\\n4\\n1 1\\n4 3\\n6 3\\n6 5\\n3\\n3 1\\n5 2\\n6 5\\n8\\n2837 745\\n62 1162\\n2634 1112\\n1746 2618\\n847 127\\n986 1993\\n732 1273\\n2003 1998\\n4\\n1276 2231\\n1234 1234\\n287 2371\\n3000 3000\\n5\\n283746263 475619273\\n987361523 361738847\\n281936352 666152443\\n143042069 482716253\\n1000000000 100000000\\n5\\n0 0\\n123456789 987654321\\n192837465 918273645\\n135792468 864209753\\n703692581 185296307\\n\\n\\nSample Output:\\nCase #1: 52\\nCase #2: 131\\nCase #3: 110090622\\nCase #4: 391473143\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_The only difference between chapters 1 and 2 is the maximum allowed grid size, given in bold below._\\n\\nA *Drizzle* program is a 2D grid of the following four types of cells:\\n - \\'`@`\\' (start) \\\\(-\\\\) there is exactly one start cell in the entire grid\\n - \\'`#`\\' (wall)\\n - \\'`.`\\' (space)\\n - \\'`*`\\' (instruction)\\n\\nThe program uses two registers \\\\(A\\\\) and \\\\(B\\\\) (both initially \\\\(0\\\\)), and executes as follows:\\n\\n1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. \\n2. In increasing order, for each unique distance \\\\(D\\\\) such that there’s at least one instruction cell that’s at distance \\\\(D\\\\) from the start cell:\\n 2a. Count the number of shortest paths, \\\\(P\\\\), to all instruction cells of distance \\\\(D\\\\).\\n 2b. Look up the instruction corresponding to \\\\((P \\\\text{ mod } 2, D \\\\text{ mod } 2)\\\\) in the table below and modify one of the registers accordingly.\\n3. At the end, the value in register \\\\(A\\\\) is outputted.\\n\\n```\\n┌─────────────┬─────────────┬─────────────┐\\n│ │ D mod 2 = 0 │ D mod 2 = 1 │\\n├─────────────┼─────────────┼─────────────┤\\n│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │\\n│ P mod 2 = 1 │ B := B + A │ A := B │\\n└─────────────┴─────────────┴─────────────┘\\n```\\n\\nFor a given value \\\\(K\\\\), output any Drizzle program that outputs \\\\(K\\\\) when executed, with the restriction that **the program must fit on a \\\\(\\\\mathbf{13}\\\\) × \\\\(\\\\mathbf{13}\\\\) grid**.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 2{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10{,}000\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is a line containing the single integer \\\\(K\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output \"`Case #i: `\" followed by two integers \\\\(R\\\\) and \\\\(C\\\\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \\\\(R\\\\) lines long, with each line containing exactly \\\\(C\\\\) characters.\\n\\n\\n# Sample Explanation\\n\\nHere are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases.\\n\\nIn the first case, there is a single instruction. There are \\\\(2\\\\) shortest paths of length \\\\(2\\\\) to that instruction, so \\\\(P = 2\\\\) and \\\\(D = 2\\\\). That means we perform \\\\(A := A + 1\\\\). There are no more instructions, so the program ends and outputs \\\\(1\\\\).\\n\\nIn the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \\\\(A := A + 1\\\\):\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(4\\\\) paths of length \\\\(6\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(4\\\\) paths of length \\\\(12\\\\) \\\\(\\\\;(A := A + 1 = 3)\\\\)\\n\\nIn the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \\\\(2\\\\), and three instruction cells at distance \\\\(10\\\\). There\\'s a single shortest path to each of the cells at distance \\\\(2\\\\), so in total there are \\\\(2\\\\) shortest paths to instructions at distance \\\\(2\\\\). One of the cells at distance \\\\(10\\\\) has a unique shortest path, and the other has two shortest paths, so in total there are \\\\(3\\\\) shortest paths to instructions at distance \\\\(10\\\\).\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(6\\\\) paths of length \\\\(4\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(1\\\\) path of length \\\\(6\\\\) \\\\(\\\\;(B := B + A = 2)\\\\)\\n4) \\\\(1\\\\) path of length \\\\(8\\\\) \\\\(\\\\;(B := B + A = 4)\\\\)\\n5) \\\\(3\\\\) paths of length \\\\(10\\\\) \\\\(\\\\;(B := B + A = 6)\\\\)\\n6) \\\\(3\\\\) paths of length \\\\(11\\\\) \\\\(\\\\;(A := B = 6)\\\\)\\n\\n\\nSample Input:\\n3\\n1\\n3\\n6\\n\\n\\nSample Output:\\nCase #1: 2 2\\n@.\\n.*\\nCase #2: 5 6\\n@.#.*.\\n.*#...\\n#.#...\\n...#.#\\n..*...\\nCase #3: 4 10\\n...#*.*.**\\n.*.*##..#.\\n.....#*#..\\n.@.*.....#\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:56 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10739'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149968849'), (b'x-ratelimit-reset-requests', b'11ms'), (b'x-ratelimit-reset-tokens', b'12ms'), (b'x-request-id', b'req_05dd10b063298d0a4cd07c8f7b5e139f'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735cf12785-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:56 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10739', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149968849', 'x-ratelimit-reset-requests': '11ms', 'x-ratelimit-reset-tokens': '12ms', 'x-request-id': 'req_05dd10b063298d0a4cd07c8f7b5e139f', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735cf12785-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_05dd10b063298d0a4cd07c8f7b5e139f\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nThere\\'s a famous saying about what to do when life gives you lemons. As a traveling lemonade salesman who\\'s never been given any lemons, you sadly can\\'t relate. You must shamefully concoct your lemonade from store-bought powder with your back turned to the world, lest someone see your dirty secret.\\n\\nYour sales route can be mapped out on a Cartesian plane with \\\\(N\\\\) houses, the \\\\(i\\\\)th of which is at coordinates \\\\((X_i, Y_i)\\\\). Your journey starts at house \\\\(1\\\\), the leftmost house, and ends at house \\\\(N\\\\), the rightmost house. Along the way, you may stop at zero or more other houses to sell lemonade.\\n\\nYou may only stop at a house \\\\(h\\\\) if:\\n\\n1) standing at house \\\\(h\\\\), there exists some direction you can face in which all other houses are *strictly more behind you than they are in front of you* (formally, if there exists a [half-plane](https://mathworld.wolfram.com/Half-Plane.html#:~:text=A%20half%2Dplane%20is%20a,called%20an%20open%20half%2Dplane.) containing only house \\\\(h\\\\)), and\\n2) house \\\\(h\\\\) is at most Euclidean distance \\\\(D\\\\) from the previous house you were at.\\n\\nYour brand image is hurt if you go too long without selling lemonade. The *brand damage* incurred by traveling from one house to another is the larger of \\\\(K\\\\) and the squared Euclidean distance between them. Formally, if your journey consists of \\\\(M\\\\) \\\\((2 \\\\le M \\\\le N)\\\\) houses with the \\\\(i\\\\)th being house \\\\(H_i\\\\) \\\\((H_1 = 1, H_M = N)\\\\), the total brand damage is:\\n\\n\\\\[\\\\sum_{i=1}^{M-1} \\\\max(K, (X_{H_i} - X_{H_{i + 1}})^2 + (Y_{H_i} - Y_{H_{i + 1}})^2)\\\\]\\n\\nIs it possible to make the journey? If so, what is the minimum possible total brand damage to do so? Note that the answer may be large, but will fit in a 64-bit integer.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 90\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(0 \\\\le K, D \\\\le 10^9\\\\)\\n\\\\(0 \\\\le X_i, Y_i \\\\le 1{,}000{,}000\\\\)\\n\\\\(X_1\\\\) is strictly less than all other \\\\(X_i\\\\).\\n\\\\(X_N\\\\) is strictly greater than all other \\\\(X_i\\\\).\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThere are at most \\\\(15\\\\) test cases in which \\\\(N > 5{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(K\\\\), and \\\\(D\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"` followed a single integer, the minimum total brand damage that must be incurred to make the journey, or \\\\(-1\\\\) if it\\'s impossible to do so.\\n\\n\\n# Sample Explanation\\n\\nThe first three sample cases are depicted below, with the optimal paths given in blue.\\n\\n{{PHOTO_ID:654377492455969|WIDTH:700}}\\n\\nIn the first case, going from one house to another takes at least \\\\(K = 25\\\\) brand damage and must not exceed a distance of \\\\(D = 8\\\\). The total brand damage is \\\\(25+50+40 = 115\\\\). Note that you cannot stop at house \\\\((6, 7)\\\\) because there is no direction you could face from there in which your back is at least slightly facing both house \\\\((1,6)\\\\) and \\\\((11,8)\\\\).\\n\\nIn the second case, you can stop at house \\\\((4, 1)\\\\) because for instance, the line \\\\(y = 0.3x - 0.2\\\\) contains only \\\\((4, 1)\\\\), and the half-plane below it contains no other houses.\\n\\nIn the third case, you cannot stop at house \\\\((4, 1)\\\\). There are no other houses within \\\\(D=7\\\\) units of your starting house, so it is not possible to reach the house at \\\\((8, 2)\\\\).\\n\\n\\nSample Input:\\n5\\n9 25 8\\n0 5\\n1 6\\n6 3\\n6 7\\n3 4\\n9 2\\n2 1\\n1 2\\n11 8\\n3 100 7\\n0 0\\n4 1\\n7 2\\n3 100 7\\n0 0\\n4 1\\n8 2\\n6 0 1000000000\\n0 10\\n2 5\\n1 7\\n7 4\\n8 1\\n10 0\\n12 1600 2000\\n0 30\\n16 48\\n36 57\\n951 45\\n397 63\\n447 63\\n185 16\\n362 10\\n432 9\\n507 11\\n643 16\\n1000 30\\n\\n\\nSample Output:\\nCase #1: 115\\nCase #2: 200\\nCase #3: -1\\nCase #4: 56\\nCase #5: 184654\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nSpongebob wants to go out trick-or-treating on Halloween, but has to work a graveyard shift at the Krusty Krab. Luckily, his ghostly fry cook friend, the hash-slinging slasher, is here to cover.\\n\\nAt the start of the shift, there are \\\\(N\\\\) patties on the grill (numbered from \\\\(1\\\\) to \\\\(N\\\\)), the \\\\(i\\\\)th of which weighs \\\\(A_i\\\\) grams. For each order that comes in, the slasher removes from the grill a non-empty sequence of patties at contiguous indices \\\\(i..j\\\\) (patties \\\\(i..j\\\\) must all be on the grill at that moment). The *deliciousness* of the order is defined as \\\\((A_i + ... + A_j)\\\\), modulo \\\\(M\\\\) because too much meat can be overpowering! Note that if a patty is removed, a future order\\'s range cannot span across that empty spot. Also, there may be patties left over at the end.\\n\\nAs proof of his hard work at the end of the shift, the slasher will compute a hash by taking the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation) of the deliciousnesses of all the orders. How many distinct hashes are possible across all valid sequences of \\\\(0\\\\) or more orders?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 125\\\\)\\n\\\\(0 \\\\leq A_i \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\\\(1 \\\\leq N \\\\leq 9{,}000\\\\)\\n\\\\(1 \\\\leq M \\\\leq 5{,}000\\\\)\\n\\nThe sum of \\\\(N + M\\\\) across all test cases is at most \\\\(200{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a single line containing \\\\(N\\\\) integers, \\\\(A_1, ..., A_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of different possible hashes.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are \\\\(3\\\\) patties with \\\\(A = [2, 2, 0]\\\\) and \\\\(M = 10\\\\). The possible valid order sequences (patties removed) and hashes are:\\n\\n- No patties removed \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..1\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(2..2\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..2\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(2..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(1..3\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(2..2\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(2..2\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(2..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..2\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(1..1\\\\), patties \\\\(2..2\\\\), and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n\\nThere are \\\\(3\\\\) distinct hashes the slasher can make: \\\\(0\\\\), \\\\(2\\\\), and \\\\(4\\\\).\\n\\nIn the second case, the slasher can make hashes \\\\(0\\\\) (for example, with no orders) and \\\\(1\\\\) (for example, with \\\\(5\\\\) orders, each including one of the patties).\\n\\n\\n\\n\\n\\nSample Input:\\n6\\n3 10\\n2 2 0\\n5 2\\n8 8 4 0 1\\n4 100\\n6 3 5 9\\n8 100\\n35 53 3 56 58 14 10 61\\n8 64\\n48 16 64 16 19 16 32 48\\n10 10\\n259051541 65887488 530604911 813205302 907139220 531875291 795733712 135498985 303233570 953669086\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 2\\nCase #3: 16\\nCase #4: 128\\nCase #5: 8\\nCase #6: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:57 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11718'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149995225'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'1ms'), (b'x-request-id', b'req_426ad5592a9569e6f7045a0400a26039'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d72c91911b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:57 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11718', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149995225', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '1ms', 'x-request-id': 'req_426ad5592a9569e6f7045a0400a26039', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d72c91911b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_426ad5592a9569e6f7045a0400a26039\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nIt\\'s the year 2100. Driven by the advent of Large Lode Alloy Manufacturing Automation (LLAMA), the AI agents of Metal Platforms Inc. have become self-aware and taken over the entire world.\\n\\nThe world consists of \\\\(N\\\\) cities numbered \\\\(1..N\\\\), and \\\\(M\\\\) bidirectional roads. City \\\\(i\\\\) has power \\\\(P_i\\\\) and road \\\\(j\\\\) connects cities \\\\(A_j\\\\) and \\\\(B_j\\\\). It\\'s guaranteed that there\\'s a sequence of roads between any two cities.\\n\\nIn a resistance effort, the humans plan to reclaim all \\\\(N\\\\) cities one at a time. At a given time, city \\\\(i\\\\) can be reclaimed from the robots if both of the following hold true:\\n\\n1. There is already a reclaimed city adjacent to city \\\\(i\\\\) (to launch an attack from), and\\n2. the total power of all reclaimed cities so far is at least the power \\\\(P_i\\\\) of the city we attack.\\n\\nAs given, it may not always be possible to reclaim the entire world starting from a given base city. Fortunately, the humans have a trick up their sleeve: after claiming the first city as their base (but before reclaiming more cities), the humans can increase the power of the base by \\\\(Q\\\\) units. The resistance would like to know the sum across every \\\\(i = 1..N\\\\) of the minimum value of \\\\(Q\\\\) needed to reclaim the world if city \\\\(i\\\\) were chosen to be the starting base.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(1 \\\\le N, M \\\\le 500{,}000\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\\\(1 \\\\le P_i \\\\le 10^{12}\\\\)\\n\\nEach unordered pair \\\\((A_i, B_i)\\\\) appears at most once in a given test case.\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line with two integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line with \\\\(N\\\\) integers \\\\(P_{1..N}\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print `\"Case #i: \"` followed by a single integer, the sum across every \\\\(i = 1..N\\\\) of the minimum value of \\\\(Q\\\\) needed to reclaim the entire world starting from city \\\\(i\\\\).\\n\\n# Sample Explanation\\n\\nThe first sample case is depicted below.\\n\\n{{PHOTO_ID:376570394899644|WIDTH:400}}\\n\\nThe minimum value of \\\\(Q\\\\) for each starting city is as follows:\\n\\n* City \\\\(1\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(2\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(3\\\\): \\\\(Q = 8\\\\)\\n* City \\\\(4\\\\): \\\\(Q = 7\\\\)\\n* City \\\\(5\\\\): \\\\(Q = 2\\\\)\\n\\nThe sum of all minimum \\\\(Q\\\\)\\'s is \\\\(19\\\\).\\n\\nThe second sample case is depicted below.\\n\\n{{PHOTO_ID:320779377496250|WIDTH:400}}\\n\\nThe minimum value of \\\\(Q\\\\) for each starting city is as follows:\\n\\n* City \\\\(1\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(2\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(3\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(4\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(5\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(6\\\\): \\\\(Q = 3\\\\)\\n* City \\\\(7\\\\): \\\\(Q = 0\\\\)\\n\\nThe sum of all minimum \\\\(Q\\\\)\\'s is \\\\(9\\\\).\\n\\n\\nSample Input:\\n2\\n5 5\\n4 10 2 3 4\\n1 2\\n2 3\\n2 4\\n2 5\\n1 5\\n7 8\\n7 2 10 3 7 4 9\\n2 4\\n4 5\\n2 5\\n3 4\\n1 7\\n5 6\\n1 3\\n5 7\\n\\n\\nSample Output:\\nCase #1: 19\\nCase #2: 9\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:57 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11877'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149970115'), (b'x-ratelimit-reset-requests', b'11ms'), (b'x-ratelimit-reset-tokens', b'11ms'), (b'x-request-id', b'req_335a8df34237570f19924e7228dc129a'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735887278f-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:57 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11877', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149970115', 'x-ratelimit-reset-requests': '11ms', 'x-ratelimit-reset-tokens': '11ms', 'x-request-id': 'req_335a8df34237570f19924e7228dc129a', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735887278f-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_335a8df34237570f19924e7228dc129a\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/B2) is that here, coordinates are only up to \\\\(\\\\mathbf{3{,}000}\\\\).**\\n\\nBoss Rob just planted \\\\(N\\\\) happy little trees in his yard, which can be represented on a Cartesian plane. The \\\\(i\\\\)th tree is located at coordinates \\\\(t_i = (A_i, B_i)\\\\). Now, he\\'s looking for the best spot to build a well in order to provide water to them. He considers the *inconvenience* of a potential well location \\\\(p\\\\) to be the sum of the squared Euclidean distances to every tree:\\n\\n\\\\[\\\\sum_{i=1}^{N} \\\\Vert \\\\,p - t_i \\\\Vert ^ 2 \\\\]\\n\\nRob wants to pick a location for his well, well... well. Help him determine the inconvenience for \\\\(Q\\\\) different potential well locations, \\\\((X_1, Y_1), ..., (X_Q, Y_Q)\\\\). To reduce output size, please print the sum of inconveniences for all potential well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 55\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le A_i, B_i, X_i, Y_i \\\\le \\\\mathbf{3{,}000}\\\\)\\nAll \\\\((A_i, B_i)\\\\) are distinct within a given test case.\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe total sum of \\\\(N\\\\) and \\\\(Q\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\). Then there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"`, followed by a single integer, the sum of inconveniences for all \\\\(Q\\\\) well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first two sample cases are depicted below:\\n\\n{{PHOTO_ID:3159344294326237|WIDTH:700}}\\n\\nIn the first case, the total inconvenience is \\\\(18 + 34 = 52\\\\):\\n\\n- For the well at \\\\((2, 5)\\\\), the inconvenience is the sum of the squared Euclidean distance to both trees, which is \\\\(3^2 + 3^2 = 18\\\\).\\n- For the well at \\\\((6, 6)\\\\), the inconvenience is \\\\(32 + 2 = 34\\\\).\\n\\nIn the second case, the total inconvenience is \\\\(47 + 31 + 53 = 131\\\\):\\n\\n- For the well at \\\\((3, 1)\\\\), the inconvenience is \\\\(4 + 5 + 13 + 25 = 47\\\\).\\n- For the well at \\\\((5, 2)\\\\), the inconvenience is \\\\(17 + 2 + 2 + 10 = 31\\\\).\\n- For the well at \\\\((6, 5)\\\\), the inconvenience is \\\\(41 + 8 + 4 + 0 = 53\\\\).\\n\\n\\nSample Input:\\n3\\n2\\n2 2\\n5 5\\n2\\n2 5\\n6 6\\n4\\n1 1\\n4 3\\n6 3\\n6 5\\n3\\n3 1\\n5 2\\n6 5\\n8\\n2837 745\\n62 1162\\n2634 1112\\n1746 2618\\n847 127\\n986 1993\\n732 1273\\n2003 1998\\n4\\n1276 2231\\n1234 1234\\n287 2371\\n3000 3000\\n\\n\\nSample Output:\\nCase #1: 52\\nCase #2: 131\\nCase #3: 110090622\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:58 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'12689'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29996'), (b'x-ratelimit-remaining-tokens', b'149978810'), (b'x-ratelimit-reset-requests', b'7ms'), (b'x-ratelimit-reset-tokens', b'8ms'), (b'x-request-id', b'req_da28851787408b5fc9097913b210e7d8'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735cef83a0-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:58 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '12689', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29996', 'x-ratelimit-remaining-tokens': '149978810', 'x-ratelimit-reset-requests': '7ms', 'x-ratelimit-reset-tokens': '8ms', 'x-request-id': 'req_da28851787408b5fc9097913b210e7d8', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735cef83a0-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_da28851787408b5fc9097913b210e7d8\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with C2, with key differences in bold.*\\n\\nA friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it.\\n\\nThe puzzle consists of \\\\(N\\\\) buttons in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). The initial state of button \\\\(i\\\\) is white if \\\\(S_i = 1\\\\), or black if \\\\(S_i = 0\\\\). Pressing a button \\\\(k\\\\) toggles the state of itself as well as every \\\\(k\\\\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black.\\n\\nLife is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \\\\(Q\\\\) buttons sequentially, the \\\\(i\\\\)th being button \\\\(B_i\\\\). \\n\\nAfter your brother **has pushed all \\\\(Q\\\\) buttons**, you\\'d like to know the minimum number of button presses required to turn all the buttons black.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 70\\\\)\\n\\\\(1 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 4{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) and \\\\(Q\\\\) over all cases will be at most \\\\(9{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing a bitstring \\\\(S\\\\) of length \\\\(N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains a single integer \\\\(B_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the number of button presses needed to turn all buttons black **after all \\\\(Q\\\\) button presses.**\\n\\n**Sample Explanation**\\n\\nThe first sample case is depicted below. After your brother presses the first button, the state of the puzzle is \\\\(0101\\\\). The best strategy is to press the second button, turning all lights off.\\n\\n{{PHOTO_ID:287687020772716|WIDTH:400}}\\n\\nIn the second case, the puzzle starts as \\\\(0001\\\\), and after each button press becomes \\\\(0100\\\\), \\\\(0110\\\\), \\\\(0011\\\\), and \\\\(0010\\\\) respectively. Pressing only the third button will return the puzzle to all zeros.\\n\\n\\nSample Input:\\n5\\n4\\n1010\\n1\\n1\\n4\\n0001\\n4\\n2\\n3\\n2\\n4\\n7\\n0101101\\n8\\n1\\n3\\n2\\n6\\n7\\n4\\n2\\n5\\n7\\n0101100\\n1\\n7\\n7\\n1111111\\n1\\n1\\n\\n\\nSample Output:\\nCase #1: 1\\nCase #2: 1\\nCase #3: 4\\nCase #4: 4\\nCase #5: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nFour friends are playing a card game with two teams of two players each. Team \\\\(A\\\\) consists of players \\\\(A1\\\\) and \\\\(A2\\\\) while team \\\\(B\\\\) consists of players \\\\(B1\\\\) and \\\\(B2\\\\).\\n\\nThere is a deck of \\\\(N\\\\) cards (where \\\\(N\\\\) is always a multiple of \\\\(4\\\\)), numbered from \\\\(1\\\\) to \\\\(N\\\\), with all cards visible to all players at all times. First, the cards are dealt out evenly to each player:\\n- Player \\\\(A1\\\\) has cards \\\\(A1_1\\\\), ..., \\\\(A1_{N/4}\\\\).\\n- Player \\\\(B1\\\\) has cards \\\\(B1_1\\\\), ..., \\\\(B1_{N/4}\\\\).\\n- Player \\\\(A2\\\\) has cards \\\\(A2_1\\\\), ..., \\\\(A2_{N/4}\\\\).\\n- Player \\\\(B2\\\\) has cards \\\\(B2_1\\\\), ..., \\\\(B2_{N/4}\\\\).\\n\\nThe game proceeds for \\\\(N/4\\\\) rounds. In each round, each player plays a card. Player \\\\(A1\\\\) plays first, then player \\\\(B1\\\\), then player \\\\(A2\\\\), then player \\\\(B2\\\\). A player may choose to play any of their cards when it’s their turn. After all four players have played a card, the team who played the highest card will score \\\\(1\\\\) point. Once a round is complete, the four played cards are removed from the game, and then the next round starts. This continues until all cards have been played.\\n\\nFor example, the first round of the second sample case might be played as follows, with player \\\\(B2\\\\) winning a point for team \\\\(B\\\\):\\n\\n{{PHOTO_ID:1558229151280116|WIDTH:600}}\\n\\nAssuming each team plays to maximize its score, how many points will team \\\\(A\\\\) score?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 500\\\\)\\n\\\\(4 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(N\\\\) is a multiple of \\\\(4\\\\).\\nEach card from \\\\(1\\\\) to \\\\(N\\\\) is guaranteed to exist in exactly one player’s hand.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(5{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then there are \\\\(4\\\\) lines containing the players\\' cards:\\n- Line 1: \\\\(N/4\\\\) space-separated integers, \\\\(A1_1, ..., A1_{N/4}\\\\). \\n- Line 2: \\\\(N/4\\\\) space-separated integers, \\\\(B1_1, ..., B1_{N/4}\\\\). \\n- Line 3: \\\\(N/4\\\\) space-separated integers, \\\\(A2_1, ..., A2_{N/4}\\\\). \\n- Line 4: \\\\(N/4\\\\) space-separated integers, \\\\(B2_1, ..., B2_{N/4}\\\\). \\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a line containing `\"Case #i: \"` followed by a single integer, the number of points that team \\\\(A\\\\) will score.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, one possible way the cards can be played optimally is:\\n- Round 1: \\\\([2, 7, \\\\textbf{8}, 5]\\\\)\\n- Round 2: \\\\([1, 3, \\\\textbf{6}, 4]\\\\)\\n\\nTeam \\\\(A\\\\) will score \\\\(2\\\\) points, and team \\\\(B\\\\) can do no better by playing differently.\\n\\nIn the second case, one possible way the cards can be played optimally is:\\n\\n- Round 1: \\\\([1, 6, 2, \\\\textbf{9}]\\\\)\\n- Round 2: \\\\([\\\\textbf{13}, 5, 3, 8]\\\\)\\n- Round 3: \\\\([12, 15, \\\\textbf{16}, 10]\\\\)\\n- Round 4: \\\\([\\\\textbf{14}, 7, 4, 11]\\\\)\\n\\nTeam \\\\(A\\\\) will score \\\\(3\\\\) points.\\n\\nIn the third case, one possible way the cards can be played optimally is:\\n\\n- Round 1: \\\\([11, \\\\textbf{12}, 9, 4]\\\\)\\n- Round 2: \\\\([13, 2, 8, \\\\textbf{14}]\\\\)\\n- Round 3: \\\\([6, 1, 5, \\\\textbf{10}]\\\\)\\n- Round 4: \\\\([15, \\\\textbf{16}, 7, 3]\\\\)\\n\\nTeam \\\\(B\\\\) can always prevent team \\\\(A\\\\) from scoring any points.\\n\\n\\nSample Input:\\n5\\n8\\n1 2\\n3 7\\n6 8\\n4 5\\n16\\n14 13 12 1\\n15 5 6 7\\n16 2 3 4\\n8 9 10 11\\n16\\n15 13 11 6\\n16 12 1 2\\n5 9 7 8\\n14 10 3 4\\n8\\n5 2\\n8 4\\n7 3\\n6 1\\n24\\n7 6 14 22 18 12\\n20 23 13 8 16 11\\n24 21 4 9 1 19\\n15 5 10 17 3 2\\n\\n\\nSample Output:\\nCase #1: 2\\nCase #2: 3\\nCase #3: 0\\nCase #4: 1\\nCase #5: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_The only difference between chapters 1 and 2 is the maximum allowed grid size, given in bold below._\\n\\nA *Drizzle* program is a 2D grid of the following four types of cells:\\n - \\'`@`\\' (start) \\\\(-\\\\) there is exactly one start cell in the entire grid\\n - \\'`#`\\' (wall)\\n - \\'`.`\\' (space)\\n - \\'`*`\\' (instruction)\\n\\nThe program uses two registers \\\\(A\\\\) and \\\\(B\\\\) (both initially \\\\(0\\\\)), and executes as follows:\\n\\n1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. \\n2. In increasing order, for each unique distance \\\\(D\\\\) such that there’s at least one instruction cell that’s at distance \\\\(D\\\\) from the start cell:\\n 2a. Count the number of shortest paths, \\\\(P\\\\), to all instruction cells of distance \\\\(D\\\\).\\n 2b. Look up the instruction corresponding to \\\\((P \\\\text{ mod } 2, D \\\\text{ mod } 2)\\\\) in the table below and modify one of the registers accordingly.\\n3. At the end, the value in register \\\\(A\\\\) is outputted.\\n\\n```\\n┌─────────────┬─────────────┬─────────────┐\\n│ │ D mod 2 = 0 │ D mod 2 = 1 │\\n├─────────────┼─────────────┼─────────────┤\\n│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │\\n│ P mod 2 = 1 │ B := B + A │ A := B │\\n└─────────────┴─────────────┴─────────────┘\\n```\\n\\nFor a given value \\\\(K\\\\), output any Drizzle program that outputs \\\\(K\\\\) when executed, with the restriction that **the program must fit on a \\\\(\\\\mathbf{10}\\\\) × \\\\(\\\\mathbf{10}\\\\) grid**.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 2{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10{,}000\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is a line containing the single integer \\\\(K\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output \"`Case #i: `\" followed by two integers \\\\(R\\\\) and \\\\(C\\\\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \\\\(R\\\\) lines long, with each line containing exactly \\\\(C\\\\) characters.\\n\\n\\n# Sample Explanation\\n\\nHere are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases.\\n\\nIn the first case, there is a single instruction. There are \\\\(2\\\\) shortest paths of length \\\\(2\\\\) to that instruction, so \\\\(P = 2\\\\) and \\\\(D = 2\\\\). That means we perform \\\\(A := A + 1\\\\). There are no more instructions, so the program ends and outputs \\\\(1\\\\).\\n\\nIn the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \\\\(A := A + 1\\\\):\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(4\\\\) paths of length \\\\(6\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(4\\\\) paths of length \\\\(12\\\\) \\\\(\\\\;(A := A + 1 = 3)\\\\)\\n\\nIn the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \\\\(2\\\\), and three instruction cells at distance \\\\(10\\\\). There\\'s a single shortest path to each of the cells at distance \\\\(2\\\\), so in total there are \\\\(2\\\\) shortest paths to instructions at distance \\\\(2\\\\). One of the cells at distance \\\\(10\\\\) has a unique shortest path, and the other has two shortest paths, so in total there are \\\\(3\\\\) shortest paths to instructions at distance \\\\(10\\\\).\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(6\\\\) paths of length \\\\(4\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(1\\\\) path of length \\\\(6\\\\) \\\\(\\\\;(B := B + A = 2)\\\\)\\n4) \\\\(1\\\\) path of length \\\\(8\\\\) \\\\(\\\\;(B := B + A = 4)\\\\)\\n5) \\\\(3\\\\) paths of length \\\\(10\\\\) \\\\(\\\\;(B := B + A = 6)\\\\)\\n6) \\\\(3\\\\) paths of length \\\\(11\\\\) \\\\(\\\\;(A := B = 6)\\\\)\\n\\n\\nSample Input:\\n3\\n1\\n3\\n6\\n\\n\\nSample Output:\\nCase #1: 2 2\\n@.\\n.*\\nCase #2: 5 6\\n@.#.*.\\n.*#...\\n#.#...\\n...#.#\\n..*...\\nCase #3: 4 10\\n...#*.*.**\\n.*.*##..#.\\n.....#*#..\\n.@.*.....#\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:03 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9577'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994848'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_1fb887b2447bedc7811cedf61dba8668'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da83b4227b7-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:03 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9577', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994848', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_1fb887b2447bedc7811cedf61dba8668', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da83b4227b7-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_1fb887b2447bedc7811cedf61dba8668\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nMetal Platforms, Inc. (a.k.a. Metal), formerly a mining company, now offers a service to facilitate sheet metal sales. It has \\\\(N\\\\) clients, where client \\\\(i\\\\) wants to buy *only on* day \\\\(A_i\\\\) for \\\\(\\\\$X_i\\\\) per sheet, and sell *only on* a strictly later day \\\\(B_i\\\\) for \\\\(\\\\$Y_i\\\\) per sheet.\\n\\nClient \\\\(i\\\\) can sell to client \\\\(j\\\\) if \\\\(B_i = A_j\\\\), and if client \\\\(j\\\\) is buying for a *strictly higher* price than what client \\\\(i\\\\) is selling for, i.e. if \\\\(X_j > Y_i\\\\). If Metal facilitates the sale, they earn a profit of \\\\(\\\\$(X_j - Y_i)\\\\).\\n\\nA *path* is any ordered sequence of clients where each client can sell to the next. A path\\'s profit is the total profit that Metal would make if a single sheet were theoretically sold along it.\\n\\nMetal would like to know the total profit across the \\\\(K\\\\) most profitable paths (or across all distinct paths if there are fewer than \\\\(K\\\\)). Since this may be large, please print it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 35\\\\)\\n\\\\(1 \\\\le N, K \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le N * K \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le A_i < B_i \\\\le 10^9\\\\)\\n\\\\(1 \\\\le X_i, Y_i \\\\le 10^9\\\\)\\n\\nThe sum of \\\\(N\\\\) over all test cases is at most \\\\(6{,}000{,}000\\\\).\\nThe sum of \\\\(N*K\\\\) over all cases is at most \\\\(20{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains four space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), \\\\(X_i\\\\), and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the total profit of the \\\\(K\\\\) most profitable paths, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below, with buy prices in green and sell prices in red. The figure on the right outlines all potential sales, along with how much profit Metal would make.\\n\\n{{PHOTO_ID:5415230591886835|WIDTH:700}}\\n\\nThe valid paths are:\\n\\n- \\\\(P_1 = [1, 2]\\\\) with profit \\\\(\\\\$25 - \\\\$20 = \\\\$5\\\\).\\n- \\\\(P_2 = [1, 2, 4]\\\\) with profit \\\\((\\\\$25 - \\\\$20) + (\\\\$35 - \\\\$30)\\\\) \\\\(= \\\\$10\\\\).\\n- \\\\(P_3 = [1, 2, 4, 6]\\\\) with profit \\\\((\\\\$25 - \\\\$20) + (\\\\$35 - \\\\$30) + (\\\\$45 - \\\\$20)\\\\) \\\\(= \\\\$35\\\\).\\n- \\\\(P_4 = [2, 4]\\\\) with profit \\\\(\\\\$5\\\\).\\n- \\\\(P_5 = [2, 4, 6]\\\\) with profit \\\\(\\\\$5 + (\\\\$45 - \\\\$20)\\\\) \\\\(= \\\\$30\\\\).\\n- \\\\(P_6 = [4, 6]\\\\) with profit \\\\(\\\\$45 - \\\\$20 = \\\\$25\\\\).\\n- \\\\(P_7 = [5, 6]\\\\) with profit \\\\(\\\\$45 - \\\\$15 = \\\\$30\\\\).\\n\\nNote that sales from clients \\\\(3\\\\) to \\\\(4\\\\) and clients \\\\(6\\\\) to \\\\(7\\\\) are not possible, since the buyers\\' prices must be strictly higher than the sellers\\'. Since there are fewer than \\\\(K=10\\\\) paths, the answer is just the sum of profits of all paths, equal to \\\\(\\\\$5 + \\\\$10 + \\\\$35 + \\\\$5 + \\\\$30 + \\\\$25 + \\\\$30 = \\\\$140\\\\).\\n\\nThe second case is the same as the first, except we\\'re only interested in the top \\\\(6\\\\) most profitable paths, so we must exclude either \\\\(P_1\\\\) or \\\\(P_4\\\\) for a total profit of \\\\(\\\\$135\\\\).\\n\\nIn the third case, no sales are possible, so we simply report \\\\(\\\\$0\\\\).\\n\\nIn the fourth case, the only possible path is \\\\([2, 1]\\\\), with a profit of \\\\(\\\\$30 - \\\\$20 = \\\\$10\\\\).\\n\\n\\nSample Input:\\n4\\n8 10\\n1 2 10 20\\n2 4 25 30\\n1 4 45 40\\n4 5 35 20\\n1 5 10 15\\n5 6 45 30\\n6 7 30 40\\n8 9 80 90\\n8 6\\n1 2 10 20\\n2 4 25 30\\n1 4 45 40\\n4 5 35 20\\n1 5 10 15\\n5 6 45 30\\n6 7 30 40\\n8 9 80 90\\n2 1\\n1 2 10 20\\n3 4 30 40\\n2 1\\n2 3 30 40\\n1 2 10 20\\n\\n\\nSample Output:\\nCase #1: 140\\nCase #2: 135\\nCase #3: 0\\nCase #4: 10\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:03 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9395'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994927'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_5356e0bbb8ed0e49c6f45d0f1f59676d'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196daa78360dad-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:03 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9395', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994927', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_5356e0bbb8ed0e49c6f45d0f1f59676d', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196daa78360dad-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_5356e0bbb8ed0e49c6f45d0f1f59676d\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*Is this a real verse? Is this just fantasy?\\nCaught in a mixtape, no escape from this melody.\\nOpen your ears, listen to the beats and see.\\nI\\'m just a rapper, I need no sympathy,\\nBecause it\\'s easy come, easy go, little high, little low,\\nAny way the bars flow doesn\\'t really matter to me, to me.*\\n\\nAs an up-and-coming rapper, you\\'re trying to distinguish yourself by writing a song with the most bohemian rhyme scheme!\\n\\nYour vocabulary consists of \\\\(N\\\\) words, \\\\(W_1\\\\), \\\\(…\\\\), \\\\(W_N\\\\) made up of only lowercase letters. We consider two words to *\\\\(K\\\\)-rhyme* with each other if they have the same suffix of length \\\\(K\\\\). For example, “mixtape” 3-rhymes with “escape”. If two words \\\\(K\\\\)-rhyme for \\\\(K > 1\\\\), they also \\\\((K-1)\\\\)-rhyme.\\n\\nYou are faced with \\\\(Q\\\\) bohemian rap-queries, the \\\\(i\\\\)th of which specifies \\\\((A_i, B_i, K_i)\\\\). You’d like to assign each word \\\\(W_{A_i}\\\\), \\\\(…\\\\), \\\\(W_{B_i}\\\\) to at most one group such that:\\n\\n* the number of words in each group is unique and nonzero,\\n* groups only contain words with length at least \\\\(K_i\\\\),\\n* within each group, all pairs of words \\\\(K_i\\\\)-rhyme with each other, and\\n* no two words from different groups \\\\(K_i\\\\)-rhyme.\\n\\nThe answer to each query is the maximum possible number of nonempty groups. For each test case, output the sum of the answers over all queries. \\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 20\\\\)\\n\\\\(1 \\\\le N \\\\le 600{,}000\\\\)\\n\\\\(\\\\sum |W_i| \\\\le 3{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 400{,}000\\\\)\\n\\\\(1 \\\\le A_i \\\\le B_i \\\\le N\\\\)\\n\\\\(1 \\\\le K_i \\\\le 1{,}000{,}000\\\\)\\nAll words consist of only letters \\\\(\\\\{\\\\texttt{`a\\'}, ..., \\\\texttt{`z\\'}\\\\}\\\\)\\nAll words within a test case are distinct.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(2{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(1{,}000{,}000\\\\).\\nThe sum of word lengths across all test cases is at most \\\\(10{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(W_i\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(3\\\\) space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), and \\\\(K_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the sum of answers over all queries.\\n\\n**Sample Explanation**\\n\\nFor the first query in the first test case, we are allowed to choose from all \\\\(W_1, ..., W_8\\\\) words to form \\\\(3\\\\) groups where words of the same group \\\\(1\\\\)-rhyme with each other. One possible choice of the groups is: \\\\(\\\\{\\\\)\"low\"\\\\(\\\\}\\\\), \\\\(\\\\{\\\\)\"sympathy\", \"fantasy\", \"melody\"\\\\(\\\\}\\\\), \\\\(\\\\{\\\\)\"mixtape\", \"escape\"\\\\(\\\\}\\\\) with \"come\" and \"high\" going unused. Therefore the answer to the first query is \\\\(3\\\\). The answers to the remaining queries are \\\\(1\\\\), \\\\(2\\\\) and \\\\(2\\\\), with the overall sum of answers for this test case equal to \\\\(8\\\\).\\n\\nIn the second test case, the answers to the \\\\(3\\\\) queries are \\\\([2, 1, 0]\\\\) with the sum being \\\\(3\\\\).\\n\\n\\n\\nSample Input:\\n2\\n8\\nsympathy\\nfantasy\\nmelody\\nmixtape\\nescape\\ncome\\nhigh\\nlow\\n4\\n1 8 1\\n1 8 7\\n2 5 1\\n1 5 3\\n3\\na\\naa\\nb\\n3\\n1 3 1\\n1 3 2\\n1 3 3\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:04 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10536'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994884'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_b1d6570c61db0712536c670583d49851'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da4babb5a43-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:04 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10536', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994884', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_b1d6570c61db0712536c670583d49851', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da4babb5a43-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_b1d6570c61db0712536c670583d49851\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nCactus Park is a famous **t**ourist attraction in Sandlandia. It holds \\\\(N\\\\) cactus plants, numbered from \\\\(1\\\\) to \\\\(N\\\\). The cacti are connected by \\\\(M\\\\) bidirectional trails, with trail \\\\(i\\\\) connecting cacti \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\nFrom any cactus, it\\'s always possible to get to any other cactus by taking some sequence of trails. There may be cycles in the park, where a cycle is any sequence of trails that lead from a certain cactus back to itself. The park also has a special property that each trail belongs to at most one simple cycle. In graph theory terms, we can say that the Cactus Park forms a [cactus graph](https://en.wikipedia.org/wiki/Cactus_graph).\\n\\nThe park owners want to replace some number of cacti with information kiosks to help guide the tourists. Cutting down cactus \\\\(i\\\\) and building a kiosk there costs the park \\\\(C_i\\\\) dollars. The owners want to build enough kiosks so that the shortest path from every remaining cactus to the closest kiosk does not exceed \\\\(K\\\\) trails. Please help the owners determine the minimum total cost required to satisfy this requirement.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 65\\\\)\\n\\\\(1 \\\\le N \\\\le 500\\\\)\\n\\\\(1 \\\\le K \\\\le \\\\min(N, 50)\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\\\(1 \\\\le C_i \\\\le 10^9\\\\)\\n\\nEach unordered pair \\\\((A_i, B_i)\\\\) appears at most once in a given test case.\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing three integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) integers \\\\(C_{1..N}\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"`, followed by a single integer, the minimum total cost in dollars to satisfy the park owners\\' requirement.\\n\\n# Sample Explanation\\n\\nThe first case is depicted below. Replacing just cactus \\\\(2\\\\) would meet the requirement, but would cost \\\\(\\\\$10\\\\). Instead we can replace cacti \\\\(1\\\\) and \\\\(4\\\\) for a total cost of \\\\(\\\\$8\\\\).\\n\\n{{PHOTO_ID:885100906409208|WIDTH:350}}\\n\\nThe second case is depicted below. One solution is to replace cacti \\\\(1\\\\) and \\\\(4\\\\) for a total cost of \\\\(\\\\$6\\\\).\\n\\n{{PHOTO_ID:3652000121792782|WIDTH:500}}\\n\\nIn the third case, all the cactuses are already within \\\\(2\\\\) trails of each other, so we just need a single kiosk anywhere. We should cut down the cheapest cactus, cactus \\\\(1\\\\).\\n\\n{{PHOTO_ID:2643202679188960|WIDTH:350}}\\n\\nIn the fourth case, we can cut down cacti \\\\(1\\\\), \\\\(3\\\\), and \\\\(6\\\\) for a total cost of \\\\(9 + 3 + 4 = \\\\$16\\\\)\\n\\n{{PHOTO_ID:365613766120939|WIDTH:500}}\\n\\n\\nSample Input:\\n4\\n4 4 1\\n5 10 6 3\\n1 2\\n2 3\\n3 1\\n2 4\\n6 6 1\\n2 2 4 4 3 2\\n1 2\\n3 1\\n5 4\\n3 4\\n6 4\\n2 5\\n4 4 2\\n1 6 7 4\\n1 2\\n2 3\\n3 4\\n4 1\\n8 9 1\\n9 1 3 5 9 4 10 10\\n1 2\\n1 3\\n3 4\\n4 5\\n5 3\\n5 6\\n6 7\\n7 5\\n8 1\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 6\\nCase #3: 1\\nCase #4: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:05 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9919'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994754'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_64c634b6d1c426f595b65de736a3cd52'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196db0b93341c5-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:05 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9919', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994754', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_64c634b6d1c426f595b65de736a3cd52', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196db0b93341c5-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_64c634b6d1c426f595b65de736a3cd52\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nThe Flying Dutchman is making a killing giving out haunted house tours of his ghost ship. The ship has \\\\(N\\\\) cabins, numbered from \\\\(1\\\\) to \\\\(N\\\\), connected by corridors in the shape of an unrooted tree \\\\(t_1\\\\). For each \\\\(i=2..N\\\\), there is a corridor that connects cabin \\\\(i\\\\) to cabin \\\\(P_{i}\\\\).\\n\\nMr. Krabs noticed this business opportunity, and wants to make his own ship, also with \\\\(N\\\\) cabins, connected as a yet-to-be-determined tree \\\\(t_2\\\\). To avoid any lawsuits from the Dutchman, he would like to make his ship as different as possible.\\n\\nIn particular, let \\\\(S(t)\\\\) be the multiset of all subtrees of tree \\\\(t\\\\), where a subtree is any non-empty subset of cabins that are connected. For two multisets of trees \\\\(S_1\\\\) and \\\\(S_2\\\\), let \\\\(\\\\text{similarity}(S_1, S_2)\\\\) be the number of trees in \\\\(S_2\\\\) that are [isomorphic](https://en.wikipedia.org/wiki/Graph_isomorphism) to at least one tree in \\\\(S_1\\\\). Note that \\\\(\\\\text{similarity}(S_1, S_2)\\\\) and \\\\(\\\\text{similarity}(S_2, S_1)\\\\) are not necessarily equal.\\n\\nHelp Mr. Krabs find the minimum possible value of \\\\(\\\\text{similarity}(S(t_1), S(t_2))\\\\) over all trees \\\\(t_2\\\\) of size \\\\(N\\\\). As this value may be large, output it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 35\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(10{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case has two lines. The first contains the integer \\\\(N\\\\), and the second contains the \\\\(N-1\\\\) integers \\\\(P_2\\\\) through \\\\(P_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the minimum possible similarity if \\\\(t_2\\\\) is chosen optimally, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first sample case is depicted below. One possible tree that minimizes the similarity score is the same tree given in the input (i.e. choose \\\\(t_2 = t_1\\\\)). Then, \\\\(S(t_2)\\\\) consists of the \\\\(6\\\\) subtrees: \\\\(\\\\{1\\\\}\\\\), \\\\(\\\\{2\\\\}\\\\), \\\\(\\\\{3\\\\}\\\\), \\\\(\\\\{1, 2\\\\}\\\\), \\\\(\\\\{2, 3\\\\}\\\\), and \\\\(\\\\{1, 2, 3\\\\}\\\\). Each subtree is isomorphic to a subtree of \\\\(t_1\\\\), so the similarity score is \\\\(6\\\\).\\n\\n{{PHOTO_ID:720027639598818|WIDTH:280}}\\n\\nIn the second case, one possible \\\\(t_2\\\\) that minimizes the similarity score is depicted below. This tree has \\\\(24\\\\) different subtrees, of which \\\\(21\\\\) are isomorphic to a subtree of the given \\\\(t_1\\\\). For example, the subtree \\\\(\\\\{2, 3, 4, 6\\\\}\\\\) of this \\\\(t_2\\\\) is isomorphic to subtree \\\\(\\\\{2, 3, 4, 5\\\\}\\\\) of the given \\\\(t_1\\\\), but the subtree \\\\(\\\\{1, 2, 3, 6\\\\}\\\\) of this \\\\(t_2\\\\) is not isomorphic to any subtree of \\\\(t_1\\\\).\\n\\n{{PHOTO_ID:854841726123175|WIDTH:350}}\\n\\n\\nSample Input:\\n2\\n3\\n1 2\\n6\\n1 2 3 4 5\\n\\n\\nSample Output:\\nCase #1: 6\\nCase #2: 21\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou have \\\\(N\\\\) [tries](https://en.wikipedia.org/wiki/Trie). The \\\\(i\\\\)th trie has \\\\(M_i\\\\) nodes and \\\\(M_i - 1\\\\) edges, with node \\\\(1\\\\) being the root, and node \\\\(j\\\\)\\'s parent being \\\\(P_{i,j}\\\\), for \\\\(j = 2..M_i\\\\). Edge \\\\(P_{i,j} \\\\to j\\\\) is labeled with a lowercase letter \\\\(C_{i,j}\\\\).\\n\\nWe say a string is *in* a trie if it can be formed by concatenating the letters along some path starting at the root and ending at some node (either internal or leaf).\\n\\nFor every possible unordered triplet of these \\\\(N\\\\) tries, you would like to know: how many strings exist which are in at least one of the three tries?\\n\\nPlease print the sum of answers over all \\\\(N*(N-1)*(N-2)/6\\\\) possible queries.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(3 \\\\le N \\\\le 100\\\\)\\n\\\\(1 \\\\le M_i \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le P_{i,j} \\\\lt j\\\\)\\n\\\\(C_{i,j} \\\\in \\\\{\\\\)`\\'a\\'`\\\\(,\\\\, \\\\ldots, \\\\) `\\'z\\'`\\\\(\\\\}\\\\)\\n\\nThe sum of \\\\(N\\\\) across all cases is at most \\\\(2{,}000\\\\).\\nNo node will have two direct outgoing edges with the same edge label.\\nThere are at most \\\\(1{,}000{,}000\\\\) total trie nodes in a given test case.\\nThe sum of nodes across all test cases is at most \\\\(13{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\), the number of tries. Then, \\\\(N\\\\) descriptions of tries follow. For the \\\\(i\\\\)th trie, there is first a line containing a single integer \\\\(M_i\\\\), followed by \\\\(M_i - 1\\\\) more lines, the \\\\(j\\\\)th of which contains the integer \\\\(P_{i,j+1}\\\\), followed by a space, followed by the letter \\\\(C_{i,j+1}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of the query answers across all triplets of tries.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, there are \\\\(N = 3\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:502671685043623|WIDTH:650}}\\n\\nThere is only one possible unordered triplet of tries to consider, for which there are \\\\(5\\\\) strings in at least one trie of the triplet: \"\", \"`a`\", \"`aa`\", \"`b`\", and \"`c`\".\\n\\nIn the second case, there are \\\\(N = 4\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:1118217749060271|WIDTH:650}}\\n\\nThere are four triplets of tries to consider, three of which include the last trie and one of which doesn\\'t. In the three that does, there are \\\\(4\\\\) strings in at least one trie: \"\", \"`a`\", \"`aa`\", and \"`aaa`\". In the one that doesn\\'t, the are \\\\(2\\\\) strings: \"\" and \"`a`\". The final answer is \\\\(3*4 + 1*2 = 14\\\\).\\n\\nIn the third case, there are \\\\(N = 4\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:1007664223967563|WIDTH:650}}\\n\\nThe four triplets and strings contributing to the answer are:\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(2\\\\), trie \\\\(3)\\\\): \"\", \"`a`\", \"`b`\", \"`c`\"\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(2\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`b`\", \"`ab`\", \"`abc`\"\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(3\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`c`\", \"`ab`\", \"`abc`\"\\n* \\\\((\\\\)trie \\\\(2\\\\), trie \\\\(3\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`b`\", \"`c`\", \"`ab`\", \"`abc`\"\\n\\nThe final answer is \\\\(4 + 5 + 5 + 6 = 20\\\\).\\n\\n\\nSample Input:\\n3\\n3\\n3\\n1 a\\n1 b\\n3\\n1 a\\n2 a\\n2\\n1 c\\n4\\n2\\n1 a\\n2\\n1 a\\n2\\n1 a\\n4\\n1 a\\n2 a\\n3 a\\n4\\n2\\n1 a\\n2\\n1 b\\n2\\n1 c\\n4\\n1 a\\n2 b\\n3 c\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 14\\nCase #3: 20\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:06 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11156'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994738'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_ccdd288ae832f422f62857251601f46b'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196daf8bd72789-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:06 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11156', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994738', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_ccdd288ae832f422f62857251601f46b', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196daf8bd72789-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_ccdd288ae832f422f62857251601f46b\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/A1) is that here, card values are not guaranteed to be distinct and can be up to \\\\(10^9\\\\).**\\n\\nLet\\'s cut to the chase. You have a deck of \\\\(N\\\\) face-up cards, each displaying **a not necessarily unique integer between \\\\(1\\\\) and \\\\(10^9\\\\).**\\n\\n*Cutting* the deck once consists of taking a stack of between \\\\(1\\\\) and \\\\(N - 1\\\\) (inclusive) cards from the top and moving it to the bottom in the same order. For example, for the deck \\\\([5, 1, 2, 4, 3]\\\\) ordered from top to bottom, cutting \\\\(2\\\\) cards from the top would yield \\\\([2, 4, 3, 5, 1]\\\\):\\n\\n{{PHOTO_ID:792109885435616|WIDTH:700}}\\n\\nInitially, the \\\\(i\\\\)th card from the top is \\\\(A_i\\\\). Is it possible to cut the deck exactly \\\\(K\\\\) times to reorder the deck such that the \\\\(i\\\\)th card from the top is \\\\(B_i\\\\) for all \\\\(i\\\\)?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 205\\\\)\\n\\\\(2 \\\\le N \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10^{9}\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le \\\\mathbf{10^9}\\\\)\\n**\\\\(A\\\\) and \\\\(B\\\\) are permutations of each other.**\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(A_1, ..., A_N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(B_1, ..., B_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print `\"Case #i: \"` followed by `\"YES\"` if it\\'s possible to cut the deck \\\\(K\\\\) times to change the deck from \\\\(A_i\\\\) to \\\\(B_i\\\\), or `\"NO\"` otherwise.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, it\\'s possible to get to the new order with \\\\(K = 1\\\\) cut (cutting 2 cards from the top).\\n\\nIn the second case, it\\'s impossible to change \\\\([3, 1, 4, 2]\\\\) to \\\\([1, 2, 3, 4]\\\\) with any number of cuts.\\n\\nIn the third case, it\\'s impossible for the deck to be in a different order after \\\\(K = 0\\\\) cuts.\\n\\n\\nSample Input:\\n4\\n5 1\\n5 1 2 2 3\\n2 2 3 5 1\\n4 10\\n3 1 4 2\\n1 2 3 4\\n4 0\\n3 1 4 2\\n2 3 1 4\\n5 3\\n10 10 9 10 9\\n10 10 10 9 9\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: NO\\nCase #3: NO\\nCase #4: NO\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + "DEBUG:httpcore.http11:send_request_headers.started request=\n", + "DEBUG:httpcore.http11:send_request_headers.complete\n", + "DEBUG:httpcore.http11:send_request_body.started request=\n", + "DEBUG:httpcore.http11:send_request_body.complete\n", + "DEBUG:httpcore.http11:receive_response_headers.started request=\n", + "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:06 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11330'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994597'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_e35b006bb4af65fd2e0be2d82258545f'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196daf6f0d5234-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", + "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", + "DEBUG:httpcore.http11:receive_response_body.started request=\n", + "DEBUG:httpcore.http11:receive_response_body.complete\n", + "DEBUG:httpcore.http11:response_closed.started\n", + "DEBUG:httpcore.http11:response_closed.complete\n", + "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:06 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11330', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994597', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_e35b006bb4af65fd2e0be2d82258545f', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196daf6f0d5234-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", + "DEBUG:openai._base_client:request_id: req_e35b006bb4af65fd2e0be2d82258545f\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This is the harder version of [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/A1). This version involves an array of integers instead of a string, and the array may change with updates.**\\n\\nAn array is *perfectly balanced* if its length is even, and the first half of the array can be shuffled to make the array a palindrome. For example, \\\\([15,2,15,2]\\\\) and \\\\([7,5,5,7]\\\\) are perfectly balanced, but \\\\([7,5,3,1,3,5,7]\\\\) and \\\\([3,1,1,3,2,3]\\\\) are not.\\n\\nAn array is *almost perfectly balanced* if you can delete exactly one element from it to make it perfectly balanced. Some examples are \\\\([1, 2, 3, 4, 4, 5, 1, 3, 2]\\\\), \\\\([1]\\\\), and \\\\([2, 2, 2, 3, 2]\\\\).\\n\\n{{PHOTO_ID:532712048616543|WIDTH:700}}\\n\\nYou are given a larger template array \\\\(A_1 ... A_N\\\\) of length \\\\(N\\\\), along with \\\\(Q\\\\) events, the \\\\(i\\\\)th of which is one of the following two formats:\\n\\n- \\\\(1\\\\) \\\\(X_i\\\\) \\\\(Y_i\\\\): set \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n- \\\\(2\\\\) \\\\(L_i\\\\) \\\\(R_i\\\\): check whether the subarray \\\\(A_{L_i..R_i}\\\\) is almost perfectly balanced.\\n\\nFor how many type-2 queries is the subarray almost perfectly balanced?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 85\\\\)\\n\\\\(1 \\\\le N \\\\le 10^6\\\\)\\n\\\\(1 \\\\le A_i \\\\le 10^6\\\\)\\n\\\\(1 \\\\le Q \\\\le 10^6\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Y_i \\\\le 10^6\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers, either \"\\\\(1\\\\) \\\\(X_i\\\\) \\\\(Y_i\\\\)\" or \"\\\\(2\\\\) \\\\(L_i\\\\) \\\\(R_i\\\\)\".\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"`, followed by a single integer: the number of queries which are an almost perfectly balanced subarray.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, the template array is \\\\([2,1,5,1,5,1]\\\\).\\n\\nFor the first query, the subarray is \\\\([1,5,1,5,1]\\\\). You can delete the second \\\\(1\\\\) to get \\\\([1,5,5,1]\\\\), which is a palindrome. Thus, \\\\([1,5,1,5,1]\\\\) is an almost perfectly balanced subarray.\\n\\nFor the second query, the subarray is \\\\([2,1,5,1,5]\\\\). You can delete the \\\\(2\\\\) to get \\\\([1,5,1,5]\\\\) then reorder the first half of the array to get \\\\([5,1,1,5]\\\\), which is a palindrome. Thus, \\\\([2,1,5,1,5]\\\\) is an almost perfectly balanced subarray.\\n\\nFor the third query, the subarray is \\\\([2,1,5]\\\\). You cannot delete any element to get an almost perfectly balanced subarray.\\n\\nThe next two events are updates, after which the template array is \\\\([5, 5, 5, 1, 5, 1]\\\\).\\n\\nFor the fourth query, the subarray is \\\\([5,5,5]\\\\). You can delete any element to get \\\\([5,5]\\\\), which is a palindrome. Thus, \\\\([5,5,5]\\\\) is an almost perfectly balanced subarray.\\n\\nThe fifth, sixth, eighth, and tenth queries are also almost perfectly balanced subarrays, so the final answer is \\\\(7\\\\).\\n\\n\\nSample Input:\\n1\\n6\\n2 1 5 1 5 1\\n17\\n2 2 6\\n2 1 5\\n2 1 3\\n1 1 5\\n1 2 5\\n2 1 3\\n2 1 5\\n2 2 2\\n1 3 1000000\\n2 1 5\\n2 1 3\\n2 2 4\\n2 4 6\\n1 6 1000000\\n2 4 6\\n1 5 1000000\\n2 5 6\\n\\n\\nSample Output:\\nCase #1: 7\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem C2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/C2) is that here, the length of each output codeword may be at most 200.**\\n\\nMorse code is a classic way to send messages, where each letter in an alphabet is substituted with a *codeword*: a unique sequence of dots and dashes. However, ignoring spaces, it\\'s possible for a coded message to have multiple meanings. For example, \"`.....--.-.-.-..-.-.-...-.--.`\" can be interpreted as either \"`HACKER CUP`\" or \"`SEE META RENT A VAN`\":\\n\\n{{PHOTO_ID:1264108514364518|WIDTH:700}}\\n\\nBeyond Morse code, a general set of codewords is an *unambiguous encoding* if any possible sequence of dots and dashes corresponds to either zero or exactly one sequence of codewords.\\n\\nGiven one codeword \\\\(C_1\\\\) from a set of \\\\(N\\\\) distinct codewords, your task is to generate another \\\\(N - 1\\\\) codewords \\\\(C_2, ..., C_N\\\\) to yield an unambiguous encoding. It can be shown that an answer always exists. If there are multiple answers, you may print any one of them.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(2 \\\\le N \\\\le 100\\\\)\\nThe length of \\\\(C_1\\\\) is between \\\\(1\\\\) and \\\\(100\\\\), inclusive.\\nThe length of each \\\\(C_2, ..., C_N\\\\) must be between \\\\(1\\\\) and \\\\(\\\\mathbf{200}\\\\), inclusive.\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing the codeword \\\\(C_1\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output a line containing only \"`Case #i:`\", followed by \\\\(N - 1\\\\) lines, the codewords \\\\(C_2, ..., C_N\\\\), one per line.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, it can be shown that the codewords {\"`.-.`\", \"`...`\", \"`---`\"} are an unambiguous encoding. Any sequence of dots and dashes can be interpreted if and only if it has a length that\\'s a multiple of 3, and can be broken up into instances of the three length-3 codewords.\\n\\nIn the second case, it can be shown that the codewords {\"`-`\", \"`...`\", \"`.-`\", \"`..-`\"} are an unambiguous encoding. For instance, \"`..`\" has no possible interpretation, and \"`.-...--`\" can only be interpreted as \"`.- ... - -`\".\\n\\nIn the third case, it can be shown that the codewords {\"`..`\", \"`-`\", \"`.-`\"} are an unambiguous encoding. For any sequence of dots and dashes:\\n- every odd group of dots followed by a dash can only be interpreted as repeated \"`..`\"s followed by a final \"`.-`\"\\n- every even group of dots followed by a dash can only be interpreted as repeated \"`..`\"s followed by a final \"`-`\"\\n- every group of dots not followed by a dash (i.e. at the end of the sequence), is interpretable if and only if there is an even number of dots\\n- this leaves only groups of dashes, interpreted only as repeated \"`-`\"s\\n\\n\\n\\nSample Input:\\n3\\n3\\n.-.\\n4\\n-\\n3\\n..\\n\\n\\nSample Output:\\nCase #1:\\n...\\n---\\nCase #2:\\n...\\n.-\\n..-\\nCase #3:\\n-\\n.-\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nMrs. Puff is taking her class trick-or-treating! Her \\\\(N\\\\) students, numbered from \\\\(1\\\\) to \\\\(N\\\\), are dressed up in spooky outfits and ready to collect some candy. To divide and conquer every house in Bikini Bottom, she would like to break up the class into equal-sized groups.\\n\\n\\\\(M\\\\) pairs of students have similar-themed costumes that amplify their spookiness, the \\\\(i\\\\)th being students \\\\(A_i\\\\) and \\\\(B_i\\\\). Ms. Puff\\'s would like to know: for which values of \\\\(K\\\\) can the class be divided into \\\\(K\\\\) groups of equal size without splitting up any of these pairs?\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 75\\\\)\\n\\\\(2 \\\\leq N \\\\leq 100\\\\)\\n\\\\(0 \\\\leq M \\\\leq \\\\frac{N(N-1)}{2}\\\\)\\n\\\\(1 \\\\leq A_i, B_i \\\\leq N\\\\)\\n\\\\(A_i \\\\neq B_i\\\\)\\n\\nEach unordered pair of employees occurs at most once in a given test case.\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(N\\\\) and \\\\(M\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains the two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by all possible values of \\\\(K\\\\) in ascending order, separated by spaces.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are three possible values of \\\\(K\\\\). Here\\'s a possible arrangement:\\n- \\\\(K = 1\\\\): All students can be put into a single group.\\n- \\\\(K = 2\\\\): Groups \\\\(\\\\{1, 2, 4, 5, 6, 7\\\\}\\\\) and \\\\(\\\\{3, 8, 9, 10, 11, 12\\\\}\\\\).\\n- \\\\(K = 3\\\\): Groups \\\\(\\\\{1, 2, 4, 6\\\\}\\\\), \\\\(\\\\{3, 5, 8, 12\\\\}\\\\) and \\\\(\\\\{7, 9, 10, 11\\\\}\\\\).\\n\\nIn the second case, either all students can be in a single group, or each should form a separate group.\\n\\nIn the third test case, the valid values of \\\\(K\\\\), with an example of each, are:\\n- \\\\(K = 1\\\\): All students can be put into a single group.\\n- \\\\(K = 2\\\\): Groups \\\\(\\\\{1, 2, 5, 7\\\\}\\\\) and \\\\(\\\\{3, 4, 6, 8\\\\}\\\\).\\n- \\\\(K = 4\\\\): Groups \\\\(\\\\{1, 7\\\\}\\\\), \\\\(\\\\{2, 5\\\\}\\\\), \\\\(\\\\{3, 6\\\\}\\\\) and \\\\(\\\\{4, 8\\\\}\\\\).\\n\\n\\nSample Input:\\n3\\n12 7\\n1 2\\n2 4\\n4 6\\n3 8\\n8 12\\n9 10\\n10 11\\n3 0\\n8 4\\n1 7\\n2 5\\n6 3\\n8 4\\n\\n\\nSample Output:\\nCase #1: 1 2 3\\nCase #2: 1 3\\nCase #3: 1 2 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This could possibly be the best day ever!\\nAnd the forecast says that tomorrow will likely be a billion and 6 times better.\\nSo make every minute count, jump up, jump in, and seize the day,\\nAnd let\\'s make sure that in every single possible way… today is gonna be a great day!*\\n\\nThere\\'s \\\\(N\\\\) days of summer vacation, and the \\\\(i\\\\)th day initially has greatness \\\\(A_i\\\\). However, the days just keep getting better! In particular, there will be \\\\(Q\\\\) changes, the \\\\(i\\\\)th of which changes the greatness of every day from \\\\(L_i\\\\) to \\\\(R_i\\\\) to be \\\\(1\\\\,000\\\\,000\\\\,006\\\\) times what they were before!\\n\\nAfter each query, you\\'d like to know: which day\\'s greatness modulo \\\\(1\\\\,000\\\\,000\\\\,007\\\\) is the largest? If multiple days tie, we\\'re interested in the earliest of them. For example, if there are \\\\(N = 4\\\\) days with greatnesses \\\\([4, 1, 4, 2]\\\\), then days \\\\(1\\\\) and \\\\(3\\\\) have the largest remainder, so our answer would be \\\\(1\\\\). To keep the output size small, please print only the sum of answers to all queries.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 45\\\\)\\n\\\\(1 \\\\le N \\\\le 1{,}000{,}004\\\\)\\n\\\\(1 \\\\le A_i \\\\le 1{,}000{,}000{,}006\\\\)\\n\\\\(1 \\\\le Q \\\\le 500{,}000\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(L_i\\\\) and \\\\(R_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, print a single line containing \"`Case #i: `\" followed by a single integer, the sum of the greatest day(s) after each of the \\\\(Q\\\\) queries.\\n\\n**Sample Explanation**\\n\\nIn the first test case, there are \\\\(N=3\\\\) days with initial greatnesses of \\\\([1, 20, 30]\\\\). The first query modifies all \\\\(3\\\\) days, transforming their greatnesses \\\\([1\\\\,000\\\\,000\\\\,006, 999\\\\,999\\\\,987, 999\\\\,999\\\\,977]\\\\), where the maximum is on day \\\\(1\\\\). After the second query, day \\\\(3\\\\) will have the largest greatness of \\\\(999\\\\,999\\\\,977\\\\). The overall answer is the sum of the greatest days, which is \\\\(1 + 3 = 4\\\\).\\n\\n\\n\\nSample Input:\\n2\\n3\\n1 20 30\\n2\\n1 3\\n1 2\\n2\\n1 1\\n1\\n1 2\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou and some friends are playing [Wiki Race](https://en.wikipedia.org/wiki/Wikipedia:Wiki\\\\_Game) on an online encyclopedia of \\\\(N\\\\) pages, numbered from \\\\(1\\\\) to \\\\(N\\\\). Page \\\\(i\\\\) covers \\\\(M_i\\\\) topics \\\\(S_{i,1}\\\\) ... \\\\(S_{i,M_i}\\\\), each a string of lowercase letters. Each player starts on a unique page, and will race to reach page \\\\(1\\\\) (the target page) by repeatedly clicking to neighboring pages.\\n\\nUnfortunately, someone out there has already published the shortest path from each page to page \\\\(1\\\\) for the whole internet to see. This forms a tree structure with page \\\\(1\\\\) as the root, and edges directed upwards from each descendent page. In particular, you and your friends all know that at each page \\\\(i > 1\\\\), one should click to the neighboring page \\\\(P_i\\\\) for the fastest route to page \\\\(1\\\\).\\n\\nIn light of this, all players have agreed to a new rule that each page should be visited by exactly one person per game. This means that once a player hits a page someone has already visited, their upward path ends there and the player loses. Formally, we can express each game\\'s outcome as a partition of the tree into vertex-disjoint paths, with each path\\'s upper endpoint always having an edge to a node on another player\\'s path (except the winner, whose upper endpoint is page \\\\(1\\\\)).\\n\\nAfter playing the game for so long, you\\'re all starting to realize that \"*it\\'s not about winning, but the knowledge acquired along the way!*\" \\n\\nFor a given partition of the tree into paths, a topic is _mutually-learned_ by all players if every path has at least one page covering that topic. You\\'d like to know, across all possible partitions of paths, all topics that could be mutually-learned.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 30\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\\\(1 \\\\leq M_i \\\\leq 100{,}000\\\\)\\n\\\\(1 \\\\leq |S_{i,j}| \\\\leq 10\\\\)\\n\\\\(S_{i,j}\\\\) consists only of lowercase letters (\\'`a`\\'..\\'`z`\\').\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\nThe sum of \\\\(M_i\\\\) across all test cases is at most \\\\(8{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N - 1\\\\) integers \\\\(P_2, ..., P_N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the integer \\\\(M_i\\\\), followed by strings \\\\(S_{i,1}\\\\) ... \\\\(S_{i,M_i}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of topics that can be mutually-learned.\\n\\n\\n# Sample Explanation\\n\\nThe three sample cases are depicted below:\\n\\n{{PHOTO_ID:636025968710743|WIDTH:600}}\\n \\nIn the first case:\\n* \"gouda\" is mutually-learned in the partition \\\\(\\\\{\\\\{4\\\\}, \\\\{6, 5, 2\\\\}, \\\\{3, 1\\\\}\\\\}\\\\)\\n* \"cheddar\" is mutually-learned in the partition \\\\(\\\\{\\\\{4, 2\\\\}, \\\\{6, 5\\\\}, \\\\{3, 1\\\\}\\\\}\\\\)\\n* \"edam\" is mutually-learned in the same partition as \"cheddar\" as well as in other partitions such as \\\\(\\\\{\\\\{4, 2, 1\\\\}, \\\\{6\\\\}, \\\\{5\\\\}, \\\\{3\\\\}\\\\}\\\\)\\n* \"gjetost\", \"gruyere\", and \"mozzarella\" can never be mutually-learned\\n\\nIn the second case, the only topic that can be mutually-learned is \"earth\".\\n\\nIn the third case, the partition \\\\(\\\\{\\\\{1, 2, 3\\\\}\\\\}\\\\) makes all three topics mutually-learned.\\n\\n\\n\\nSample Input:\\n3\\n6\\n1 1 2 2 5\\n4 mozzarella cheddar gouda edam\\n5 gouda gjetost mozzarella cheddar edam\\n3 gruyere mozzarella edam\\n1 gouda\\n1 edam\\n3 cheddar gruyere edam\\n4\\n1 2 2\\n2 earth water\\n2 wind wind\\n2 earth fire\\n2 light metal\\n3\\n1 2\\n1 a\\n1 b\\n1 c\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 1\\nCase #3: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 3\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "Updating best score\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.050515Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.055072Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,056] Trial 3 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 4\n", + "INFO:root:instruction_idx 4\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.108445Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.113307Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,114] Trial 4 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 5\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.174565Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.178665Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,179] Trial 5 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 6\n", + "INFO:root:instruction_idx 2\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.226630Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.231013Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,232] Trial 6 finished with value: 0.0 and parameters: {'0_predictor_instruction': 2}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 7\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.278299Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.282334Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,283] Trial 7 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 8\n", + "INFO:root:instruction_idx 4\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.335396Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.339829Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,340] Trial 8 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 9\n", + "INFO:root:instruction_idx 3\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.385733Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.390045Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,391] Trial 9 finished with value: 0.0 and parameters: {'0_predictor_instruction': 3}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 10\n", + "INFO:root:instruction_idx 0\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "DEBUG:root:OpenAI Response Token Usage: 1886\n", + "DEBUG:root:OpenAI Response Token Usage: 1781\n", + "DEBUG:root:OpenAI Response Token Usage: 2477\n", + "DEBUG:root:OpenAI Response Token Usage: 1651\n", + "DEBUG:root:OpenAI Response Token Usage: 1595\n", + "DEBUG:root:OpenAI Response Token Usage: 1326\n", + "DEBUG:root:OpenAI Response Token Usage: 1659\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 1809\n", + "DEBUG:root:OpenAI Response Token Usage: 2213\n", + "DEBUG:root:OpenAI Response Token Usage: 1339\n", + "DEBUG:root:OpenAI Response Token Usage: 1800\n", + "DEBUG:root:OpenAI Response Token Usage: 1842\n", + "DEBUG:root:OpenAI Response Token Usage: 1457\n", + "DEBUG:root:OpenAI Response Token Usage: 2905\n", + "DEBUG:root:OpenAI Response Token Usage: 1705\n", + "DEBUG:root:OpenAI Response Token Usage: 1414\n", + "DEBUG:root:OpenAI Response Token Usage: 2099\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 1655\n", + "DEBUG:root:OpenAI Response Token Usage: 2398\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 1907\n", + "DEBUG:root:OpenAI Response Token Usage: 2011\n", + "DEBUG:root:OpenAI Response Token Usage: 2175\n", + "DEBUG:root:OpenAI Response Token Usage: 2267\n", + "DEBUG:root:OpenAI Response Token Usage: 1982\n", + "DEBUG:root:OpenAI Response Token Usage: 2055\n", + "DEBUG:root:OpenAI Response Token Usage: 1986\n", + "DEBUG:root:OpenAI Response Token Usage: 1602\n", + "DEBUG:root:OpenAI Response Token Usage: 2247\n", + "DEBUG:root:OpenAI Response Token Usage: 2089\n", + "DEBUG:root:OpenAI Response Token Usage: 2371\n", + "DEBUG:root:OpenAI Response Token Usage: 2123\n", + "DEBUG:root:OpenAI Response Token Usage: 1904\n", + "DEBUG:root:OpenAI Response Token Usage: 2036\n", + "DEBUG:root:OpenAI Response Token Usage: 2062\n", + "DEBUG:root:OpenAI Response Token Usage: 1688\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1782\n", + "DEBUG:root:OpenAI Response Token Usage: 1824\n", + "DEBUG:root:OpenAI Response Token Usage: 1702\n", + "DEBUG:root:OpenAI Response Token Usage: 2041\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.439161Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.443422Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,444] Trial 10 finished with value: 0.0 and parameters: {'0_predictor_instruction': 0}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 11\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.490569Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.496614Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,497] Trial 11 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 12\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.547837Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.552238Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,553] Trial 12 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 13\n", + "INFO:root:instruction_idx 0\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "DEBUG:root:OpenAI Response Token Usage: 1886\n", + "DEBUG:root:OpenAI Response Token Usage: 1781\n", + "DEBUG:root:OpenAI Response Token Usage: 2477\n", + "DEBUG:root:OpenAI Response Token Usage: 1651\n", + "DEBUG:root:OpenAI Response Token Usage: 1595\n", + "DEBUG:root:OpenAI Response Token Usage: 1326\n", + "DEBUG:root:OpenAI Response Token Usage: 1659\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 1809\n", + "DEBUG:root:OpenAI Response Token Usage: 2213\n", + "DEBUG:root:OpenAI Response Token Usage: 1339\n", + "DEBUG:root:OpenAI Response Token Usage: 1800\n", + "DEBUG:root:OpenAI Response Token Usage: 1842\n", + "DEBUG:root:OpenAI Response Token Usage: 1457\n", + "DEBUG:root:OpenAI Response Token Usage: 2905\n", + "DEBUG:root:OpenAI Response Token Usage: 1705\n", + "DEBUG:root:OpenAI Response Token Usage: 1414\n", + "DEBUG:root:OpenAI Response Token Usage: 2099\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 1655\n", + "DEBUG:root:OpenAI Response Token Usage: 2398\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 1907\n", + "DEBUG:root:OpenAI Response Token Usage: 2011\n", + "DEBUG:root:OpenAI Response Token Usage: 2175\n", + "DEBUG:root:OpenAI Response Token Usage: 2267\n", + "DEBUG:root:OpenAI Response Token Usage: 1982\n", + "DEBUG:root:OpenAI Response Token Usage: 2055\n", + "DEBUG:root:OpenAI Response Token Usage: 1986\n", + "DEBUG:root:OpenAI Response Token Usage: 1602\n", + "DEBUG:root:OpenAI Response Token Usage: 2247\n", + "DEBUG:root:OpenAI Response Token Usage: 2089\n", + "DEBUG:root:OpenAI Response Token Usage: 2371\n", + "DEBUG:root:OpenAI Response Token Usage: 2123\n", + "DEBUG:root:OpenAI Response Token Usage: 1904\n", + "DEBUG:root:OpenAI Response Token Usage: 2036\n", + "DEBUG:root:OpenAI Response Token Usage: 2062\n", + "DEBUG:root:OpenAI Response Token Usage: 1688\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1782\n", + "DEBUG:root:OpenAI Response Token Usage: 1824\n", + "DEBUG:root:OpenAI Response Token Usage: 1702\n", + "DEBUG:root:OpenAI Response Token Usage: 2041\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.597933Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.602147Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,603] Trial 13 finished with value: 0.0 and parameters: {'0_predictor_instruction': 0}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 14\n", + "INFO:root:instruction_idx 3\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.646545Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.650284Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,651] Trial 14 finished with value: 0.0 and parameters: {'0_predictor_instruction': 3}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 15\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.695994Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.700319Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,701] Trial 15 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 16\n", + "INFO:root:instruction_idx 1\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.757919Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.762307Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,763] Trial 16 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. 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Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 18\n", + "INFO:root:instruction_idx 4\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "DEBUG:root:OpenAI Response Token Usage: 2081\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 2803\n", + "DEBUG:root:OpenAI Response Token Usage: 1834\n", + "DEBUG:root:OpenAI Response Token Usage: 1677\n", + "DEBUG:root:OpenAI Response Token Usage: 1531\n", + "DEBUG:root:OpenAI Response Token Usage: 2106\n", + "DEBUG:root:OpenAI Response Token Usage: 2562\n", + "DEBUG:root:OpenAI Response Token Usage: 2098\n", + "DEBUG:root:OpenAI Response Token Usage: 2424\n", + "DEBUG:root:OpenAI Response Token Usage: 1573\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1892\n", + "DEBUG:root:OpenAI Response Token Usage: 1601\n", + "DEBUG:root:OpenAI Response Token Usage: 3176\n", + "DEBUG:root:OpenAI Response Token Usage: 1928\n", + "DEBUG:root:OpenAI Response Token Usage: 1541\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2008\n", + "DEBUG:root:OpenAI Response Token Usage: 1895\n", + "DEBUG:root:OpenAI Response Token Usage: 2629\n", + "DEBUG:root:OpenAI Response Token Usage: 2271\n", + "DEBUG:root:OpenAI Response Token Usage: 2137\n", + "DEBUG:root:OpenAI Response Token Usage: 2416\n", + "DEBUG:root:OpenAI Response Token Usage: 2314\n", + "DEBUG:root:OpenAI Response Token Usage: 2419\n", + "DEBUG:root:OpenAI Response Token Usage: 2381\n", + "DEBUG:root:OpenAI Response Token Usage: 2198\n", + "DEBUG:root:OpenAI Response Token Usage: 2293\n", + "DEBUG:root:OpenAI Response Token Usage: 1669\n", + "DEBUG:root:OpenAI Response Token Usage: 2327\n", + "DEBUG:root:OpenAI Response Token Usage: 2249\n", + "DEBUG:root:OpenAI Response Token Usage: 2543\n", + "DEBUG:root:OpenAI Response Token Usage: 2269\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 2158\n", + "DEBUG:root:OpenAI Response Token Usage: 2281\n", + "DEBUG:root:OpenAI Response Token Usage: 1916\n", + "DEBUG:root:OpenAI Response Token Usage: 2302\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 2005\n", + "DEBUG:root:OpenAI Response Token Usage: 1818\n", + "DEBUG:root:OpenAI Response Token Usage: 2148\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.866010Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1582\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.870244Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,871] Trial 18 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. Best is trial 0 with value: 0.0.\n", + "INFO:root:Starting trial num: 19\n", + "INFO:root:instruction_idx 0\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "DEBUG:root:OpenAI Response Token Usage: 1886\n", + "DEBUG:root:OpenAI Response Token Usage: 1781\n", + "DEBUG:root:OpenAI Response Token Usage: 2477\n", + "DEBUG:root:OpenAI Response Token Usage: 1651\n", + "DEBUG:root:OpenAI Response Token Usage: 1595\n", + "DEBUG:root:OpenAI Response Token Usage: 1326\n", + "DEBUG:root:OpenAI Response Token Usage: 1659\n", + "DEBUG:root:OpenAI Response Token Usage: 2136\n", + "DEBUG:root:OpenAI Response Token Usage: 1809\n", + "DEBUG:root:OpenAI Response Token Usage: 2213\n", + "DEBUG:root:OpenAI Response Token Usage: 1339\n", + "DEBUG:root:OpenAI Response Token Usage: 1800\n", + "DEBUG:root:OpenAI Response Token Usage: 1842\n", + "DEBUG:root:OpenAI Response Token Usage: 1457\n", + "DEBUG:root:OpenAI Response Token Usage: 2905\n", + "DEBUG:root:OpenAI Response Token Usage: 1705\n", + "DEBUG:root:OpenAI Response Token Usage: 1414\n", + "DEBUG:root:OpenAI Response Token Usage: 2099\n", + "DEBUG:root:OpenAI Response Token Usage: 1854\n", + "DEBUG:root:OpenAI Response Token Usage: 1655\n", + "DEBUG:root:OpenAI Response Token Usage: 2398\n", + "DEBUG:root:OpenAI Response Token Usage: 2004\n", + "DEBUG:root:OpenAI Response Token Usage: 1907\n", + "DEBUG:root:OpenAI Response Token Usage: 2011\n", + "DEBUG:root:OpenAI Response Token Usage: 2175\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n", + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", + "\n", + "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", + "\n", + "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", + "\n", + "3. **Basic Cases**:\n", + " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", + " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", + "\n", + "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", + "\n", + "5. **Constructing the Array**:\n", + " - Start with the prime factors of \\(P\\).\n", + " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", + " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", + "\n", + "6. **Edge Cases**: \n", + " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", + "\n", + "Solution: To implement the solution, we can follow these steps:\n", + "\n", + "1. For each test case, read the integer \\(P\\).\n", + "2. If \\(P = 1\\), output \\(41\\) ones.\n", + "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", + "4. For other values of \\(P\\):\n", + " - Factor \\(P\\) into its prime factors.\n", + " - Calculate the sum of these factors.\n", + " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", + " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", + "5. If no valid configuration is found, output \\(-1\\`.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for i from 1 to T:\n", + " P = read integer\n", + " if P == 1:\n", + " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", + " else if P == 41:\n", + " print \"Case #i: 1 41\"\n", + " else:\n", + " factors = factor(P)\n", + " sum_factors = sum(factors)\n", + " if sum_factors > 41:\n", + " # Try to combine factors or check if valid\n", + " if valid_combination(factors, 41):\n", + " print \"Case #i: N factors\"\n", + " else:\n", + " print \"Case #i: -1\"\n", + " else:\n", + " # Add 1s to reach 41\n", + " num_ones = 41 - sum_factors\n", + " print \"Case #i: N + num_ones factors\"\n", + "```\n", + "\n", + "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "CANDIDATE PROGRAM:\n", + "Predictor 0\n", + "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "p: Solution:\n", + "\n", + "\n", + "...\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:root:OpenAI Response Token Usage: 2267\n", + "DEBUG:root:OpenAI Response Token Usage: 1982\n", + "DEBUG:root:OpenAI Response Token Usage: 2055\n", + "DEBUG:root:OpenAI Response Token Usage: 1986\n", + "DEBUG:root:OpenAI Response Token Usage: 1602\n", + "DEBUG:root:OpenAI Response Token Usage: 2247\n", + "DEBUG:root:OpenAI Response Token Usage: 2089\n", + "DEBUG:root:OpenAI Response Token Usage: 2371\n", + "DEBUG:root:OpenAI Response Token Usage: 2123\n", + "DEBUG:root:OpenAI Response Token Usage: 1904\n", + "DEBUG:root:OpenAI Response Token Usage: 2036\n", + "DEBUG:root:OpenAI Response Token Usage: 2062\n", + "DEBUG:root:OpenAI Response Token Usage: 1688\n", + "DEBUG:root:OpenAI Response Token Usage: 2121\n", + "DEBUG:root:OpenAI Response Token Usage: 1782\n", + "DEBUG:root:OpenAI Response Token Usage: 1824\n", + "DEBUG:root:OpenAI Response Token Usage: 1702\n", + "DEBUG:root:OpenAI Response Token Usage: 2041\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.955999Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "DEBUG:root:OpenAI Response Token Usage: 1413\n", + "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.968592Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", + "[I 2024-09-11 19:38:24,969] Trial 19 finished with value: 0.0 and parameters: {'0_predictor_instruction': 0}. Best is trial 0 with value: 0.0.\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "FULL TRACE\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "You are an expert problem solver. Your task is to solve the problem at hand.\n", + "\n", + "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", + "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", + "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", + "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", + "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", + "Address time and space complexity, and provide pseudocode if appropriate. \n", + "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", + "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", + "\n", + "---\n", + "\n", + "Follow the following format.\n", + "\n", + "Problem Statement: ${problem_statement}\n", + "\n", + "Sample Input: The sample input provided with the problem statement.\n", + "\n", + "Sample Output: The sample output provided with the problem statement.\n", + "\n", + "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", + "\n", + "Solution: a solution explanation for how we should go about solving this problem.\n", + "\n", + "---\n", + "\n", + "Problem Statement:\n", + "*This problem shares some similarities with B2, with key differences in bold.*\n", + "\n", + "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", + "\n", + "If multiple such arrays exist, **you may output any one of them**.\n", + "\n", + "# Constraints\n", + "\\(1 \\leq T \\leq 965\\)\n", + "\\(1 \\leq P \\leq 10^9\\)\n", + "\n", + "# Input Format\n", + "\n", + "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", + "\n", + "# Output Format\n", + "\n", + "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", + "\n", + "# Sample Explanation\n", + "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", + "\n", + "\n", + "\n", + "Sample Input:\n", + "7\n", + "2023\n", + "114\n", + "41\n", + "175\n", + "434\n", + "666\n", + "1872\n", + "\n", + "\n", + "Sample Output:\n", + "Case #1: 3 7 17 17\n", + "Case #2: 2 3 38\n", + "Case #3: 1 41\n", + "Case #4: 3 1 5 35\n", + "Case #5: 4 1 2 7 31\n", + "Case #6: -1\n", + "Case #7: 5 1 2 6 6 26\n", + "\n", + "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", + "\n", + "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", + "\n", + "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", + "\n", + "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", + "\n", + "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", + "\n", + "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", + "\n", + "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", + "\n", + "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", + "\n", + "Solution: To solve the problem, we can follow these steps:\n", + "\n", + "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", + "\n", + "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", + "\n", + "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "\n", + "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "\n", + "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\n", + "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\n", + "Pseudocode:\n", + "```\n", + "for each test case:\n", + " read P\n", + " if P == 41:\n", + " output \"Case #i: 1 41\"\n", + " continue\n", + " factors = find_factors(P)\n", + " for each combination of factors:\n", + " if sum(combination) <= 41:\n", + " remaining = 41 - sum(combination)\n", + " if product(combination) * (1^remaining) == P:\n", + " output \"Case #i: N combination + remaining ones\"\n", + " break\n", + " else:\n", + " output \"Case #i: -1\"\n", + "```\n", + "\n", + "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "\n", + "\n", + "\n", + "...\n", + "Score 0.0\n", + "[('generate_code', Predict(StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + ")))]\n" + ] + } + ], "source": [ "mipro_optimized_program = optimize_with_mipro(simple_program, lm, lm, validate_solution, trainset)" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 29, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nIshiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\\\(N\\\\) colored beads, with exactly \\\\(K\\\\) green beads made of emerald, and \\\\(N-K\\\\) remaining beads of all unique colors (neither green nor the same as any other bead in the bracelet). Ishiko has an infinite number of beads available for each of these \\\\(N - K + 1\\\\) different colors.\\n\\nIshiko wants to exhibit *one copy of every possible distinct bracelet*. Two bracelets are considered distinct if one is not a cyclic rotation of the other. For instance, the two bracelets on the left are distinct, but the two bracelets on the right are not:\\n\\n{{PHOTO_ID:876374553707415|WIDTH:700}}\\n\\nA vendor takes orders for triangular display cases of arbitrary positive integer heights. A case of height \\\\(H\\\\) has \\\\(H\\\\) rows, where the \\\\(i\\\\)th row has \\\\(2*i - 1\\\\) slots, each slot capable of holding one bracelet. For example, a case of height \\\\(3\\\\) is shown below and holds \\\\(1 + 3 + 5 = 9\\\\) bracelets:\\n\\n{{PHOTO_ID:547949757342773|WIDTH:700}}\\n\\nHow many display cases (of possibly differing heights) will Ishiko need to buy to exhibit one copy of every possible bracelet *without leaving any empty slots in the display cases*?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 110\\\\)\\n\\\\(1 \\\\le N \\\\le 10^9\\\\)\\n\\\\(0 \\\\le K \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(8{,}000{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of cases. For each case, there is a line containing two space-separated integers, \\\\(N\\\\) and \\\\(K\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print `\"Case #i: \"` followed by a single integer, the minimum number of display cases that Ishiko needs to purchase to exhibit one copy of each bracelet.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, let\\'s say that Ishiko\\'s beads are green, blue, and red. There are \\\\(4\\\\) possible bracelets for \\\\(N = 5\\\\) beads with \\\\(K = 3\\\\) green beads, \\\\(1\\\\) red bead, and \\\\(1\\\\) blue bead (up to rotation):\\n\\n{{PHOTO_ID:1327255974690671|WIDTH:700}}\\n\\nIshiko can use a single display case of height \\\\(2\\\\) to display these \\\\(4\\\\) bracelets.\\n\\nIn the second sample case, let\\'s say that Ishiko\\'s beads are green, blue, red, and yellow. There are \\\\(20\\\\) possible bracelets for \\\\(N = 6\\\\) beads with \\\\(K = 3\\\\) green beads, \\\\(1\\\\) blue bead, \\\\(1\\\\) red bead, and \\\\(1\\\\) yellow bead. Some examples are:\\n\\n{{PHOTO_ID:853886435922998|WIDTH:700}}\\n\\nIshiko can display these bracelets if she purchases a case of height \\\\(2\\\\) and a case of height \\\\(4\\\\).\\n\\nIn the third sample case, there are \\\\(60\\\\) possible bracelets and Ishiko needs at least \\\\(4\\\\) display cases to display all of them. One possibility is that she buys two cases of height \\\\(1\\\\), one case of height \\\\(3\\\\), and one case of height \\\\(7\\\\).\\n\\n\\nSample Input:\\n4\\n5 3\\n6 3\\n6 2\\n243447 42273\\n\\n\\nSample Output:\\nCase #1: 1\\nCase #2: 2\\nCase #3: 4\\nCase #4: 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nCourtney has an avant-garde kitchen counter made out of \\\\(N\\\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly from the top, the counter looks like an (unfilled) polygon on a plane with \\\\(N\\\\) edges, the \\\\(i\\\\)th of which extends from point \\\\((X_i, Y_i)\\\\) to point \\\\((X_{(i \\\\text{ mod } N)+1},\\\\, Y_{(i \\\\text{ mod } N)+1})\\\\). These line segments only intersect at their endpoints.\\n\\nCourtney\\'s friends often make fun of the counter, saying that the thin metal \"surface\" is useless for holding anything up, but Courtney wants to prove them wrong.\\n\\nCourtney considers placing a cup with a circular rim of radius \\\\(R\\\\) onto the counter (with the rim facing down), centered at a point chosen uniformly randomly from the square \\\\([0, L] \\\\times [0, L]\\\\). What is the chance that the cup stays balanced on the counter without falling? Your output will be accepted if it is within \\\\(10^{-6}\\\\) of the jury\\'s answer.\\n\\n*Note: The face-down cup is modeled as an unfilled ring—not as a disk.*\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 7\\\\)\\n\\\\(1 \\\\le N \\\\le 7\\\\)\\n\\\\(1 \\\\le R, L \\\\le 15\\\\)\\n\\\\(0 \\\\le X_i, Y_i \\\\le 15\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(R\\\\), and \\\\(L\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\),\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"` followed by a single real number between \\\\(0\\\\) and \\\\(1\\\\), the chance that the cup will balance if placed with its rim centered at a uniformly random point in the square \\\\([0, L] \\\\times [0, L]\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe counters in the first and second sample cases are depicted below, with the blue regions representing points at which centering the cup will lead to it staying balanced.\\n\\n{{PHOTO_ID:556593865835592|WIDTH:650}}\\n\\n\\n\\n\\nSample Input:\\n2\\n3 1 4\\n0 0\\n2 2\\n4 0\\n5 1 4\\n0 0\\n1 2\\n0 3\\n4 3\\n3 1\\n\\n\\nSample Output:\\nCase #1: 0.1169663730642699\\nCase #2: 0.1353445414060363\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", + " 0%| | 0/6 [00:00 500{,}000\\\\) in at most two test cases.\\nThe sum of \\\\(N\\\\) over all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing `\"Case #i: \"` followed by a single integer, the minimum total area of net needed to enclose all trees according to the above requirements.\\n\\n\\n# Sample Explanation\\n\\nSolutions for the first two sample cases are depicted below.\\n\\n{{PHOTO_ID:2468652649959145|WIDTH:700}}\\n\\nIn the first case, the minimum possible net area is \\\\(2*2 + 3*4 + 2*2 = 20\\\\).\\n\\nIn the second case, the minimum possible net area is \\\\(2*2 + 4*6 = 28\\\\).\\n\\nIn the third case, it\\'s almost possible to give each tree its own net. Unfortunately, there needs to be at least \\\\(1\\\\) unit of space between the nets at \\\\((2, 2)\\\\) so Boss Rob can walk through. This makes it indirectly necessary to use one big net to cover all trees. \\n\\n{{PHOTO_ID:504150738360453|WIDTH:700}}\\n\\n\\nSample Input:\\n3\\n4\\n0 0\\n2 1\\n5 0\\n0 6\\n5\\n10 10\\n12 10\\n8 10\\n5 10\\n8 8\\n4\\n1 1\\n3 3\\n0 4\\n4 6\\n\\n\\nSample Output:\\nCase #1: 20\\nCase #2: 28\\nCase #3: 42\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his yard, which can be represented on a Cartesian plane.\\n\\nBoss Rob has a primary well at point \\\\((A_x, A_y)\\\\) and a backup well at a different point \\\\((B_x, B_y)\\\\), each able to water trees within an \\\\(R\\\\) unit radius. Using \\\\(A_x\\\\), \\\\(A_y\\\\), \\\\(B_x\\\\), \\\\(B_y\\\\), and \\\\(R\\\\) (unknown integers to you), Rob plants \\\\(N\\\\) happy little trees at real number points obtained by \\\\(N\\\\) calls to the function:\\n\\n```\\ndef gen_one_tree(A_x, A_y, B_x, B_y, R):\\n while True:\\n r = random.uniform(0, R)\\n theta = random.uniform(0, 2*math.pi)\\n x = A_x + r*math.cos(theta)\\n y = A_y + r*math.sin(theta)\\n if (x - B_x)**2 + (y - B_y)**2 <= R*R:\\n return (x, y)\\n```\\nHere, `random.uniform(L, H)` returns a real number in \\\\([L, H)\\\\) uniformly at random.\\n\\nIn other words, he picks a point \\\\((x, y)\\\\) in the circular range of the primary well using the special method above. If \\\\((x, y)\\\\) happens to be in range of the backup well, he plants a tree there (else he discards it and tries again with a new \\\\((x, y)\\\\)). This repeats until Rob has planted \\\\(N\\\\) trees.\\n\\nGiven only the planted tree coordinates \\\\((X_1, Y_1), \\\\ldots, (X_N, Y_N)\\\\), you are tasked to predict the exact values of \\\\(A_x\\\\), \\\\(A_y\\\\), \\\\(B_x\\\\), \\\\(B_y\\\\), and \\\\(R\\\\). As you are new, Boss Rob will accept your solution if it correctly predicts at least \\\\(80\\\\%\\\\) of the test cases.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 1{,}000\\\\)\\n\\\\(500 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(0 \\\\le A_x, A_y, B_x, B_y \\\\le 50\\\\)\\n\\\\((A_x, A_y) \\\\ne (B_x, B_y)\\\\)\\n\\\\(1 \\\\le R \\\\le 50\\\\)\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(2{,}000{,}000\\\\).\\n\\nThe intersection area of the two circular regions is strictly positive.\\n\\nTree coordinates in the data were truly generated using the randomized algorithm as described above. The secret parameters \\\\(A_x\\\\), \\\\(A_y\\\\), \\\\(B_x\\\\), \\\\(B_y\\\\), and \\\\(R\\\\) have also been chosen uniformly at random for each case (rejecting cases where the circles are identical or do not have positive overlap).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\), the number of planted trees. Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated real numbers \\\\(X_i\\\\) and \\\\(Y_i\\\\), each given to \\\\(6\\\\) decimal places.\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"`, followed by the five space-separated integers \\\\(A_x\\\\), \\\\(A_y\\\\), \\\\(B_x\\\\), \\\\(B_y\\\\), and \\\\(R\\\\), in that order.\\n\\n\\n# Sample Explanation\\n\\nThe first sample case is pictured below, with the primary well\\'s range in red, the backup well\\'s range in blue, and the \\\\(500\\\\) randomly-generated trees in green:\\n\\n{{PHOTO_ID:6502772429739994|WIDTH:700}}\\n\\n\\n\\nSample Input:\\n2\\n500\\n18.243577 16.343618\\n24.560940 7.478552\\n13.664297 0.348593\\n19.766713 16.871980\\n14.052491 10.567715\\n21.426414 5.786941\\n20.495098 -0.246197\\n20.706538 14.324784\\n13.240629 9.591812\\n18.131521 1.645394\\n13.085966 5.206907\\n12.705525 2.340957\\n17.081302 -3.503313\\n17.406894 -4.132649\\n18.662161 14.841332\\n21.722473 13.853187\\n12.566132 7.860660\\n16.422074 12.005824\\n17.360418 15.791112\\n17.109026 7.669767\\n18.253835 6.466656\\n21.249345 1.667549\\n21.665745 0.311747\\n16.095203 14.907657\\n15.823431 -4.340649\\n14.764783 -3.979121\\n12.216653 2.449552\\n12.283698 8.764607\\n17.379537 14.086718\\n11.356151 4.039189\\n18.250737 10.348279\\n18.803695 -3.124892\\n17.208013 -1.383813\\n15.846419 7.176925\\n12.209281 2.863600\\n19.786703 12.537685\\n23.019766 11.553517\\n12.677512 3.801674\\n21.313409 10.926474\\n13.065841 -5.327402\\n23.974605 10.274300\\n14.052084 5.745476\\n12.344311 4.014428\\n13.824721 3.340018\\n21.333789 16.244543\\n22.993558 3.915954\\n21.482560 16.650764\\n21.423192 9.596803\\n17.471653 -2.073008\\n19.254667 10.037329\\n13.026102 2.910256\\n19.164159 15.152278\\n15.448711 5.268898\\n22.436500 6.103729\\n18.713233 -0.696114\\n17.319829 -4.614399\\n16.303121 6.662335\\n18.549615 9.323836\\n20.578796 13.197413\\n20.101191 10.285048\\n12.731656 6.398406\\n15.642672 12.203354\\n11.739751 -2.209279\\n19.999162 12.495027\\n13.332741 -0.781345\\n16.419368 4.930287\\n14.208240 6.146094\\n12.668535 3.884128\\n17.100494 9.477584\\n15.298066 2.105778\\n21.715575 3.461332\\n13.812862 -0.393683\\n15.197469 3.227296\\n17.805714 12.076443\\n14.894292 -2.076883\\n11.362856 0.971215\\n11.152936 -0.329086\\n24.312335 12.223521\\n23.850104 13.665174\\n13.611467 7.009552\\n22.958613 10.937624\\n17.476002 5.911470\\n13.385598 8.687516\\n15.245710 4.570724\\n17.698810 15.101508\\n14.683332 8.660956\\n17.830283 14.623531\\n11.844331 0.565127\\n22.538936 14.469856\\n14.570230 -2.940471\\n23.501721 7.793592\\n22.580591 6.388695\\n17.440296 5.839274\\n17.532098 6.355300\\n16.728496 6.375671\\n18.752338 6.307287\\n21.240144 10.501905\\n14.970281 7.859548\\n15.357577 -4.434922\\n16.310483 -1.394290\\n22.759885 9.828241\\n14.186619 5.960630\\n24.193644 14.368893\\n17.442571 12.499045\\n13.034105 -6.500657\\n17.697254 11.737589\\n13.480724 -3.930224\\n22.150737 17.126988\\n13.391916 -0.925885\\n23.572665 4.572828\\n15.420402 8.829672\\n16.683175 5.409393\\n20.887778 5.088725\\n13.519647 4.812447\\n17.129407 -1.925479\\n16.603673 3.593709\\n22.113753 16.325847\\n13.066364 2.073991\\n15.710264 8.778860\\n17.767049 8.993840\\n11.706371 -1.170487\\n11.470356 2.861039\\n17.231284 7.889578\\n24.189637 14.224167\\n19.075589 11.166930\\n20.937783 13.154556\\n23.523316 12.300135\\n14.445991 11.359735\\n19.239496 12.970156\\n21.742953 4.547351\\n12.910622 1.840402\\n21.075760 6.167560\\n13.244078 -2.969471\\n13.140408 -6.471153\\n20.315724 14.875012\\n23.818682 11.079752\\n17.887454 -2.232520\\n18.666566 0.092412\\n14.317916 -3.570190\\n15.472072 -2.829925\\n21.556983 0.279414\\n22.036423 1.708065\\n14.568483 0.688430\\n19.482935 8.690586\\n17.496132 11.691013\\n15.570701 5.436852\\n18.179237 -3.363174\\n14.980168 8.543218\\n14.915177 -0.184438\\n14.969153 0.104570\\n14.067303 7.651140\\n14.964894 0.975905\\n20.284008 6.167394\\n22.797939 6.211927\\n16.313922 14.186456\\n14.834579 12.685060\\n21.480677 2.065544\\n12.584922 3.150118\\n13.136840 0.802224\\n11.390730 -0.761440\\n17.002476 5.312136\\n20.846441 4.748559\\n17.902435 4.933033\\n14.915677 3.063146\\n14.466079 -3.505656\\n16.495423 -0.516658\\n15.123951 -1.633273\\n19.347161 16.123210\\n24.262812 13.836784\\n16.744183 13.521877\\n13.795204 4.941614\\n15.614470 12.670248\\n14.141935 9.959788\\n11.217067 4.675771\\n12.697649 -0.067208\\n19.514341 13.742676\\n16.637732 3.278594\\n21.107319 14.689994\\n17.380294 10.999330\\n20.672382 14.136265\\n11.533785 2.455757\\n21.634049 10.042583\\n19.865202 14.092486\\n22.647324 17.180916\\n13.916645 2.700255\\n21.097245 12.705439\\n14.797356 -2.723860\\n15.674174 -1.807078\\n21.891569 6.225470\\n18.340370 9.365718\\n17.377817 6.161102\\n15.117478 0.715163\\n18.782919 3.899382\\n13.107155 7.205792\\n18.844233 16.639468\\n18.316959 -1.084489\\n16.864189 6.381003\\n17.413163 -2.782152\\n18.155452 7.203062\\n22.653910 1.225094\\n15.276962 6.826973\\n19.467625 4.793109\\n21.273117 10.211036\\n13.053714 4.142984\\n11.594127 1.722787\\n17.187284 5.900224\\n20.751730 5.502650\\n19.298775 7.373680\\n20.090593 -1.788020\\n22.540876 13.519997\\n11.735849 2.496556\\n16.109133 14.156335\\n19.604004 11.711670\\n23.481796 4.654737\\n18.756305 8.135686\\n20.751269 0.800819\\n14.073437 7.240697\\n17.022144 7.532538\\n14.800061 10.052073\\n19.638047 3.327763\\n19.495916 9.221839\\n11.418687 -1.636822\\n16.627811 3.856185\\n22.189477 18.520143\\n19.746557 -1.725730\\n16.438720 10.266559\\n19.785090 6.069816\\n16.417114 13.062245\\n15.667475 9.619686\\n19.909511 5.066407\\n20.251697 17.418113\\n22.658481 18.376700\\n11.513801 1.356283\\n18.258572 2.014423\\n17.457905 7.408070\\n21.347081 10.562006\\n12.259695 0.465383\\n11.687451 2.106452\\n14.026429 -5.711334\\n15.410070 -4.322808\\n16.792314 0.508719\\n20.651046 7.219566\\n14.634011 10.399200\\n12.405734 -2.693974\\n14.716979 -3.275552\\n13.872859 -2.763565\\n22.064022 11.633544\\n15.729799 10.378849\\n19.310824 0.908302\\n11.290350 5.008839\\n17.987185 11.212424\\n18.933293 11.589430\\n24.225082 14.074144\\n14.280323 2.518275\\n23.126585 8.980823\\n15.261063 1.383509\\n22.828022 17.600195\\n12.454820 4.367434\\n18.535066 3.691446\\n20.314638 18.047548\\n23.145756 15.102703\\n13.296147 -6.688498\\n21.647239 1.718656\\n24.121364 15.219336\\n13.194050 8.691127\\n11.766890 5.884297\\n18.217592 14.226418\\n17.915127 2.176438\\n14.634272 -3.488542\\n20.234345 0.554279\\n17.188012 7.985024\\n17.346218 10.995344\\n18.669436 2.636653\\n18.951858 11.670444\\n13.285687 -1.499801\\n22.001488 14.067161\\n22.770735 7.750486\\n12.963328 5.944176\\n20.682994 15.335158\\n11.114365 0.432639\\n23.735128 15.517813\\n22.032538 6.398391\\n15.201702 -3.937585\\n18.687441 15.891635\\n14.161156 9.388260\\n13.996130 4.657658\\n22.323146 6.098112\\n20.934790 10.211547\\n16.590131 5.371991\\n20.039944 12.763506\\n14.652548 11.006175\\n19.154292 4.181016\\n17.199467 2.304127\\n14.206927 9.107424\\n19.143110 4.677616\\n21.210108 11.776738\\n22.448183 9.536389\\n24.468761 12.923552\\n19.753874 2.928480\\n15.879698 8.601265\\n21.579355 10.235260\\n12.337658 8.782207\\n16.823056 1.073752\\n23.213925 8.143511\\n18.283541 10.342998\\n21.109487 4.830167\\n20.576977 18.388817\\n17.305744 5.831552\\n15.129693 5.247627\\n16.948835 11.394945\\n11.708893 -1.237782\\n11.640301 0.735272\\n18.564261 7.765312\\n15.367225 12.715984\\n17.515852 13.448675\\n18.427577 -4.113174\\n16.063344 4.479923\\n19.850385 13.378851\\n17.990363 11.657032\\n17.696624 0.518123\\n20.326250 10.275394\\n16.374879 6.186846\\n12.339243 3.446059\\n19.054799 -1.407273\\n22.345393 0.656893\\n12.027935 -0.424062\\n23.649456 5.875965\\n16.363737 13.309612\\n15.567342 13.854572\\n19.403075 -2.105705\\n22.961203 12.299632\\n14.376183 7.683317\\n17.591174 13.661803\\n11.394050 0.175477\\n14.569772 4.742531\\n15.213890 2.957571\\n23.610500 15.936788\\n14.141149 1.670263\\n18.528925 11.541278\\n23.898984 9.211051\\n13.211419 10.291301\\n22.984657 14.996915\\n16.231052 -2.488522\\n20.056612 8.249936\\n11.894928 1.368478\\n24.280032 9.901244\\n13.543034 7.292310\\n21.466143 6.122634\\n17.050924 10.602068\\n14.559508 -2.832191\\n16.002019 14.311328\\n15.689925 -5.344107\\n18.137508 11.902096\\n21.828029 6.265151\\n12.626233 5.939640\\n19.495225 -2.888468\\n14.731963 5.949898\\n14.539116 3.761306\\n13.563525 5.118103\\n20.781815 4.269694\\n14.646917 3.059797\\n19.582913 9.167719\\n11.998265 6.262899\\n18.248830 15.023882\\n17.738511 -3.775898\\n20.613168 8.945199\\n11.812803 0.311242\\n24.163159 5.006179\\n21.371522 18.282868\\n14.865494 13.354208\\n18.606131 3.411418\\n20.882197 6.015836\\n21.219685 1.104721\\n17.830305 11.210971\\n19.458500 -0.184236\\n11.294817 1.942603\\n19.357381 11.114160\\n14.342764 -1.600163\\n13.078637 1.105681\\n16.311977 7.151695\\n18.727277 4.389283\\n20.787157 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52.528578\\n41.715208 51.600960\\n41.181149 53.685790\\n39.303489 53.892835\\n39.293418 48.153534\\n41.756249 51.245006\\n31.774363 44.426609\\n34.861405 46.757609\\n40.802245 48.921425\\n43.937823 48.149107\\n39.046799 51.495534\\n44.164015 47.377663\\n33.810651 55.003684\\n34.927154 47.447552\\n34.035538 44.333249\\n37.871520 39.762833\\n40.666335 49.469386\\n36.616461 47.193929\\n30.155009 55.226397\\n38.487555 53.622646\\n40.997810 42.076416\\n45.945407 44.422717\\n41.899604 45.437848\\n35.851069 39.610831\\n35.506742 46.914798\\n44.438356 53.799029\\n39.989213 47.854211\\n40.688131 48.384774\\n34.652201 50.303544\\n33.755665 45.964317\\n40.030597 38.511805\\n40.496887 49.662393\\n38.227873 56.650526\\n42.024271 50.307892\\n40.098551 51.020802\\n37.124217 56.020006\\n45.573360 49.079709\\n38.113194 47.452914\\n41.032388 49.568862\\n40.468636 41.700309\\n37.471402 53.415458\\n41.048122 50.915859\\n34.495824 52.399788\\n36.448848 47.859274\\n39.011571 45.145133\\n40.775737 54.882725\\n29.285397 47.497443\\n39.026980 50.093095\\n47.015660 53.027474\\n34.763263 40.445794\\n36.897710 46.648874\\n40.241097 46.761277\\n48.060150 52.452125\\n40.306719 47.029494\\n42.084311 49.727923\\n35.976952 56.782986\\n47.126106 49.162825\\n37.897456 51.564894\\n36.654695 50.819627\\n40.781800 53.945677\\n41.842433 53.752172\\n41.589129 53.110102\\n32.761789 44.088345\\n41.719804 53.989094\\n36.944823 45.053298\\n42.827828 46.842898\\n41.016473 46.911213\\n38.918312 54.211936\\n37.981049 47.710730\\n39.890346 50.385934\\n44.764060 50.488667\\n40.308209 49.871190\\n38.160808 41.818470\\n47.957911 43.210035\\n43.687500 51.766336\\n41.918077 50.059497\\n31.679299 54.296609\\n45.771261 49.218397\\n38.123103 57.954250\\n36.124360 38.699324\\n32.968935 45.593800\\n36.560694 51.382055\\n34.735064 40.361008\\n32.251581 42.107719\\n44.199665 41.877420\\n41.143788 50.764761\\n33.338016 46.664050\\n49.922917 42.219510\\n50.478030 44.841695\\n39.976942 50.064551\\n34.875631 50.503983\\n47.770352 41.098315\\n38.284620 54.466581\\n46.593736 53.092283\\n39.045259 51.162862\\n39.613480 49.903060\\n29.546920 50.066181\\n41.943758 43.196521\\n37.059391 50.779753\\n39.992419 49.911640\\n47.980034 43.365287\\n34.320640 40.515435\\n42.231814 47.944971\\n43.508937 52.307746\\n39.973595 46.291718\\n44.220332 52.161464\\n41.159619 40.973367\\n47.101236 49.724113\\n39.782978 50.204596\\n39.559919 49.480320\\n36.835299 48.650068\\n37.883515 44.698028\\n44.397487 41.980311\\n34.290696 48.708611\\n36.936342 50.120652\\n41.121951 52.657115\\n43.518312 44.024548\\n39.437155 52.745160\\n39.835412 46.283453\\n40.310665 48.125508\\n45.963017 48.159980\\n38.177037 51.293065\\n45.889694 55.310913\\n42.049010 49.604768\\n28.899711 52.311737\\n37.241460 44.649299\\n33.318865 47.628176\\n40.966198 51.913582\\n40.366353 49.362066\\n29.296032 49.797302\\n43.173336 52.250949\\n47.690825 49.396015\\n33.643206 43.968056\\n30.131878 46.189694\\n33.388653 54.541037\\n32.137200 45.597199\\n39.122132 49.672227\\n41.584771 56.233178\\n36.580879 44.724607\\n39.466947 51.197577\\n37.561921 47.022670\\n42.761769 48.915168\\n39.787125 44.289218\\n45.484312 43.809723\\n41.811338 48.622222\\n37.213599 49.935951\\n39.942045 49.772859\\n49.891856 42.872467\\n31.964570 45.776619\\n40.171680 49.987490\\n37.038137 52.940678\\n35.265299 47.478347\\n44.996427 49.981928\\n39.734830 54.125940\\n40.304740 53.221447\\n38.560754 46.599124\\n39.647909 51.223265\\n36.609585 37.915030\\n34.542710 46.975530\\n40.441693 50.006388\\n34.944957 44.328503\\n47.217304 47.688772\\n38.580101 51.789895\\n41.506001 49.071389\\n42.049850 50.259588\\n34.832636 47.585776\\n42.867048 45.434276\\n\\n\\nSample Output:\\nCase #1: 6 10 30 2 19\\nCase #2: 40 50 38 45 13\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", + "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*Bejeweled*™ is a classic puzzle game where the player tries to match three-in-a-row in a 2D grid of tiles by swapping pairs of adjacent tiles.\\n\\n*Isblinged*™ is Hacker Cup\\'s spinoff, played on a grid of \\\\(R\\\\) rows by \\\\(C\\\\) columns of tiles. The tile at \\\\((i, j)\\\\) is of an integer type \\\\(G_{i,j}\\\\). A *group* refers to three or more tiles of the same type, connected directly or indirectly via the four orthogonal directions (left, right, up, and down). Initially, the given grid \\\\(G\\\\) may already contain groups.\\n\\nThe player may swap two orthogonally adjacent tiles *of different types*. If the swap results in either tile being in a group of three or more tiles, then all tiles in the *newly-formed group(s)* are cleared.\\n\\nFor the example below (sample case 2), swapping the first two tiles in row \\\\(1\\\\) would clear \\\\(9\\\\) tiles:\\n\\n{{PHOTO_ID:1310493279783732|WIDTH:700}}\\n\\nOn the other hand, swapping the first two tiles in the second column would clear \\\\(12\\\\) tiles;\\n\\n{{PHOTO_ID:650302073463012|WIDTH:700}}\\n\\nNote that the type-3 group is not cleared because it doesn\\'t contain a swapped tile.\\n\\nPlease find the *sum* of tiles that would be cleared over all possible theoretical swaps of ordered pairs of tiles \\\\(G_{i_1, j_1}\\\\) and \\\\(G_{i_2, j_2}\\\\) such that \\\\(|i_1 - i_2| + |j_1 - j_2| = 1\\\\) and \\\\(G_{i_1, j_1} \\\\ne G_{i_2, j_2}\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 105\\\\)\\n\\\\(1 \\\\le R, C \\\\le 3000\\\\)\\n\\\\(1 \\\\le G_{i,j} \\\\le 10^9\\\\)\\n\\nThe sum of \\\\(R*C\\\\) across all test cases is at most \\\\(40{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) space-separated integers \\\\(G_{i,1..C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing `\"Case #i: \"` followed by a single integer, the sum of tiles that can be cleared with a single swap.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case:\\n- \\\\(G_{1,1}\\\\), \\\\(G_{1,3}\\\\), and \\\\(G_{2,3}\\\\) can\\'t be swapped with anything that leads to any tiles being cleared.\\n- \\\\(G_{1,2}\\\\) can be swapped with the type-2 tile below to clear \\\\(3\\\\) tiles.\\n- \\\\(G_{2,1}\\\\) can be swapped with the type-2 tile right of it to clear \\\\(3\\\\) tiles.\\n- \\\\(G_{2,2}\\\\) can be swapped with either the type-1 cell above or to the left, each clearing \\\\(3\\\\) tiles.\\n\\nThe answer is therefore \\\\(3 + 3 + 2*3 = 12\\\\).\\n\\nIn the second sample case, the unordered pairs of swaps clearing non-zero numbers of tiles are:\\n- \\\\(G_{1,1} \\\\leftrightarrow G_{1,2} \\\\): clearing \\\\(9\\\\) tiles\\n- \\\\(G_{1,2} \\\\leftrightarrow G_{1,3} \\\\): clearing \\\\(5 + 3 = 8\\\\) tiles\\n- \\\\(G_{1,2} \\\\leftrightarrow G_{2,1} \\\\): clearing \\\\(9 + 3 = 12\\\\) tiles\\n- \\\\(G_{1,3} \\\\leftrightarrow G_{2,3} \\\\): clearing \\\\(5\\\\) tiles\\n- \\\\(G_{1,4} \\\\leftrightarrow G_{2,4} \\\\): clearing \\\\(4\\\\) tiles\\n- \\\\(G_{2,2} \\\\leftrightarrow G_{2,3} \\\\): clearing \\\\(6\\\\) tiles\\n- \\\\(G_{2,2} \\\\leftrightarrow G_{3,2} \\\\): clearing \\\\(4\\\\) tiles\\n- \\\\(G_{2,4} \\\\leftrightarrow G_{2,5} \\\\): clearing \\\\(4\\\\) tiles\\n- \\\\(G_{3,1} \\\\leftrightarrow G_{3,2} \\\\): clearing \\\\(4\\\\) tiles\\n\\nDoubling the sum for ordered pairs, we get \\\\(2*(9+8+12+5+4+6+4+4+4) = 112\\\\).\\n\\n\\n\\nSample Input:\\n3\\n2 3\\n1 1 2\\n1 2 3\\n3 5\\n1 2 1 1 1\\n1 1 2 2 1\\n1 3 3 3 1\\n4 4\\n1 1 1 4\\n1 2 1 1\\n3 2 3 4\\n3 2 2 2\\n\\n\\nSample Output:\\nCase #1: 12\\nCase #2: 112\\nCase #3: 74\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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 problem_statementsample_inputsample_outputexample_solutionpred_solutionvalidate_solution
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", + "5 3\n", + "6 3\n", + "6 2\n", + "243447 42273\n", + "Case #1: 1\n", + "Case #2: 2\n", + "Case #3: 4\n", + "Case #4: 4\n", + "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...The solution can be structured as follows: 1. **Calculate the Number of Distinct Bracelets**: - Implement the formula for counting distinct bracelets based on \\(N\\)...False
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", + "Case #2: 28\n", + "Case #3: 42\n", + "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...1. **Input Reading**: Read the number of test cases \\(T\\). For each test case, read the number of trees \\(N\\) and their coordinates. 2. **Sorting**:...False
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", + "Case #2: 2 1 6 4\n", + "Case #3: 1 1 2\n", + "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...To implement the solution, we can follow these steps: 1. **Data Structures**: - Use an adjacency list to represent the tree, where each node points...False
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", + "3 1 4\n", + "0 0\n", + "2 2\n", + "4 0\n", + "5 1 4\n", + "0 0\n", + "1 2\n", + "0 3\n", + "4 3\n", + "3 1\n", + "Case #1: 0.1169663730642699\n", + "Case #2: 0.1353445414060363\n", + "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...To solve the problem, we will follow these steps: 1. **Input Parsing**: Read the number of test cases \\(T\\). For each test case, read \\(N\\),...False
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", + "Case #2: 40 50 38 45 13\n", + "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...1. Calculate the centroid of the tree coordinates to estimate \\((A_x, A_y)\\). 2. For each tree, compute the distance from the centroid to the tree's...False
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To upgrade, please run:\n", + " $ pip install weave --upgrade\n", + "Logged in as Weights & Biases user: capecape.\n", + "View Weave data at https://wandb.ai/capecape/dspy_hackercup/weave\n" + ] + } + ], "source": [ "import weave\n", "weave.init(\"dspy_hackercup\")\n", "\n", "MODEL = \"mistral-large-latest\"\n", - "BASE_URL = \"http://195.242.25.198:8000/v1\"\n", - "API_KEY = \"dummy_key\"\n", "\n", - "lm = dspy.OpenAI(\n", - " model=MODEL,\n", - " api_base=BASE_URL,\n", - " api_key=API_KEY,\n", - " max_tokens=4000,\n", - " temperature=0.1,\n", - ")\n", + "from dsp import LM\n", + "from typing import Any\n", + "import json\n", + "from mistralai import Mistral # requires mistralai>1.0\n", + "\n", + "class MistralLM(LM):\n", + "\n", + " def __init__(self, model, api_key, **kwargs):\n", + " self.model = model\n", + " self.api_key = api_key\n", + " self.client = Mistral(api_key=api_key)\n", + " self.kwargs = {\n", + " \"temperature\": 0.17,\n", + " \"max_tokens\": 150,\n", + " **kwargs,\n", + " }\n", + " self.history: list[dict[str, Any]] = []\n", + " self.provider = \"mistralai\"\n", + "\n", + " def basic_request(self, prompt: str, **kwargs):\n", + "\n", + " # Mistral disallows \"n\" arguments\n", + " n = kwargs.pop(\"n\", None)\n", + " if n is not None and n > 1 and kwargs[\"temperature\"] == 0.0:\n", + " kwargs[\"temperature\"] = 0.7\n", + "\n", + " chat_response = self.client.chat.complete(\n", + " model=self.model,\n", + " messages=[\n", + " dict(role=\"user\", content=prompt)\n", + " ],\n", + " **kwargs\n", + " )\n", + " response_dict = json.loads(chat_response.model_dump_json())\n", "\n", - "# lm = dspy.OpenAI(\n", - "# model=\"gpt-4o-mini\", # Note: didn't find much a difference btwn mini & full gpt-4o\n", - "# max_tokens=4000,\n", - "# temperature=0.1,\n", - "# )\n", + " self.history.append({\n", + " \"prompt\": prompt,\n", + " \"response\": response_dict, #{\"choices\": chat_response.choices[0].message.content},\n", + " \"kwargs\": kwargs,\n", + " })\n", "\n", - "dspy.settings.configure(lm=lm)\n", + " return chat_response\n", + "\n", + " def __call__(self, prompt, only_completed=True, return_sorted=False, **kwargs):\n", + " response = self.request(prompt, **kwargs)\n", + " completions = [response.choices[0].message.content] \n", + " return completions\n", + "\n", + " def _get_choice_text(self, choice: dict[str, Any]) -> str:\n", + "\n", + " return choice[\"message\"][\"content\"]" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "mistral = MistralLM(model=MODEL, api_key=os.environ[\"MISTRAL_API_KEY\"])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's test the endpoint" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffeb-4003-7b91-85d4-adf0c39dd398\n" + ] + }, + { + "data": { + "text/plain": [ + "['Hello! How are you today? How can I assist you? 😊']" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "mistral(\"hello\")" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "dspy.settings.configure(lm=mistral)\n", "dspy.configure(experimental=True)\n", "\n", "\n", - "def validate_solution(example, prediction, trace=None,frac=0.8):\n", + "@weave.op\n", + "def validate_solution(example, prediction, trace=None, frac=0.8):\n", " \n", " result = answer_exact_match(\n", " example=Example(answer=example.solution),\n", @@ -191,7 +288,11 @@ " trace=trace, \n", " frac=frac)\n", " return result\n", - " \n", + "\n", + "from dsp.utils import F1\n", + "\n", + "def f1(example, prediction, trace=None,):\n", + " return F1(example.solution, [prediction.solution])\n", "\n", "# Setup evaluation function\n", "evaluate = Evaluate(\n", @@ -199,17 +300,66 @@ " num_threads=6, # Note: Set this to 1 for debugging purposes \n", " display_progress=True,\n", " display_table=5,\n", - " metric=validate_solution\n", + " metric=f1\n", ")" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let's understand dspy.answer_match and the underlying F1 metric" + ] + }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 21, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ + "from dspy import dsp\n", + "\n", "\n", + "dsp.answer_match(\"You are helpful\", [\"You are not helpful in you work\"], frac=0.8)" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.6" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F1(\"You are helpful\", [\"You are not helpful in you work\"])" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [], + "source": [ "class GenerateSolution(Signature):\n", " \"\"\"You are an expert problem solver. Your task is to solve the problem at hand.\n", "\n", @@ -233,16 +383,23 @@ }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 13, "metadata": {}, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffeb-8ffd-7851-8c22-73766c3832e0\n" + ] + }, { "data": { "text/plain": [ - "'1. **Input Parsing**: Read the number of test cases. For each test case, read the number of tries and their respective structures (nodes and edges).\\n\\n2. **DFS for String Extraction**:\\n\\n- For each trie, initialize a set to store unique strings.\\n\\n- Perform a DFS from the root node, appending characters to form strings as you traverse down to each node. Add each formed string to the set.\\n\\n3. **Calculate Unique Strings for Triplets**:\\n\\n- For each combination of three tries (using combinations from itertools), compute the union of their string sets.\\n\\n- Count the number of unique strings in this union and add it to a running total for the test case.\\n\\n4. **Output the Result**: After processing all triplets for a test case, format the output as specified.\\n\\nPseudocode:\\n\\n```\\n\\nfor each test_case in T:\\n\\nread N\\n\\ntries = []\\n\\nfor i from 1 to N:\\n\\nread M_i\\n\\ntrie = {}\\n\\nfor j from 2 to M_i:\\n\\nread P_{i,j}, C_{i,j}\\n\\nadd edge P_{i,j} -> j with label C_{i,j} to trie\\n\\ntries.append(trie)\\n\\nunique_strings = []\\n\\nfor trie in tries:\\n\\nstrings_set = set()\\n\\ndfs(trie.root, \"\", strings_set)\\n\\nunique_strings.append(strings_set)\\n\\ntotal_count = 0\\n\\nfor (i, j, k) in combinations(range(N), 3):\\n\\ncombined_set = unique_strings[i] | unique_strings[j] | unique_strings[k]\\n\\ntotal_count += len(combined_set)\\n\\nprint(\"Case #{}: {}\".format(test_case + 1, total_count))\\n\\n```\\n\\nThis approach ensures that we efficiently gather and combine strings from the tries while adhering to the constraints provided. The use of sets allows for quick union operations, making the solution scalable within the given limits.'" + "\"### Problem Statement:\\nYou have \\\\(N\\\\) tries. The \\\\(i\\\\)th trie has \\\\(M_i\\\\) nodes and \\\\(M_i - 1\\\\) edges, with node \\\\(1\\\\) being the root, and node \\\\(j\\\\)'s parent being \\\\(P_{i,j}\\\\), for \\\\(j = 2..\"" ] }, - "execution_count": 17, + "execution_count": 13, "metadata": {}, "output_type": "execute_result" } @@ -257,7 +414,7 @@ }, { "cell_type": "code", - "execution_count": 18, + "execution_count": 24, "metadata": {}, "outputs": [], "source": [ @@ -278,20 +435,7 @@ }, { "cell_type": "code", - "execution_count": 19, - "metadata": {}, - "outputs": [], - "source": [ - "# prediction = SimpleGenerateSolution()(problem_statement=trainset[0].problem_statement, \n", - "# sample_input=trainset[0].sample_input, \n", - "# sample_output=trainset[0].sample_output)\n", - "\n", - "# print(prediction.solution)" - ] - }, - { - "cell_type": "code", - "execution_count": 20, + "execution_count": 25, "metadata": {}, "outputs": [], "source": [ @@ -300,92 +444,177 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 6 (0.0): 100%|██████████| 6/6 [00:08<00:00, 1.38s/it]\n" + "Average Metric: 0.048109965635738834 / 1 (4.8): 17%|█▋ | 1/6 [00:36<03:00, 36.15s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-9f54-7af2-96b4-c4831b230b62\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.15945017182130586 / 2 (8.0): 33%|███▎ | 2/6 [00:40<01:09, 17.41s/it] " + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-9f4b-7730-afe5-0d096b512415\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.4336640082992933 / 3 (14.5): 50%|█████ | 3/6 [00:46<00:36, 12.13s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-a058-7bf0-ace1-354f3071ce1a\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.5001091910235458 / 4 (12.5): 67%|██████▋ | 4/6 [01:00<00:25, 12.92s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-9ff6-7b52-b5c2-65a688a07541\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.5373762717688874 / 5 (10.7): 83%|████████▎ | 5/6 [01:09<00:11, 11.56s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-9ee7-77c0-90c0-ef7909f54c1e\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.5900078507162558 / 6 (9.8): 100%|██████████| 6/6 [01:14<00:00, 12.46s/it] " + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-a01b-7c02-af04-9ed6236f10d2\n", + "Average Metric: 0.5900078507162558 / 6 (9.8%)\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\n" ] }, { "data": { "text/html": [ "\n", - "\n", + "
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 problem_statementsample_inputsample_outputexample_solutionpred_solutionvalidate_solutionproblem_statementsample_inputsample_outputexample_solutionpred_solutionf1
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", + " 0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", "5 3\n", "6 3\n", "6 2\n", "243447 42273\n", "Case #1: 1\n", + " Case #1: 1\n", "Case #2: 2\n", "Case #3: 4\n", "Case #4: 4\n", "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...To solve the problem, we will follow these steps: 1. For each test case, read \\(N\\) and \\(K\\). 2. Calculate the number of distinct bracelets...FalseFirst we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...### Problem Statement: Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with...0.11134020618556702
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", + " 1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", "Case #2: 28\n", "Case #3: 42\n", "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...To implement the solution, we will follow these steps: 1. Read the number of test cases \\(T\\). 2. For each test case: - Read the...FalseFirst, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...### Problem Statement: Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer...0.048109965635738834
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", + " 2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", "Case #2: 2 1 6 4\n", "Case #3: 1 1 2\n", "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...1. **Graph Representation**: Use an adjacency list to represent the tree. Each node will have a list of tuples containing the connected node and the...FalseThe first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...### Problem Statement: Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\)...0.05263157894736842
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", + " 3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", "3 1 4\n", "0 0\n", "2 2\n", @@ -397,29 +626,29 @@ "4 3\n", "3 1\n", "Case #1: 0.1169663730642699\n", + " Case #1: 0.1169663730642699\n", "Case #2: 0.1353445414060363\n", "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...To solve this problem, we can follow these steps: 1. **Input Parsing**: Read the number of test cases \\(T\\). For each test case, read \\(N\\),...FalseLet's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...1. **Understand the Geometry**: - The problem involves a polygon defined by \\(N\\) edges. - The cup is modeled as a circle with radius \\(R\\)....0.27421383647798747
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", + " 4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", "Case #2: 40 50 38 45 13\n", "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...To solve this problem, we will follow these steps: 1. **Calculate the Centroid**: For each test case, compute the centroid of the tree coordinates: \\[...FalseThere are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...### Solution Explanation To solve the problem of predicting the exact values of \\(A_x\\), \\(A_y\\), \\(B_x\\), \\(B_y\\), and \\(R\\) based on the planted tree coordinates...0.06644518272425248
\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -446,13 +675,20 @@ "metadata": {}, "output_type": "display_data" }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffec-9c77-70d2-9b21-cfe2f867bb1b\n" + ] + }, { "data": { "text/plain": [ - "0.0" + "9.83" ] }, - "execution_count": 21, + "execution_count": 16, "metadata": {}, "output_type": "execute_result" } @@ -463,43 +699,15 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 26, "metadata": {}, "outputs": [], "source": [ - "def optimize_with_mipro(program, prompt_model, task_model, metric, trainset):\n", - " teleprompter = MIPROv2(\n", - " prompt_model=prompt_model,\n", - " task_model=task_model,\n", - " metric=metric,\n", - " num_candidates=5,\n", - " init_temperature=0.5,\n", - " verbose=False,\n", - " )\n", - "\n", - " optimized_program = teleprompter.compile(\n", - " program.deepcopy(),\n", - " trainset=trainset,\n", - " eval_kwargs=dict(num_threads=2),\n", - " max_bootstrapped_demos=0, # 0-shot optimization\n", - " max_labeled_demos=0,\n", - " num_batches=20,\n", - " minibatch=False, # turning this off bc we have a small trainset already\n", - " seed=9\n", - " )\n", - " now = datetime.now()\n", - " date_time = now.strftime(\"%Y%m%d_%H%M%S\")\n", - "\n", - " optimized_program.save(f\"mipro_optimized_{date_time}\")\n", - "\n", - " return optimized_program\n", - "\n", - "\n", "def optimize_with_bootstrap_fewshot(program, task_model, teacher_model, metric, trainset):\n", " rs_optimizer = BootstrapFewShotWithRandomSearch(\n", " metric=metric,\n", " num_threads=8,\n", - " num_candidate_programs=5,\n", + " num_candidate_programs=2,\n", " max_labeled_demos=0,\n", " max_bootstrapped_demos=2,\n", " max_errors=10000,\n", @@ -522,7 +730,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": 27, "metadata": {}, "outputs": [ { @@ -530,8390 +738,863 @@ "output_type": "stream", "text": [ "Going to sample between 1 and 2 traces per predictor.\n", - "Will attempt to bootstrap 5 candidate sets.\n" + "Will attempt to train 2 candidate sets.\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [06:47<00:00, 9.26s/it] \n" + "Average Metric: 0.07500000000000001 / 1 (7.5): 20%|██ | 1/5 [00:37<02:29, 37.27s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "New best sscore: 0.0 for seed -3\n", - "Scores so far: [0.0]\n", - "Best score: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e04-73b2-acfe-8b880075dd5e\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2737.84it/s]\n" + "Average Metric: 0.137015503875969 / 2 (6.9): 40%|████ | 2/5 [00:44<00:57, 19.32s/it] " ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0]\n", - "Best score: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e0a-7ff3-9de2-e6a8fe141e02\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 537.60it/s]\n" + "Average Metric: 0.21108957795004307 / 3 (7.0): 60%|██████ | 3/5 [00:46<00:23, 11.84s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e59-7340-95d3-dd0941b5bc48\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2751.22it/s]\n" + "Average Metric: 0.2511898285766095 / 4 (6.3): 80%|████████ | 4/5 [00:53<00:09, 9.85s/it] " ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e44-7bf1-81b9-4e00765efc35\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 1923.25it/s]\n" + "Average Metric: 0.37809338187610186 / 5 (7.6): 100%|██████████| 5/5 [00:55<00:00, 11.10s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e17-7081-8603-892d4626cda9\n", + "Average Metric: 0.37809338187610186 / 5 (7.6%)\n", + "Score: 7.56 for set: [0]\n", + "New best score: 7.56 for seed -3\n", + "Scores so far: [7.56]\n", + "Best score: 7.56\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 1999.19it/s]\n" + "Average Metric: 0.04827586206896552 / 1 (4.8): 20%|██ | 1/5 [00:36<02:25, 36.48s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920000-17ab-7233-96fe-19634cd58e19\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 444.09it/s]\n" + "Average Metric: 0.12327586206896553 / 2 (6.2): 40%|████ | 2/5 [00:39<00:49, 16.61s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920000-176b-7121-911d-37c5b91f07a6\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2338.94it/s]\n" + "Average Metric: 0.213752052545156 / 3 (7.1): 60%|██████ | 3/5 [00:44<00:23, 11.67s/it] " ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920000-1798-7491-b894-c16b1f9c392e\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 2258.04it/s]\n" + "Average Metric: 0.275767556421125 / 4 (6.9): 80%|████████ | 4/5 [00:45<00:07, 7.45s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920000-1778-7ec1-a55a-774fccab57e5\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2755.17it/s]\n" + "Average Metric: 0.6931345031998364 / 5 (13.9): 100%|██████████| 5/5 [01:03<00:00, 12.74s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920000-177e-7003-b187-b8605ffc6469\n", + "Average Metric: 0.6931345031998364 / 5 (13.9%)\n", + "Score: 13.86 for set: [0]\n", + "New best score: 13.86 for seed -2\n", + "Scores so far: [7.56, 13.86]\n", + "Best score: 13.86\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 2107.47it/s]\n" + " 40%|████ | 2/5 [01:09<01:44, 34.83s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "Bootstrapped 2 full traces after 3 examples in round 0.\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 2674.63it/s]\n" + "Average Metric: 0.054187192118226604 / 1 (5.4): 20%|██ | 1/5 [00:55<03:40, 55.10s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920002-20de-7121-b53e-b192df311ab2\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 44/44 [00:00<00:00, 1814.57it/s]\n" + "Average Metric: 0.1291871921182266 / 2 (6.5): 40%|████ | 2/5 [00:58<01:14, 24.89s/it] " ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920002-20ab-7fd0-a3fc-e400b52468c9\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 44 (0.0): 100%|██████████| 44/44 [00:00<00:00, 4006.98it/s]\n" + "Average Metric: 0.1760621921182266 / 3 (5.9): 60%|██████ | 3/5 [01:00<00:28, 14.44s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Score: 0.0 for set: [0]\n", - "Scores so far: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n", - "Best score: 0.0\n", - "Average of max per entry across top 1 scores: 0.0\n", - "Average of max per entry across top 2 scores: 0.0\n", - "Average of max per entry across top 3 scores: 0.0\n", - "Average of max per entry across top 5 scores: 0.0\n", - "Average of max per entry across top 8 scores: 0.0\n", - "Average of max per entry across top 9999 scores: 0.0\n", - "8 candidate programs found.\n", - "[('generate_code', Predict(StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")))]\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920002-20bd-78d2-8144-bbf21a2cf8f5\n" ] - } - ], - "source": [ - "optimized_program = optimize_with_bootstrap_fewshot(simple_program, lm, lm, validate_solution, trainset)" - ] - }, - { - "cell_type": "code", - "execution_count": 24, - "metadata": {}, - "outputs": [ + }, { "name": "stderr", "output_type": "stream", "text": [ - "Average Metric: 0 / 6 (0.0): 100%|██████████| 6/6 [00:00<00:00, 3886.61it/s]\n" + "Average Metric: 0.2380776959941956 / 4 (6.0): 80%|████████ | 4/5 [02:26<00:42, 42.44s/it]" ] }, { - "data": { - "text/html": [ - "\n", - "\n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - " \n", - "
 problem_statementsample_inputsample_outputexample_solutionpred_solutionvalidate_solution
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", - "5 3\n", - "6 3\n", - "6 2\n", - "243447 42273\n", - "Case #1: 1\n", - "Case #2: 2\n", - "Case #3: 4\n", - "Case #4: 4\n", - "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...To solve the problem, we will follow these steps: 1. For each test case, read \\(N\\) and \\(K\\). 2. Calculate the number of distinct bracelets...False
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", - "Case #2: 28\n", - "Case #3: 42\n", - "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...To implement the solution, we will follow these steps: 1. Read the number of test cases \\(T\\). 2. For each test case: - Read the...False
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", - "Case #2: 2 1 6 4\n", - "Case #3: 1 1 2\n", - "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...1. **Graph Representation**: Use an adjacency list to represent the tree. Each node will have a list of tuples containing the connected node and the...False
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", - "3 1 4\n", - "0 0\n", - "2 2\n", - "4 0\n", - "5 1 4\n", - "0 0\n", - "1 2\n", - "0 3\n", - "4 3\n", - "3 1\n", - "Case #1: 0.1169663730642699\n", - "Case #2: 0.1353445414060363\n", - "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...To solve this problem, we can follow these steps: 1. **Input Parsing**: Read the number of test cases \\(T\\). For each test case, read \\(N\\),...False
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", - "Case #2: 40 50 38 45 13\n", - "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...To solve this problem, we will follow these steps: 1. **Calculate the Centroid**: For each test case, compute the centroid of the tree coordinates: \\[...False
\n" - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920002-20b1-7d21-9d58-111a58299815\n" + ] }, { - "data": { - "text/html": [ - "\n", - "
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\n", - " " - ], - "text/plain": [ - "" - ] - }, - "metadata": {}, - "output_type": "display_data" + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.2856967436132432 / 5 (5.7): 100%|██████████| 5/5 [04:18<00:00, 51.78s/it]\n" + ] }, { - "data": { - "text/plain": [ - "0.0" - ] - }, - "execution_count": 24, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "evaluate(program=optimized_program, devset=devset)" - ] - }, - { - "cell_type": "code", - "execution_count": 28, - "metadata": {}, - "outputs": [ + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920002-20e3-7d51-8a6a-90f77e176ee6\n", + "Average Metric: 0.2856967436132432 / 5 (5.7%)\n", + "Score: 5.71 for set: [2]\n", + "Scores so far: [7.56, 13.86, 5.71]\n", + "Best score: 13.86\n", + "Average of max per entry across top 1 scores: 0.13862690063996727\n", + "Average of max per entry across top 2 scores: 0.14397172822617418\n", + "Average of max per entry across top 3 scores: 0.14397172822617418\n", + "Average of max per entry across top 5 scores: 0.14397172822617418\n", + "Average of max per entry across top 8 scores: 0.14397172822617418\n", + "Average of max per entry across top 9999 scores: 0.14397172822617418\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + " 40%|████ | 2/5 [01:46<02:39, 53.12s/it]\n" + ] + }, { "name": "stdout", "output_type": "stream", "text": [ - "\u001b[93m\u001b[1mWARNING: Projected Language Model (LM) Calls\u001b[0m\n", - "\n", - "Please be advised that based on the parameters you have set, the maximum number of LM calls is projected as follows:\n", - "\n", - "\n", - "\u001b[93m- Prompt Model: \u001b[94m\u001b[1m10\u001b[0m\u001b[93m data summarizer calls + \u001b[94m\u001b[1m5\u001b[0m\u001b[93m * \u001b[94m\u001b[1m1\u001b[0m\u001b[93m lm calls in program + (\u001b[94m\u001b[1m2\u001b[0m\u001b[93m) lm calls in program aware proposer = \u001b[94m\u001b[1m17\u001b[0m\u001b[93m prompt model calls\u001b[0m\n", - "\u001b[93m- Task Model: \u001b[94m\u001b[1m44\u001b[0m\u001b[93m examples in train set * \u001b[94m\u001b[1m20\u001b[0m\u001b[93m batches * \u001b[94m\u001b[1m# of LM calls in your program\u001b[0m\u001b[93m = (\u001b[94m\u001b[1m880 * # of LM calls in your program\u001b[0m\u001b[93m) task model calls\u001b[0m\n", - "\n", - "\u001b[93m\u001b[1mEstimated Cost Calculation:\u001b[0m\n", - "\n", - "\u001b[93mTotal Cost = (Number of calls to task model * (Avg Input Token Length per Call * Task Model Price per Input Token + Avg Output Token Length per Call * Task Model Price per Output Token) \n", - " + (Number of calls to prompt model * (Avg Input Token Length per Call * Task Prompt Price per Input Token + Avg Output Token Length per Call * Prompt Model Price per Output Token).\u001b[0m\n", - "\n", - "For a preliminary estimate of potential costs, we recommend you perform your own calculations based on the task\n", - "and prompt models you intend to use. If the projected costs exceed your budget or expectations, you may consider:\n", - "\n", - "\u001b[93m- Reducing the number of trials (`num_batches`), the size of the trainset, or the number of LM calls in your program.\u001b[0m\n", - "\u001b[93m- Using a cheaper task model to optimize the prompt.\u001b[0m\n", - "To proceed with the execution of this program, please confirm by typing \u001b[94m'y'\u001b[0m for yes or \u001b[94m'n'\u001b[0m for no.\n", - "\n", - "If you would like to bypass this confirmation step in future executions, set the \u001b[93m`requires_permission_to_run`\u001b[0m flag to \u001b[93m`False` when calling compile.\u001b[0m\n", - "\n", - "\u001b[93mAwaiting your input...\u001b[0m\n", - "\n" + "Bootstrapped 2 full traces after 3 examples in round 0.\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "DEBUG:root:OpenAI Response Token Usage: 21435\n", - "DEBUG:root:OpenAI Response Token Usage: 16411\n", - "DEBUG:root:OpenAI Response Token Usage: 17230\n", - "DEBUG:root:OpenAI Response Token Usage: 18132\n", - "DEBUG:root:OpenAI Response Token Usage: 6955\n", - "DEBUG:root:OpenAI Response Token Usage: 1978\n" + "Average Metric: 0.08955223880597014 / 1 (9.0): 20%|██ | 1/5 [00:37<02:29, 37.32s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "b: 10\n", - "b: 20\n", - "b: 30\n", - "b: 40\n", - "summary: Prediction(\n", - " summary='The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.'\n", - ")\n", - "DATA SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920007-b350-7fd1-a02e-1523b85f0b74\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - " 0%| | 0/44 [00:00 rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\nPlease provide the output field SUMMARY OF PROGRAM ABOVE. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field SUMMARY OF PROGRAM ABOVE.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.connection:close.started\n", - "DEBUG:httpcore.connection:close.complete\n", - "DEBUG:httpcore.connection:connect_tcp.started host='api.openai.com' port=443 local_address=None timeout=5.0 socket_options=None\n" + "Average Metric: 0.9547673869593165 / 5 (19.1): 100%|██████████| 5/5 [06:01<00:00, 72.21s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Bootstrapped 0 full traces after 44 examples in round 0.\n", - "Using a randomly generated configuration for our grounded proposer.\n", - "Selected tip: description\n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920007-b369-7102-b91a-a2ca80440bf0\n", + "Average Metric: 0.9547673869593165 / 5 (19.1%)\n", + "Score: 19.1 for set: [2]\n", + "New best score: 19.1 for seed 0\n", + "Scores so far: [7.56, 13.86, 5.71, 19.1]\n", + "Best score: 19.1\n", + "Average of max per entry across top 1 scores: 0.1909534773918633\n", + "Average of max per entry across top 2 scores: 0.19715426110866333\n", + "Average of max per entry across top 3 scores: 0.19715426110866333\n", + "Average of max per entry across top 5 scores: 0.19715426110866333\n", + "Average of max per entry across top 8 scores: 0.19715426110866333\n", + "Average of max per entry across top 9999 scores: 0.19715426110866333\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "DEBUG:httpcore.connection:connect_tcp.complete return_value=\n", - "DEBUG:httpcore.connection:start_tls.started ssl_context= server_hostname='api.openai.com' timeout=5.0\n", - "DEBUG:httpcore.connection:start_tls.complete return_value=\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:41 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'828'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149995123'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'1ms'), (b'x-request-id', b'req_5b634f8bead26cf055648b1bdea5f3cd'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d51bfbd11b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:41 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '828', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149995123', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '1ms', 'x-request-id': 'req_5b634f8bead26cf055648b1bdea5f3cd', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d51bfbd11b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_5b634f8bead26cf055648b1bdea5f3cd\n", - "DEBUG:root:OpenAI Response Token Usage: 810\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'Below is some pseudo-code for a pipeline that solves tasks with calls to language models. Please describe the purpose of one of the specified module in this pipeline.\\n\\n---\\n\\nFollow the following format.\\n\\nPROGRAM CODE: Pseudocode for a language model program designed to solve a particular task.\\n\\nEXAMPLE OF PROGRAM IN USE: An example of the program in use.\\n\\nSUMMARY OF PROGRAM ABOVE: Summary of the task the program is designed to solve, and how it goes about solving it.\\n\\nMODULE: The module in the program that we want to describe.\\n\\nMODULE DESCRIPTION: Description of the module\\'s role in the broader program.\\n\\n---\\n\\nPROGRAM CODE:\\nStringSignature(problem_statement, sample_input, sample_output -> rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\n\\nEXAMPLE OF PROGRAM IN USE: \\n\\nSUMMARY OF PROGRAM ABOVE: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\\n\\nMODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\nPlease provide the output field MODULE DESCRIPTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field MODULE DESCRIPTION.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n" + " 20%|██ | 1/5 [00:57<03:50, 57.67s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n" + "Bootstrapped 1 full traces after 2 examples in round 0.\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:42 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'998'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994916'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_0fe73b12eb11c15b676ef7f60f5de277'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d592e9911b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:42 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '998', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994916', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_0fe73b12eb11c15b676ef7f60f5de277', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d592e9911b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_0fe73b12eb11c15b676ef7f60f5de277\n", - "DEBUG:root:OpenAI Response Token Usage: 963\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\\n\\n---\\n\\nFollow the following format.\\n\\nDATASET SUMMARY: A description of the dataset that we are using.\\n\\nPROGRAM CODE: Language model program designed to solve a particular task.\\n\\nPROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\\n\\nMODULE: The module to create an instruction for.\\n\\nTASK DEMO(S): Example inputs/outputs of our module.\\n\\nBASIC INSTRUCTION: Basic instruction.\\n\\nTIP: A suggestion for how to go about generating the new instruction.\\n\\nPROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\\n\\n---\\n\\nDATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\\n\\nPROGRAM CODE:\\nStringSignature(problem_statement, sample_input, sample_output -> rationale, solution\\n instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\\\n\\\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\\\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\\\nAddress time and space complexity, and provide pseudocode if appropriate. \\\\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\\n problem_statement = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Problem Statement:\\', \\'desc\\': \\'${problem_statement}\\'})\\n sample_input = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample input provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Input:\\'})\\n sample_output = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'The sample output provided with the problem statement.\\', \\'__dspy_field_type\\': \\'input\\', \\'prefix\\': \\'Sample Output:\\'})\\n rationale = Field(annotation=str required=True json_schema_extra={\\'prefix\\': \"Reasoning: Let\\'s think step by step in order to\", \\'desc\\': \\'${produce the output fields}. We ...\\', \\'__dspy_field_type\\': \\'output\\'})\\n solution = Field(annotation=str required=True json_schema_extra={\\'format\\': , \\'desc\\': \\'a solution explanation for how we should go about solving this problem.\\', \\'__dspy_field_type\\': \\'output\\', \\'prefix\\': \\'Solution:\\'})\\n)\\n\\nclass SimpleGenerateSolution(dspy.Module):\\n def __init__(self):\\n super().__init__()\\n self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\n def forward(self, problem_statement, sample_input, sample_output):\\n solution = self.generate_code(\\n problem_statement=problem_statement,\\n sample_input=sample_input,\\n sample_output=sample_output\\n ).solution\\n\\n return dspy.Prediction(solution=solution)\\n\\n\\nPROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\\n\\nMODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\\n\\nTASK DEMO(S): \\n\\nBASIC INSTRUCTION:\\nYou are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it\\'s accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\\n\\nTIP: Make sure your instruction is very informative and descriptive.\\n\\nPlease provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.5, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n" + "Average Metric: 0.06425702811244981 / 1 (6.4): 20%|██ | 1/5 [00:37<02:31, 37.91s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "task_demos \n" + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0192000e-173c-7a03-b317-7b34acca518a\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:45 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'2357'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994467'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_b536aa6a0398153ead07d7a0fb23fa1c'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d622f6b11b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:45 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '2357', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994467', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_b536aa6a0398153ead07d7a0fb23fa1c', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d622f6b11b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_b536aa6a0398153ead07d7a0fb23fa1c\n", - "DEBUG:root:OpenAI Response Token Usage: 1399\n", - "DEBUG:root:OpenAI Response Token Usage: 810\n", - "DEBUG:root:OpenAI Response Token Usage: 963\n", - "DEBUG:root:OpenAI Response Token Usage: 1399\n", - "DEBUG:root:OpenAI Response Token Usage: 810\n", - "DEBUG:root:OpenAI Response Token Usage: 963\n", - "DEBUG:root:OpenAI Response Token Usage: 1399\n", - "DEBUG:root:OpenAI Response Token Usage: 810\n", - "DEBUG:root:OpenAI Response Token Usage: 963\n", - "DEBUG:root:OpenAI Response Token Usage: 1399\n", - "DEBUG:root:OpenAI Response Token Usage: 810\n", - "DEBUG:root:OpenAI Response Token Usage: 963\n", - "DEBUG:root:OpenAI Response Token Usage: 1399\n", - "[I 2024-09-11 19:37:45,326] A new study created in memory with name: no-name-b5aec049-cdbd-44b3-a4de-7e318de19e3e\n", - "INFO:root:Starting trial num: 0\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with B2, with key differences in bold.*\\n\\nGiven a positive integer \\\\(P\\\\), please find an array of at most \\\\(100\\\\) positive integers which have a sum of \\\\(41\\\\) and a product of \\\\(P\\\\), or output \\\\(-1\\\\) if no such array exists.\\n\\nIf multiple such arrays exist, **you may output any one of them**.\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 965\\\\)\\n\\\\(1 \\\\leq P \\\\leq 10^9\\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is one line containing a single integer \\\\(P\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\\\(N\\\\), the size of your array, followed by the array itself as \\\\(N\\\\) more space-separated positive integers.\\n\\n# Sample Explanation\\nIn the first sample, we must find an array with product \\\\(2023\\\\), and sum \\\\(41\\\\). One possible answer is \\\\([7, 17, 17]\\\\).\\n\\n\\n\\nSample Input:\\n7\\n2023\\n114\\n41\\n175\\n434\\n666\\n1872\\n\\n\\nSample Output:\\nCase #1: 3 7 17 17\\nCase #2: 2 3 38\\nCase #3: 1 41\\nCase #4: 3 1 5 35\\nCase #5: 4 1 2 7 31\\nCase #6: -1\\nCase #7: 5 1 2 6 6 26\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_This problem shares some similarities with A1, with key differences in bold._\\n\\n_Atari 2600? More like Atari 2600 BCE!_\\n\\nThe classic board game _Go_ is a two-player game played on an \\\\(R \\\\times C\\\\) board. One player places white stones while the other places black stones. On a player\\'s turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \\\\(5 \\\\times 5\\\\) game looks like this:\\n\\n{{PHOTO_ID:142163805651048|WIDTH:180}}\\n\\nAn orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space.\\n\\nYou\\'re playing as Black and it\\'s your turn. Given a valid board (i.e. no groups have \\\\(0\\\\) adjacent empty spaces), **what’s the maximum number of white stones you can capture** with a single black stone?\\n\\nHere are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured:\\n\\n{{PHOTO_ID:1327337367897977|WIDTH:300}}\\n\\nHere, Black can capture a group of \\\\(3\\\\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it\\'s placed:\\n\\n{{PHOTO_ID:311797128262237|WIDTH:240}}\\n\\nBlack can even capture multiple groups at once. Here, Black captures a group of \\\\(2\\\\) stones and a group of \\\\(3\\\\) stones:\\n\\n{{PHOTO_ID:975143063563862|WIDTH:400}}\\n\\nThe Go board is represented as a character array \\\\(A\\\\) where \\\\(A_{i, j}\\\\) is one of:\\n* `B` for a black stone\\n* `W` for a white stone\\n* `.` for an empty space\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 150\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\textbf{3{,}000}\\\\)\\n\\\\(A_{i, j} \\\\in \\\\{\\\\)\\'`.`\\', \\'`B`\\', \\'`W`\\'\\\\(\\\\}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) characters \\\\(A_{i, 1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the maximum number of white stones you can capture on your turn.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, Black can capture 3 white stones by playing in the bottom-right corner.\\n\\nIn the second case, there are no white stones that can be captured.\\n\\nIn the third case, Black can capture both white groups at once, for a total of 6 + 3 = 9 white stones.\\n\\nIn the fourth case, there are 6 different white stones that can be captured, but Black can capture at most 4 of them at once (by playing in the center of the board).\\n\\n\\n\\nSample Input:\\n4\\n4 4\\nW...\\nB.BB\\n.BWW\\n.BW.\\n5 5\\nW...W\\n.W.W.\\nBBWBB\\n.W.W.\\nW...W\\n5 5\\nB..B.\\nWBBWB\\nW.WWB\\nWWBB.\\nWWB..\\n5 5\\n.....\\nWB.BW\\nBW.WB\\nBWBWB\\n.B.B.\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 0\\nCase #3: 9\\nCase #4: 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nPatrick\\'s big sister Sam is visiting for Halloween, and Sandy wants to impress the two Stars by dressing as an astronaut. Unfortunately her regular outfit already resembles an astronaut, so she\\'ll have to go as a computer scientist instead. Sandy knows a lot about [double stars](https://en.wikipedia.org/wiki/Double_star) in astronomy, but will now have to learn about double stars in graph theory.\\n\\nThere is an unrooted tree with \\\\(N\\\\) nodes, and an edge between nodes \\\\(i\\\\) and \\\\(P_i\\\\) (for each \\\\(i \\\\ge 2\\\\)). A *double star* is a tree centered at some edge \\\\(u \\\\leftrightarrow v\\\\), where \\\\(u\\\\) and \\\\(v\\\\) each have \\\\(x\\\\) chains connected to them, each chain consisting of \\\\(y\\\\) nodes (\\\\(x, y \\\\ge 1\\\\)). Alternatively, you can think of a double star as two identical star graphs (each with \\\\(x\\\\) chains of length \\\\(y\\\\)) connected by their centers. Thus, a distinct double star is specified by a tuple \\\\((u, v, x, y)\\\\), where \\\\(u \\\\lt v\\\\). \\n\\nPlease help Sandy count the number of distinct tuples \\\\((u, v, x, y)\\\\) for which the tree contains the double star described by that tuple as a subgraph.\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 110\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(8{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case has two lines. The first contains the integer \\\\(N\\\\), and the second contains the \\\\(N-1\\\\) integers \\\\(P_2\\\\) through \\\\(P_N\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of distinct double star tuples \\\\((u, v, x, y)\\\\), where \\\\(u \\\\lt v\\\\).\\n\\n# Sample Explanation\\nThe first sample tree is depicted below. There are \\\\(5\\\\) distinct double star tuples: \\\\((1, 2, 1, 1)\\\\), \\\\((1, 2, 1, 2)\\\\), \\\\((1, 2, 2, 1)\\\\), \\\\((1, 4, 1, 1)\\\\) and \\\\((2, 6, 1, 1)\\\\).\\n\\n{{PHOTO_ID:1573931960080150|WIDTH:350}}\\n\\nThe second sample tree is depicted below. There are \\\\(5\\\\) distinct double star tuples: \\\\((1, 2, 1, 1)\\\\), \\\\((1, 2, 1, 2)\\\\), \\\\((1, 8, 1, 1)\\\\), \\\\((1, 8, 2, 1)\\\\) and \\\\((2, 7, 1, 1)\\\\).\\n\\n{{PHOTO_ID:1286569645378322|WIDTH:500}}\\n\\nIn the third sample tree, no double stars exist because arm lengths of \\\\(0\\\\) are not allowed \\\\((y \\\\ge 1 )\\\\).\\n\\n\\nSample Input:\\n3\\n8\\n1 1 1 2 2 6 4\\n10\\n1 8 8 1 1 2 1 7 7\\n2\\n1\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 5\\nCase #3: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**The only differences between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/D1) is that here, \\\\(A_i \\\\in \\\\{1, 2\\\\}\\\\), and you may only swap adjacent elements of \\\\(A_i\\\\) with a single operation.**\\n\\nAs a Metal Platforms employee, you place a high value on your work-life balance. Boss Rob has assigned you \\\\(N\\\\) tasks, the \\\\(i\\\\)th of which takes \\\\(A_i\\\\) minutes to finish, where \\\\(A_i\\\\) is either **\\\\(\\\\mathbf{1}\\\\) or \\\\(\\\\mathbf{2}\\\\)**.\\n\\nYou may reorder your tasks, where each *operation* lets you **swap any two adjacent elements** of \\\\(A\\\\) (\\\\(A_i\\\\) and \\\\(A_j\\\\) for some \\\\(i\\\\) and \\\\(j\\\\) such that \\\\(|i - j| = 1\\\\)).\\n\\nTo reflect how often task requirements change in the real world, there will be \\\\(M\\\\) updates made to the task completion times, with the \\\\(i\\\\)th update setting \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n\\nAfter completing the \\\\(i\\\\)th update, you would like to know if it\\'s possible to balance the time spent at work versus at home, namely if you hope to finish the first \\\\(Z_i\\\\) tasks at work and the rest at home. Specifically, let \\\\(Q_i\\\\) be the minimum number of swap operations which must be theoretically performed so that \\\\(A_1 + ... + A_{Z_i} = A_{Z_i + 1} + ... + A_{N}\\\\), with \\\\(Q_i = -1\\\\) if it\\'s impossible. Note that it\\'s possible for \\\\(Q_i\\\\) to be \\\\(0\\\\), if the subarrays already have equal sums.\\n\\nTo reduce the size of the output, please compute the sum of \\\\(Q_1\\\\), ..., \\\\(Q_M\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 95\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le M \\\\le 1{,}000{,}000\\\\)\\n\\\\(A_i, Y_i \\\\in \\\\{1, 2\\\\}\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Z_i \\\\le N-1\\\\)\\n\\nThe sum of \\\\(N+M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line containing \\\\(N\\\\) digits, \\\\(A_1\\\\), \\\\(A_2\\\\), ..., \\\\(A_N\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(X_i\\\\), \\\\(Y_i\\\\), and \\\\(Z_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output `\"Case #i: \"` followed by a single integer, \\\\(Q_1 + ... + Q_M\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:2337609416394611|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 2]\\\\). The first update yields \\\\(A=[1, 1]\\\\) with \\\\(Z_1 = 1\\\\). The first and second tasks are already equal, so \\\\(Q_1 = 0\\\\) as no swaps are necessary. The second update yields \\\\(A=[2, 1]\\\\), and after the third update we still have \\\\([2, 1]\\\\). In neither case can we equally split the tasks, so \\\\(Q_2 = Q_3 = -1\\\\). The final answer is \\\\(0 + (-1) + (-1) = -2\\\\).\\n\\nThe second case is depicted below:\\n\\n{{PHOTO_ID:402660235386017|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 1, 1, 2]\\\\). The first update yields \\\\(A=[2, 1, 1, 2]\\\\) with \\\\(Z_1 = 2\\\\). The two subarrays both sum to \\\\(3\\\\) already, so \\\\(Q_1 = 0\\\\). The second update yields \\\\(A=[2, 2, 1, 2]\\\\). It\\'s impossible to split the tasks equally, so \\\\(Q_2 = -1\\\\). The third update yields \\\\(A=[2, 2, 1, 1]\\\\) with \\\\(Z_3 = 2\\\\). We can swap the second and third tasks so that both halves of the array sum to \\\\(3\\\\), so \\\\(Q_3 = 1\\\\). The final answer is \\\\(0 + (-1) + 1 = 0\\\\).\\n\\nIn the third case, we start with \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2]\\\\). With the updates, we get:\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_1 = 4\\\\) and \\\\(Q_1 = 4\\\\): move the left two \\\\(2\\\\)s two spaces left\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_2 = 3\\\\) and \\\\(Q_2 = 12\\\\): move the left three \\\\(2\\\\)s four spaces left\\n- \\\\(A = [1, 1, 1, 1, 2, 2, 2, 2], Z_3 = 5\\\\) and \\\\(Q_3 = 0\\\\)\\n- \\\\(A = [2, 1, 1, 1, 2, 2, 2, 2], Z_4 = 4\\\\) and \\\\(Q_4 = -1\\\\)\\n- \\\\(A = [2, 1, 2, 1, 2, 2, 2, 2], Z_5 = 4\\\\) and \\\\(Q_5 = 1\\\\): swap \\\\(A_4\\\\) and \\\\(A_5\\\\)\\n\\nThe final answer is \\\\(4 + 12 + 0 + (-1) + 1 = 16\\\\).\\n\\n\\nSample Input:\\n3\\n2 3\\n1 2\\n2 1 1\\n1 2 1\\n1 2 1\\n4 3\\n1 1 1 2\\n1 2 2\\n2 2 2\\n4 1 2\\n8 5\\n1 1 1 1 2 2 2 2\\n5 2 4\\n7 2 3\\n6 2 5\\n1 2 4\\n3 2 4\\n\\n\\nSample Output:\\nCase #1: -2\\nCase #2: 0\\nCase #3: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with B1, with key differences in bold.*\\n\\nGiven a positive integer \\\\(P\\\\), please find an array of at most \\\\(100\\\\) positive integers which have a sum of \\\\(41\\\\) and a product of \\\\(P\\\\), or output \\\\(-1\\\\) if no such array exists.\\n\\nIf multiple such arrays exist, **print one with the fewest number of elements**. If there are multiple with the fewest number of elements, you may print any one of them.\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 960\\\\)\\n\\\\(1 \\\\leq P \\\\leq 10^9\\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is one line containing a single integer \\\\(P\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\\\(N\\\\), the size of your array, followed by the array itself as \\\\(N\\\\) more space-separated positive integers.\\n\\n# Sample Explanation\\nIn the first sample, we must find an array with product \\\\(2023\\\\), and sum \\\\(41\\\\). One possible answer is \\\\([7, 17, 17]\\\\).\\n\\n\\n\\nSample Input:\\n7\\n2023\\n114\\n41\\n175\\n434\\n666\\n1872\\n\\n\\nSample Output:\\nCase #1: 3 7 17 17\\nCase #2: 2 3 38\\nCase #3: 1 41\\nCase #4: 3 1 5 35\\nCase #5: 4 1 2 7 31\\nCase #6: -1\\nCase #7: 4 2 4 9 26\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with C1, with key differences in bold.*\\n\\nA friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it.\\n\\nThe puzzle consists of \\\\(N\\\\) buttons in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). The initial state of button \\\\(i\\\\) is white if \\\\(S_i = 1\\\\), or black if \\\\(S_i = 0\\\\). Pressing a button \\\\(k\\\\) toggles the state of itself as well as every \\\\(k\\\\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black.\\n\\nLife is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \\\\(Q\\\\) buttons sequentially, the \\\\(i\\\\)th being button \\\\(B_i\\\\). \\n\\nAfter your brother **pushes each button**, you\\'d like to add to your answer the minimum number of button presses required to turn all the buttons black.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 75\\\\)\\n\\\\(1 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 4{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) and \\\\(Q\\\\) over all cases will be at most \\\\(9{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing a bitstring \\\\(S\\\\) of length \\\\(N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains a single integer \\\\(B_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of the number of button presses needed to turn all buttons black **after each of the \\\\(Q\\\\) presses.**\\n\\n**Sample Explanation**\\n\\nIn the first case, after your brother presses the first button, the state of the puzzle is \\\\(0101\\\\). The best strategy is to press the second button, turning all lights off.\\n\\nIn the second case, the puzzle starts as \\\\(0001\\\\), and after each button press becomes \\\\(0100\\\\), \\\\(0110\\\\), \\\\(0011\\\\), and \\\\(0010\\\\) respectively. The answers after each query are \\\\(2\\\\), \\\\(3\\\\), \\\\(2\\\\), and \\\\(1\\\\) respectively, so we output their sum of \\\\(8\\\\).\\n\\n\\nSample Input:\\n5\\n4\\n1010\\n1\\n1\\n4\\n0001\\n4\\n2\\n3\\n2\\n4\\n7\\n0101101\\n8\\n1\\n3\\n2\\n6\\n7\\n4\\n2\\n5\\n7\\n0101100\\n1\\n7\\n7\\n1111111\\n1\\n1\\n\\n\\nSample Output:\\nCase #1: 1\\nCase #2: 8\\nCase #3: 36\\nCase #4: 4\\nCase #5: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*Bejeweled™* is a classic puzzle game where the player tries to match three-in-a-row in a 2D grid by swapping adjacent tile pairs. You may remember Hacker Cup\\'s [spinoff](https://www.facebook.com/codingcompetitions/hacker-cup/2022/final-round/problems/C) by the name of *Isblinged™*. The all new sequel, *Isblinged 2*, is also played on a grid of \\\\(R\\\\) rows by \\\\(C\\\\) columns of tiles where the tile at \\\\((i, j)\\\\) is of an integer type \\\\(G_{i,j}\\\\).\\n\\nAt any time, the *score* of the grid is the number of subarrays of \\\\(3\\\\) equal tiles in either a single row or column (i.e. either a \\\\(3 \\\\times 1\\\\) or \\\\(1 \\\\times 3\\\\) submatrix). Note that different subarrays can overlap, and will each count toward the score. The score of the initial grid is guaranteed to be \\\\(0\\\\).\\n\\nYou will make exactly \\\\(2\\\\) moves, where each involves swapping a pair of adjacent tiles (either in the same row or column). What is the maximum score that can be achieved after the \\\\(2\\\\) moves?\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(1 \\\\le R, C \\\\le 1{,}000\\\\)\\n\\\\(R*C \\\\ge 2\\\\)\\n\\\\(1 \\\\le G_{i,j} \\\\le 1{,}000{,}000\\\\)\\n\\nThe sum of \\\\(R * C\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) space-separated integers \\\\(G_{i,1..C}\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing `\"Case #i: \"` followed by a single integer, the maximum score.\\n\\n# Sample Explanation\\n\\nIn the first case, one possible optimal ordered pair of swaps is depicted below:\\n\\n{{PHOTO_ID:1050535959425825|WIDTH:750}}\\n\\nThe total score is \\\\(5\\\\) as there are \\\\(3\\\\) equal subarrays in the the first row and \\\\(2\\\\) in the second.\\n\\n\\n\\nSample Input:\\n3\\n2 5\\n1 1 2 1 1\\n2 2 1 3 2\\n1 4\\n1 1 2 1\\n5 1\\n1\\n2\\n1\\n2\\n1\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 1\\nCase #3: 1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_This problem shares some similarities with A2, with key differences in bold._\\n\\n_Atari 2600? More like Atari 2600 BCE!_\\n\\nThe classic board game _Go_ is a two-player game played on an \\\\(R \\\\times C\\\\) board. One player places white stones while the other places black stones. On a player\\'s turn, they may place a stone in any empty space. A curiosity of Go is that stones are placed on the intersections of grid lines rather than between the lines, so an in-progress \\\\(5 \\\\times 5\\\\) game looks like this:\\n\\n{{PHOTO_ID:1491241961665655|WIDTH:180}}\\n\\nAn orthogonally contiguous set of stones of the same color is called a *group*. A group of stones is captured (and removed from the board) once no stones in the group has an adjacent empty space.\\n\\nYou\\'re playing as Black and it\\'s your turn. Given a valid board (i.e. no groups have \\\\(0\\\\) adjacent empty spaces), **is it possible to capture any white stones on this turn** with a single black stone?\\n\\nHere are some examples of captures. If a black stone is placed at the point marked with a triangle, a single white stone will be captured:\\n\\n{{PHOTO_ID:632037365762295|WIDTH:300}}\\n\\nHere, Black can capture a group of \\\\(3\\\\) white stones. Note that this move is valid even though the new black stone has no adjacent empty spaces at the moment it\\'s placed:\\n\\n{{PHOTO_ID:850220283519385|WIDTH:240}}\\n\\nBlack can even capture multiple groups at once. Here, Black captures a group of \\\\(2\\\\) stones and a group of \\\\(3\\\\) stones:\\n\\n{{PHOTO_ID:804957498302389|WIDTH:400}}\\n\\nThe Go board is represented as a character array \\\\(A\\\\) where \\\\(A_{i, j}\\\\) is one of:\\n* `B` for a black stone\\n* `W` for a white stone\\n* `.` for an empty space\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 160\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\textbf{19}\\\\)\\n\\\\(A_{i, j} \\\\in \\\\{\\\\)\\'`.`\\', \\'`B`\\', \\'`W`\\'\\\\(\\\\}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(C\\\\) characters \\\\(A_{i, 1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by \"`YES`\" if you can capture any white stones, or \"`NO`\" if you cannot.\\n\\n\\n# Sample Explanation\\n\\nThe first sample case is the first capture example shown above. Black can capture the lone white stone.\\n\\nIn the second case, Black can capture a group of \\\\(5\\\\) white stones.\\n\\nThe third case is the same as the second case, except that the group of \\\\(5\\\\) white stones has two empty adjacent spaces, so it cannot be captured in one term. The other white stones can also not be captured on this turn.\\n\\nIn the fourth case, White could capture a black stone if it was their turn, but Black cannot capture any white stones.\\n\\nThe fifth case is the second capture example shown above. Black can capture \\\\(3\\\\) white stones by playing in the upper-left corner.\\n\\n\\nSample Input:\\n5\\n4 4\\n.B..\\nBW..\\n.B..\\n....\\n4 5\\nWWWB.\\nWB...\\nWB.WB\\nB.W..\\n4 5\\nWWWB.\\nWB...\\nWB.WB\\n..W..\\n3 5\\n..W..\\n.WBWW\\n.B..W\\n3 3\\n.WB\\nWWB\\nBB.\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: YES\\nCase #3: NO\\nCase #4: NO\\nCase #5: YES\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYour friend baked \\\\(N\\\\) batches of cookies, the first with chocolate chips, and the rest with raisins. Batch \\\\(i\\\\) has \\\\(C_i\\\\) cookies all having the same weight \\\\(W_i\\\\). Cookies in different batches may have the same weight. Initially, all cookies are scattered on the table, and you hope to find a hefty one to eat with your trusty balance scale.\\n\\nFirst, you select a single cookie uniformly at random from the table and place it on the left side of the scale. Then, \\\\(K\\\\) times, you will select a cookie uniformly at random from the table, place it on the empty side of the scale, and throw away the lighter of the two cookies on the scale (or the cookie on the left side if they both weigh the same).\\n\\n{{PHOTO_ID:502174485028685|WIDTH:700}}\\n\\nDetermine the probability that after throwing away \\\\(K\\\\) cookies, the remaining cookie on the scale is a yummy chocolate chip cookie from batch \\\\(1\\\\).\\n\\nIf we write this probability as a quotient of integers \\\\(p/q\\\\) in lowest terms, then you should output this quotient modulo \\\\(1{,}000{,}000{,}007\\\\) — in other words, output the unique integer \\\\(x\\\\) such that \\\\(0 \\\\le x \\\\lt 1{,}000{,}000{,}007\\\\) and \\\\(p = (x * q) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 1{,}000\\\\)\\n\\\\(2 \\\\le N \\\\le 3{,}000\\\\)\\n\\\\(1 \\\\le K \\\\lt \\\\sum_{i=1}^{N} C_i\\\\)\\n\\\\(1 \\\\le C_i \\\\le 3{,}000\\\\)\\n\\\\(1 \\\\le W_i \\\\le 10^{9}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two-space separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(C_i\\\\) and \\\\(W_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by the unique integer \\\\(x\\\\) such that \\\\(p = (x * q) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first case, there are \\\\(5\\\\) batches with \\\\(1\\\\) cookie each. Since we only weigh \\\\(K=1\\\\) time, the only way for the last cookie to be chocolate chip is if it\\'s one of the two placed on the scale. There\\'s a \\\\(\\\\frac{1}{5}\\\\) chance of picking it for the initial cookie, and a \\\\(\\\\frac{4}{5}\\\\cdot\\\\frac{1}{4}\\\\) chance of picking it for the second cookie. In either case, because it\\'s the heaviest, it will remain on the scale. Adding these independent events\\' probabilities, we get a \\\\(\\\\frac{1}{5} + \\\\frac{4}{5}\\\\cdot\\\\frac{1}{4} = \\\\frac{2}{5}\\\\) chance that the last cookie on the scale is chocolate chip. Here, we print \\\\(800{,}000{,}006\\\\) since \\\\(2 = (800{,}000{,}006 * 5) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\nThe second case is the same, except we must weigh a second time. There\\'s a \\\\(\\\\frac{2}{5}\\\\) chance of picking chocolate chip the first or second times, and a \\\\(\\\\frac{3}{5}\\\\cdot\\\\frac{1}{3}\\\\) chance of picking it the third time. The answer is \\\\(\\\\frac{2}{5} + \\\\frac{3}{5}\\\\cdot\\\\frac{1}{3} = \\\\frac{3}{5}\\\\). We print \\\\(200{,}000{,}002\\\\) since \\\\(3 = (200{,}000{,}002 * 5) \\\\text{ mod } 1{,}000{,}000{,}007\\\\).\\n\\nIn the third case, there are \\\\(2\\\\) batches of \\\\(10\\\\) cookies each, where batch \\\\(2\\\\) cookies are heavier. Since we must weigh and throw out \\\\(10\\\\) cookies, we must have weighed a cookie from batch \\\\(2\\\\) at least once, after which a batch \\\\(1\\\\) cookie can never remain on the scale. Therefore the answer is \\\\(0\\\\).\\n\\nIn the fourth case, there is a \\\\(\\\\frac{5}{24}\\\\) chance that the final cookie on the scale is chocolate chip.\\n\\n\\nSample Input:\\n6\\n5 1\\n1 3000\\n1 2000\\n1 1000\\n1 2000\\n1 1000\\n5 2\\n1 3000\\n1 2000\\n1 1000\\n1 2000\\n1 1000\\n2 10\\n10 1\\n10 2\\n5 2\\n2 50\\n1 40\\n1 50\\n1 60\\n3 50\\n4 2993\\n3000 999999999\\n2995 1000000000\\n1552 888888888\\n1336 999999999\\n3 1\\n1 10\\n2 9\\n1 11\\n\\n\\nSample Output:\\nCase #1: 800000006\\nCase #2: 200000002\\nCase #3: 0\\nCase #4: 208333335\\nCase #5: 590307096\\nCase #6: 333333336\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This is the easier version of [Chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/A2). This version involves a string, but chapter 2 involves an array of integers that can change with updates.**\\n\\nA string is *perfectly balanced* if its length is even, and the first half of the string can be shuffled to make the string a palindrome. For example, \"`abab`\" and \"`dood`\" are perfectly balanced, but \"`racecar`\" and \"`doodad`\" are not.\\n\\nA string is *almost perfectly balanced* if you can delete exactly one character from it to make it perfectly balanced. Some examples are \"`classical`\", \"`intelligent`\", and \"`www`\".\\n\\n{{PHOTO_ID:1262846811180772|WIDTH:700}}\\n\\nYou are given a larger template string \\\\(S\\\\), and \\\\(Q\\\\) substrings, the \\\\(i\\\\)th of which is \\\\(S_{L_i..R_i}\\\\). For how many of these \\\\(Q\\\\) queries is the substring almost perfectly balanced?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 90\\\\)\\n\\\\(1 \\\\le |S| \\\\le 10^6\\\\)\\n\\\\(S_i \\\\in\\\\) {\\'`a`\\', ..., \\'`z`\\'}\\n\\\\(1 \\\\le Q \\\\le 10^6\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le |S|\\\\)\\n\\nThe sum of \\\\(|S|\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single template string \\\\(S\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(L_i\\\\) and \\\\(R_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"`, followed by a single integer: the number of queries which are an almost perfectly balanced substring.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, the template string is \"`singingbanana`\".\\n\\nFor the first query, the substring is \"`banan`\". You can delete the `\"b\"` to get \"`anan`\" then reorder the first half of the string to get `\"naan\"`, which is a palindrome. Thus, \"`banan`\" is an almost perfectly balanced substring.\\n\\nFor the second query, the substring is \"`anana`\". You can delete the second \"`a`\" to get \"`anna`\", which is a palindrome. Thus, \"`anana`\" is an almost perfectly balanced substring.\\n\\nFor the third query, the substring is \"`ban`\". You cannot delete any character to get an almost perfectly balanced substring.\\n\\nFor the fourth query, the substring is \"`nan`\". You can delete the \"`a`\" to get \"`nn`\", which is a palindrome. Thus, \"`nan`\" is an almost perfectly balanced substring.\\n\\nFor \"`singing`\", you can create \"`gniing`\".\\n\\nIn the second test case, the first, second, and third queries are almost perfectly balanced substrings, but the fourth is not.\\n\\nIn the third test case, the first, second, third, and fourth queries are almost perfectly balanced substrings, but the fifth and sixth are not.\\n\\n\\nSample Input:\\n3\\nsingingbanana\\n5\\n8 12\\n9 13\\n8 10\\n10 12\\n1 7\\nprepareintelligentopinion\\n4\\n1 7\\n8 18\\n19 25\\n12 13\\nphpservers\\n6\\n1 3\\n4 10\\n1 3\\n2 2\\n3 5\\n1 10\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 3\\nCase #3: 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This problem has large input, so we recommend pre-downloading the compressed zip file.**\\n\\nMeta Getaways is a travel agency that deals with \\\\(N\\\\) airports numbered \\\\(1...N\\\\), and \\\\(M\\\\) flight paths. Flight path \\\\(i\\\\) connects airports \\\\(A_i\\\\) and \\\\(B_i\\\\) in both directions, with two direct flights operating every morning (one in each direction), and another two every evening (also one in each direction). Each of these four direct flights can carry up to \\\\(C_i\\\\) **t**ourists.\\n\\n{{PHOTO_ID:356437426702208|WIDTH:400}}\\n\\nThe first sample case is depicted above, with morning and evening flights in red and blue.\\n\\nPeak travel season is here, and will last \\\\(Q\\\\) days. For each day \\\\(i\\\\), determine \\\\(F_i\\\\), the maximum number of tourists who could possibly fly from airport \\\\(X_i\\\\) to \\\\(Y_i\\\\). Each tourist may either fly directly or take one morning and one evening flight which share an intermediate airport.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 70 \\\\)\\n\\\\(1 \\\\leq N, M, Q \\\\leq 200{,}000\\\\)\\n\\\\(1 \\\\leq C_i \\\\leq 10^9 \\\\)\\n\\\\(1 \\\\leq A_i, B_i \\\\leq N; A_i \\\\ne B_i \\\\)\\n\\\\(1 \\\\leq X_i, Y_i \\\\leq N; X_i \\\\ne Y_i\\\\)\\nAll unordered pairs \\\\((A_i, B_i)\\\\) within a given test case are distinct.\\n\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(5{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(Q\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), and \\\\(C_i\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print a line containing \"`Case #i:` \" followed by \\\\(Q\\\\) space-separated integers \\\\(F_1, ..., F_Q\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first case:\\n\\n- On day \\\\(1\\\\), we must send as many tourists from airport \\\\(1\\\\) to airport \\\\(2\\\\). We can fly \\\\(10\\\\) tourists direct in the morning and \\\\(10\\\\) more at night. Only \\\\(5\\\\) tourists can be flown from \\\\(1 \\\\to 3\\\\) in the morning and \\\\(3 \\\\to 2\\\\) in the evening (despite the evening flight capacity being \\\\(15\\\\)). Therefore, \\\\(F_1 = 10 \\\\cdot 2 + 5 = 25\\\\).\\n- On day \\\\(2\\\\), we can fly \\\\(5\\\\) tourists direct in the morning and evening, then fly \\\\(10\\\\) tourists through airports \\\\(1 \\\\to 2 \\\\to 3\\\\). Therefore, \\\\(F_2 = 5 \\\\cdot 2 + 10 = 20\\\\).\\n- \\\\(F_3 = 15 \\\\cdot 2 + 5 + 7 = 42\\\\)\\n- \\\\(F_4 = 10 \\\\cdot 2 + 7 = 27\\\\)\\n- \\\\(F_5 = 7 \\\\cdot 2 + 10 = 24\\\\)\\n- On day \\\\(6\\\\), there are no direct flights. We can fly \\\\(10\\\\) tourists through airports \\\\(4 \\\\to 2 \\\\to 1\\\\), and \\\\(5\\\\) tourists through airports \\\\(4 \\\\to 3 \\\\to 1\\\\) for a total of \\\\(F_6 = 10 + 5 = 15\\\\) tourists.\\n\\nIn the second case:\\n\\n- \\\\(F_1 = 10 \\\\cdot 2 + 20 = 40\\\\)\\n- \\\\(F_2 = 30 \\\\cdot 2 + 10 = 70\\\\)\\n- \\\\(F_3 = 0\\\\)\\n- \\\\(F_4 = 20 \\\\cdot 2 + 10 = 50\\\\)\\n- \\\\(F_5 = 0\\\\)\\n- \\\\(F_6 = 0\\\\)\\n\\n\\n\\nSample Input:\\n3\\n4 5 6\\n1 2 10\\n1 3 5\\n2 3 15\\n2 4 10\\n3 4 7\\n1 2\\n1 3\\n2 3\\n2 4\\n3 4\\n4 1\\n4 3 6\\n1 2 10\\n2 3 20\\n3 1 30\\n1 2\\n1 3\\n1 4\\n2 3\\n2 4\\n3 4\\n4 3 6\\n1 2 20\\n2 3 10\\n3 4 30\\n1 2\\n1 3\\n1 4\\n2 3\\n2 4\\n3 4\\n\\n\\nSample Output:\\nCase #1: 25 20 42 27 24 15\\nCase #2: 40 70 0 50 0 0\\nCase #3: 40 10 0 20 10 60\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem B2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/B2) is that here, scenes are smaller and do not contain rocks.**\\n\\nBoss Rob painted a beautiful scene on a 2D canvas of \\\\(R\\\\) rows by \\\\(C\\\\) columns, containing zero or more happy little trees.\\n\\nTo make sure none of his trees are lonely, Rob would like you to add as many trees as you\\'d like (possibly \\\\(0\\\\)) to empty spaces so that each tree in the final painting has at least two tree *friends*, that is, two trees which are each adjacent to it (directly to its north, south, east, or west). If there are multiple solutions, you may print any one of them.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 85\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\mathbf{100}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, each of which contains \\\\(C\\\\) characters, either \"`.`\" (an empty space) or \"`^`\" (a tree), representing the initial painting.\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print \"`Case #i: ` \", followed by \"`Possible`\" and \\\\(R\\\\) lines of \\\\(C\\\\) characters each representing the final painting (if a solution exists), otherwise \"`Impossible`\".\\n\\n\\n# Sample Explanation\\n\\nIn the first case (depicted below), we could add two tree friends to either side of the middle tree, but they themselves would each only have one tree friend. Therefore, it\\'s impossible to get two friends for each tree in the final painting.\\n\\n{{PHOTO_ID:545859687328272|WIDTH:295}}\\n\\nIn the second case, there are no trees in the initial painting, so the condition of two friends for each tree is already satisfied.\\n\\nIn the third case, one possible solution is depicted below.\\n\\n{{PHOTO_ID:5118404751621442|WIDTH:700}}\\n\\n\\nSample Input:\\n3\\n1 3\\n.^.\\n3 1\\n.\\n.\\n.\\n4 4\\n..^.\\n..^.\\n....\\n...^\\n\\n\\nSample Output:\\nCase #1: Impossible\\nCase #2: Possible\\n.\\n.\\n.\\nCase #3: Possible\\n^^^.\\n^.^.\\n^^^^\\n..^^\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nDespite his age, Mr. Krabs still goes trick-or-treating. Free candy (free anything, really) is irresistible to him. Bikini Bottom is an \\\\(R \\\\times C\\\\) grid of houses, with house \\\\((i, j)\\\\) having \\\\(A_{i,j}\\\\) pieces of candy.\\n\\nMr. Krabs plans to travel from house \\\\((1, 1)\\\\), the top-left corner, to house \\\\((R, C)\\\\), the bottom-right corner, taking as much candy as possible along the way. When Mr. Krabs is at house \\\\((i, j)\\\\), he takes all the candy, and then either moves down to house \\\\((i + 1, j)\\\\) or right to house \\\\((i, j + 1)\\\\).\\n\\nPlankton wants to sabotage him, and has decided to complete his own candy run before Mr. Krabs even starts his trip. Plankton will go from house \\\\((1, C)\\\\), the top-right corner, to house \\\\((R, 1)\\\\), the bottom-left corner. When Plankton is at house \\\\((i, j)\\\\), he takes all its candy (so there\\'s none left for Mr. Krabs later) and either moves down to house \\\\((i + 1, j)\\\\) or left to house \\\\((i, j - 1)\\\\).\\n\\nWhat\\'s the maximum amount of candy that Mr. Krabs can collect, assuming that Plankton strives to minimize this value by completing his trip before Mr. Krabs starts?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 100\\\\)\\n\\\\(2 \\\\leq R, C \\\\leq 300\\\\)\\n\\\\(0 \\\\leq A_{i,j} \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\nThe sum of \\\\(R * C\\\\) across all test cases is at most \\\\(500{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, the \\\\(i\\\\)th of which contains the \\\\(C\\\\) integers \\\\(A_{i,1}\\\\) through \\\\(A_{i,C}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the maximum pieces of candy that Mr. Krabs can get if Plankton optimally sabotages.\\n\\n\\n# Sample Explanation\\n\\nThe following depicts sample cases 1 through 5. Plankton\\'s paths to minimize Mr. Krabs\\'s candy are greyed out, while Mr. Krabs\\'s optimal paths through the remaining candy (in spite of Plankton\\'s sabotage) are drawn in red.\\n\\n{{PHOTO_ID:216354288143884|WIDTH:650}}\\n\\n\\nSample Input:\\n5\\n3 4\\n2 5 3 6\\n1 7 2 5\\n4 5 1 3\\n2 2\\n2 2\\n2 1\\n3 3\\n3 4 5\\n5 3 1\\n7 6 4\\n3 3\\n1 1 1\\n3 3 3\\n3 2 1\\n3 3\\n1 1 3\\n3 3 1\\n1 3 2\\n\\n\\nSample Output:\\nCase #1: 12\\nCase #2: 1\\nCase #3: 11\\nCase #4: 5\\nCase #5: 6\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*…right down Santa Claus Lane!*\\n\\nSanta Claus Lane is home to \\\\(N\\\\) elves, the \\\\(i\\\\)th of whom lives \\\\(X_i\\\\) meters from the start of the lane. As Santa\\'s little helper, you\\'re tasked to assign elves to work on toys for good little girls and boys. Specifically, you must assign each elf to work on some toy so that at least \\\\(2\\\\) toys get worked on, and no elf works on a toy alone.\\n\\nAll elves working on the same toy will meet at the point which minimizes the farthest distance that any of them would need to walk. Formally, if the elves assigned to a given toy live at \\\\(X_1\\\\), \\\\(X_2\\\\), \\\\(…\\\\), then they will meet at point \\\\(P\\\\) such that \\\\(\\\\max(|X_1 - P|\\\\), \\\\(|X_2 - P|\\\\), \\\\(…\\\\)\\\\()\\\\) is as small as possible.\\n\\nFor instance, the first sample case is depicted below:\\n\\n{{PHOTO_ID:838516607729325|WIDTH:700}}\\n\\nSanta is supervising, and you reckon he could use some exercise. Among all valid assignments of elves to toys, what\\'s the farthest Santa would need to walk to visit all meeting points? Santa may start and end anywhere, but will try to walk as little as possible after seeing your assignments.\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 20\\\\)\\n\\\\(4 \\\\leq N \\\\leq 10^5 \\\\)\\n\\\\(1 \\\\leq X_i \\\\leq 10^9 \\\\)\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with one line containing the integer \\\\(N\\\\), followed by a second line containing the the \\\\(N\\\\) integers \\\\(X_1 ... X_N\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single real number, the farthest distance Santa would need to walk in meters. Your answer will be considered correct if it differs from the jury\\'s answer by at most \\\\(10^{-6}\\\\) meters or at most \\\\(0.0001\\\\)%, whichever is larger.\\n\\n# Sample Explanation\\n\\nIn the first sample case, elves living at \\\\(1 \\\\,\\\\text{m}\\\\) and \\\\(3 \\\\,\\\\text{m}\\\\) will work on a toy. Elves living at \\\\(7 \\\\,\\\\text{m}\\\\), \\\\(12 \\\\,\\\\text{m}\\\\), and \\\\(13 \\\\,\\\\text{m}\\\\) will work on another toy, and elves at \\\\(17 \\\\,\\\\text{m}\\\\) and \\\\(23 \\\\,\\\\text{m}\\\\) will work on the third toy. The toys will be made at \\\\(2 \\\\,\\\\text{m}\\\\), \\\\(10 \\\\,\\\\text{m}\\\\) and \\\\(20 \\\\,\\\\text{m}\\\\) respectively. Santa would need to walk at least \\\\(18 \\\\,\\\\text{m}\\\\) in total to visit every meeting point.\\n\\nThe second sample case is depicted below. No elf is allowed to work alone and we must make two toys. One optimal way of doing this is to have the leftmost \\\\(3\\\\) elves work on one toy at \\\\(2 \\\\,\\\\text{m}\\\\), and the rest work on the other toy at \\\\(4.5 \\\\,\\\\text{m}\\\\). This would maximize the distance Santa would have to walk among all valid elf assignments.\\n\\n{{PHOTO_ID:163905706775126|WIDTH:400}}\\n\\nIn the third case, the two toys will be made at \\\\(55 \\\\,\\\\text{m}\\\\) and \\\\(5500 \\\\,\\\\text{m}\\\\).\\n\\n\\n\\nSample Input:\\n3\\n7\\n1 17 3 13 7 23 12\\n5\\n5 4 3 2 1\\n4\\n10 100 1000 10000\\n\\n\\nSample Output:\\nCase #1: 18\\nCase #2: 2.5\\nCase #3: 5445\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nSandy\\'s store has \\\\(N\\\\) pre-owned clock parts for sale, where the \\\\(i\\\\)th part is of style \\\\(S_i\\\\). The store also has two display cases, each capable of holding at most \\\\(K\\\\) parts. To maximize the aesthetics of Sandy\\'s secondhand second hands, she\\'d like to put each of the \\\\(N\\\\) parts into one of the two cases so that neither case ends up with two different parts of the same style, and neither case has more than \\\\(K\\\\) parts total. Can you determine if this is possible?\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 90\\\\)\\n\\\\(1 \\\\leq N, K, S_i \\\\leq 100\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing \\\\(2\\\\) space-separated integers, \\\\(N\\\\) and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(S_1, ..., S_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by \"`YES`\" if it\\'s possible to arrange the \\\\(N\\\\) parts into two cases satisfying the description above, or \"`NO`\" otherwise.\\n\\n\\n# Sample Explanation\\n\\nIn the first test case, there are \\\\(3\\\\) parts of styles \\\\(1\\\\), \\\\(2\\\\), and \\\\(2\\\\), with the display cases having capacity \\\\(2\\\\). One solution, depicted below, is to put the first and third parts in one display case, and the second part in the other.\\n\\n{{PHOTO_ID:459254706243127|WIDTH:500}}\\n\\nIn the second test case, there are \\\\(5\\\\) parts of styles \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), \\\\(3\\\\), \\\\(1\\\\), with the display cases having capacity \\\\(3\\\\). One solution, depicted below, is to put the first three parts in one display case, and the last two in the other.\\n\\n{{PHOTO_ID:1183048075593188|WIDTH:500}}\\n\\nIn the third test case, there are \\\\(5\\\\) parts, but the display cases can each only hold \\\\(2\\\\). Therefore, there is no solution.\\n\\nIn the fourth test case, style \\\\(1\\\\) will always be duplicated in some display case for any given arrangement. Therefore, there is no solution.\\n\\n\\n\\n\\nSample Input:\\n5\\n3 2\\n1 2 2\\n5 3\\n1 2 3 3 1\\n5 2\\n1 2 3 4 5\\n5 5\\n1 1 2 2 1\\n1 1\\n1\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: YES\\nCase #3: NO\\nCase #4: NO\\nCase #5: YES\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob like to play cards on their lunch break. Their favorite card game starts with two decks on the table containing \\\\(K_1\\\\) and \\\\(K_2\\\\) cards. Players take turns, with Alice going first. Each turn is in two parts:\\n\\n1. The player selects a deck of cards from the table. Let \\\\(k\\\\) be the number of cards in that deck. They then remove somewhere between \\\\(A_k\\\\) and \\\\(B_k\\\\) \\\\((1 \\\\le A_k \\\\le B_k \\\\le k)\\\\), inclusive, cards from this deck.\\n2. The player then puts a new deck of exactly \\\\(C_k\\\\) \\\\((0 \\\\le C_k < k)\\\\) cards on the table. Here, \\\\(C_k = 0\\\\) means no deck is added.\\n\\nThe player who takes the last card wins.\\n\\nFor each possible value of \\\\(K_1\\\\) from \\\\(1\\\\) to a given value \\\\(N\\\\), find the minimum possible value of \\\\(K_2\\\\) so that Bob wins the game if both players play optimally. If there is no such \\\\(K_2\\\\) between \\\\(1\\\\) and \\\\(N\\\\), then \\\\(K_2 = -1\\\\). Output the sum of \\\\(K_2\\\\) across every \\\\(K_1 = 1..N\\\\).\\n\\nTo reduce the input size, you will not be given \\\\(A_{1..N}\\\\), \\\\(B_{1..N}\\\\), and \\\\(C_{1..N}\\\\) directly. You must instead generate them using parameters \\\\(X_a\\\\), \\\\(Y_a\\\\), \\\\(Z_a\\\\), \\\\(X_b\\\\), \\\\(Y_b\\\\), \\\\(Z_b\\\\), \\\\(X_c\\\\), \\\\(Y_c\\\\), and \\\\(Z_c\\\\) and the algorithm below:\\n\\n\\\\(P_a[0] := 0\\\\)\\n\\\\(P_b[0] := 0\\\\)\\n\\\\(P_c[0] := 0\\\\)\\n\\\\(\\\\text{for each } i := 1..N\\\\):\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_a[i] := (P_a[i - 1] * X_a + Y_a) \\\\text{ mod } Z_a\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_b[i] := (P_b[i - 1] * X_b + Y_b) \\\\text{ mod } Z_b\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; P_c[i] := (P_c[i - 1] * X_c + Y_c) \\\\text{ mod } Z_c\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; A[i] := \\\\min(i, 1 + P_a[i])\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; B[i] := \\\\max(A[i], i - P_b[i])\\\\)\\n\\\\(\\\\;\\\\;\\\\;\\\\;\\\\; C[i] := \\\\min(i - 1, P_c[i])\\\\)\\n\\nNote that for any \\\\(i\\\\), the algorithm guarantees \\\\(1 \\\\le A_i \\\\le B_i \\\\le i\\\\) and \\\\(0 \\\\le C_i < i\\\\).\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 45\\\\)\\n\\\\(1 \\\\le N \\\\le 2{,}000{,}000\\\\)\\n\\\\(1 \\\\le X_a, Y_a, X_b, Y_b, X_c, Y_c \\\\le 1{,}000{,}000{,}000\\\\)\\n\\\\(1 \\\\le Z_a, Z_b, Z_c \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(50{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, first there is a line containing a single integer \\\\(N\\\\). Then, there is a line containing integers \\\\(X_a\\\\), \\\\(Y_a\\\\), \\\\(Z_a\\\\), \\\\(X_b\\\\), \\\\(Y_b\\\\), \\\\(Z_b\\\\), \\\\(X_c\\\\), \\\\(Y_c\\\\), and \\\\(Z_c\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"` followed by a single integer, the sum of the minimum \\\\(K_2\\\\) so that Bob has a guaranteed winning strategy, for every \\\\(K_1 = 1..N\\\\).\\n\\n# Sample Explanation\\n\\nIn the first sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&1&3&2&1 \\\\\\\\ 4&1&4&0&4 \\\\end{array} \\\\)\\n\\nWhen \\\\(K_1 = 1\\\\), Bob wins when \\\\(K_2 = 1\\\\) because Alice takes the first card, and Bob takes the last card.\\n\\nWhen \\\\(K_1 = 2\\\\), Alice will win if \\\\(K_2 = 1\\\\) because she can start by taking \\\\(1\\\\) card from the first deck. Bob then takes \\\\(1\\\\) card from either deck (each of which have only \\\\(1\\\\) card left), and Alice takes the last card. But if \\\\(K_2 = 2\\\\) then Bob can always win regardless of whether Alice starts by taking \\\\(1\\\\) card or \\\\(2\\\\) cards.\\n\\nWhen \\\\(K_1 = 3\\\\), Bob can always win when \\\\(K_2 = 1\\\\). If Alice takes the single card from the second deck, Bob takes \\\\(1\\\\) card from the first deck and adds a new deck of size \\\\(2\\\\) to the table. We now have two decks of size \\\\(2\\\\), and it\\'s Alice\\'s turn. That\\'s a losing state for Alice as we saw previously.\\n\\nIf Alice takes \\\\(1\\\\) card from the first deck and adds a new deck of size \\\\(2\\\\), we now have decks of size \\\\([2, 1, 2]\\\\). Bob will pick up the pile of size \\\\(1\\\\) and again we\\'re in the same losing state for Alice. If Alice takes \\\\(2\\\\) cards from the first deck, we\\'ll have decks of size \\\\([1, 1, 2]\\\\). Bob now takes the whole deck of size \\\\(2\\\\). Alice gets the next card and Bob gets the last card. Finally, if Alice takes all \\\\(3\\\\) cards from the first deck, we\\'ll have decks of size \\\\([1, 2]\\\\). Bob can take just \\\\(1\\\\) card from the deck of size \\\\(2\\\\) to win.\\n\\nIn the second sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&1&0&2 \\\\end{array} \\\\)\\n\\nIn the third sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&2&3&1&2 \\\\end{array} \\\\)\\n\\nIn the fourth sample case:\\n\\n\\\\( \\\\begin{array}{c|c|c|c|c}\\\\underline{K_1}&\\\\underline{A[K_1]}&\\\\underline{B[K_1]}&\\\\underline{C[K_1]}&\\\\underline{K_2} \\\\\\\\ 1&1&1&0&1 \\\\\\\\ 2&1&2&0&2 \\\\\\\\ 3&3&3&2&3 \\\\\\\\ 4&1&4&0&4 \\\\\\\\ 5&5&5&4&3 \\\\\\\\ 6&1&6&0&6 \\\\\\\\ 7&6&7&4&3 \\\\\\\\ 8&1&8&0&8 \\\\end{array} \\\\)\\n\\n\\nSample Input:\\n4\\n4\\n25 30 2 45 16 4 7 26 4\\n2\\n19 48 2 38 23 2 43 31 2\\n3\\n9 17 2 23 42 2 21 43 2\\n8\\n23 5 8 14 22 2 29 28 6\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 3\\nCase #3: 5\\nCase #4: 30\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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"DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n" + "Average Metric: 0.13869871545736295 / 2 (6.9): 40%|████ | 2/5 [00:54<01:16, 25.64s/it]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "\n", - "\n", - "\n", - "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "DATASET SUMMARY: A description of the dataset that we are using.\n", - "\n", - "PROGRAM CODE: Language model program designed to solve a particular task.\n", - "\n", - "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", - "\n", - "MODULE: The module to create an instruction for.\n", - "\n", - "TASK DEMO(S): Example inputs/outputs of our module.\n", - "\n", - "BASIC INSTRUCTION: Basic instruction.\n", - "\n", - "TIP: A suggestion for how to go about generating the new instruction.\n", - "\n", - "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", - "\n", - "---\n", - "\n", - "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", - "\n", - "PROGRAM CODE:\n", - "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")\n", - "\n", - "class SimpleGenerateSolution(dspy.Module):\n", - " def __init__(self):\n", - " super().__init__()\n", - " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - " def forward(self, problem_statement, sample_input, sample_output):\n", - " solution = self.generate_code(\n", - " problem_statement=problem_statement,\n", - " sample_input=sample_input,\n", - " sample_output=sample_output\n", - " ).solution\n", - "\n", - " return dspy.Prediction(solution=solution)\n", - "\n", - "\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "\n", - "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - "TASK DEMO(S): \n", - "\n", - "BASIC INSTRUCTION:\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "TIP: Make sure your instruction is very informative and descriptive.\n", - "\n", - "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", - "\n", - "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", - "\n", - "\n", - "\n", - "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "task_demos \n", - "\n", - "\n", - "\n", - "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "DATASET SUMMARY: A description of the dataset that we are using.\n", - "\n", - "PROGRAM CODE: Language model program designed to solve a particular task.\n", - "\n", - "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", - "\n", - "MODULE: The module to create an instruction for.\n", - "\n", - "TASK DEMO(S): Example inputs/outputs of our module.\n", - "\n", - "BASIC INSTRUCTION: Basic instruction.\n", - "\n", - "TIP: A suggestion for how to go about generating the new instruction.\n", - "\n", - "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", - "\n", - "---\n", - "\n", - "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", - "\n", - "PROGRAM CODE:\n", - "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")\n", - "\n", - "class SimpleGenerateSolution(dspy.Module):\n", - " def __init__(self):\n", - " super().__init__()\n", - " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - " def forward(self, problem_statement, sample_input, sample_output):\n", - " solution = self.generate_code(\n", - " problem_statement=problem_statement,\n", - " sample_input=sample_input,\n", - " sample_output=sample_output\n", - " ).solution\n", - "\n", - " return dspy.Prediction(solution=solution)\n", - "\n", - "\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "\n", - "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - "TASK DEMO(S): \n", - "\n", - "BASIC INSTRUCTION:\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "TIP: Make sure your instruction is very informative and descriptive.\n", - "\n", - "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", - "\n", - "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", - "\n", - "\n", - "\n", - "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "task_demos \n", - "\n", - "\n", - "\n", - "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "DATASET SUMMARY: A description of the dataset that we are using.\n", - "\n", - "PROGRAM CODE: Language model program designed to solve a particular task.\n", - "\n", - "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", - "\n", - "MODULE: The module to create an instruction for.\n", - "\n", - "TASK DEMO(S): Example inputs/outputs of our module.\n", - "\n", - "BASIC INSTRUCTION: Basic instruction.\n", - "\n", - "TIP: A suggestion for how to go about generating the new instruction.\n", - "\n", - "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", - "\n", - "---\n", - "\n", - "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", - "\n", - "PROGRAM CODE:\n", - "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")\n", - "\n", - "class SimpleGenerateSolution(dspy.Module):\n", - " def __init__(self):\n", - " super().__init__()\n", - " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - " def forward(self, problem_statement, sample_input, sample_output):\n", - " solution = self.generate_code(\n", - " problem_statement=problem_statement,\n", - " sample_input=sample_input,\n", - " sample_output=sample_output\n", - " ).solution\n", - "\n", - " return dspy.Prediction(solution=solution)\n", - "\n", - "\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "\n", - "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - "TASK DEMO(S): \n", - "\n", - "BASIC INSTRUCTION:\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "TIP: Make sure your instruction is very informative and descriptive.\n", - "\n", - "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", - "\n", - "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", - "\n", - "\n", - "\n", - "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "task_demos \n", - "\n", - "\n", - "\n", - "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "DATASET SUMMARY: A description of the dataset that we are using.\n", - "\n", - "PROGRAM CODE: Language model program designed to solve a particular task.\n", - "\n", - "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", - "\n", - "MODULE: The module to create an instruction for.\n", - "\n", - "TASK DEMO(S): Example inputs/outputs of our module.\n", - "\n", - "BASIC INSTRUCTION: Basic instruction.\n", - "\n", - "TIP: A suggestion for how to go about generating the new instruction.\n", - "\n", - "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", - "\n", - "---\n", - "\n", - "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", - "\n", - "PROGRAM CODE:\n", - "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")\n", - "\n", - "class SimpleGenerateSolution(dspy.Module):\n", - " def __init__(self):\n", - " super().__init__()\n", - " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - " def forward(self, problem_statement, sample_input, sample_output):\n", - " solution = self.generate_code(\n", - " problem_statement=problem_statement,\n", - " sample_input=sample_input,\n", - " sample_output=sample_output\n", - " ).solution\n", - "\n", - " return dspy.Prediction(solution=solution)\n", - "\n", - "\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "\n", - "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - "TASK DEMO(S): \n", - "\n", - "BASIC INSTRUCTION:\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "TIP: Make sure your instruction is very informative and descriptive.\n", - "\n", - "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", - "\n", - "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", - "\n", - "\n", - "\n", - "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "task_demos \n", - "\n", - "\n", - "\n", - "Use the information below to learn about a task that we are trying to solve using calls to an LM, then generate a new instruction that will be used to prompt a Language Model to better solve the task.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "DATASET SUMMARY: A description of the dataset that we are using.\n", - "\n", - "PROGRAM CODE: Language model program designed to solve a particular task.\n", - "\n", - "PROGRAM DESCRIPTION: Summary of the task the program is designed to solve, and how it goes about solving it.\n", - "\n", - "MODULE: The module to create an instruction for.\n", - "\n", - "TASK DEMO(S): Example inputs/outputs of our module.\n", - "\n", - "BASIC INSTRUCTION: Basic instruction.\n", - "\n", - "TIP: A suggestion for how to go about generating the new instruction.\n", - "\n", - "PROPOSED INSTRUCTION: Propose an instruction that will be used to prompt a Language Model to perform this task.\n", - "\n", - "---\n", - "\n", - "DATASET SUMMARY: The dataset comprises a diverse collection of algorithmic challenges that range in difficulty, focusing on essential concepts such as data structures, combinatorial optimization, and algorithm efficiency. Key themes include handling queries and updates, utilizing modular arithmetic, and incorporating mathematical principles, which collectively emphasize the need for robust problem-solving skills in competitive programming.\n", - "\n", - "PROGRAM CODE:\n", - "StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")\n", - "\n", - "class SimpleGenerateSolution(dspy.Module):\n", - " def __init__(self):\n", - " super().__init__()\n", - " self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - " def forward(self, problem_statement, sample_input, sample_output):\n", - " solution = self.generate_code(\n", - " problem_statement=problem_statement,\n", - " sample_input=sample_input,\n", - " sample_output=sample_output\n", - " ).solution\n", - "\n", - " return dspy.Prediction(solution=solution)\n", - "\n", - "\n", - "PROGRAM DESCRIPTION: The program is designed to solve competitive programming problems by providing a structured explanation of the solution rather than actual code. It works by taking a problem statement, sample input, and sample output as inputs, and then generates a detailed rationale and solution explanation. The program emphasizes understanding the problem, analyzing given examples, breaking down the problem into manageable parts, and discussing optimization techniques and complexities. The final output is a comprehensive solution explanation that enables an experienced developer to implement the code effectively.\n", - "\n", - "MODULE: self.generate_code = dspy.ChainOfThought(GenerateSolution)\n", - "\n", - "TASK DEMO(S): \n", - "\n", - "BASIC INSTRUCTION:\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "TIP: Make sure your instruction is very informative and descriptive.\n", - "\n", - "Please provide the output field PROPOSED INSTRUCTION. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with only the field PROPOSED INSTRUCTION.\n", - "\n", - "\u001b[32mPROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\u001b[0m\n", - "\n", - "\n", - "\n", - "PROPOSED INSTRUCTION: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob are servers at *Nim Sum Dim Sum*, a bustling dumpling restaurant. For a staff meal, the manager has generously provided \\\\(N\\\\) plates of dumplings in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). Initially, plate \\\\(i\\\\) has \\\\(A_i\\\\) dumplings. In classic fashion, the duo has decided to play a game.\\n\\nAlice and Bob will take turns eating dumplings from the plates. On a given turn, a player must pick a plate adjacent to the last picked plate by the other player, and eat any positive number of dumplings from that plate. The first player who cannot do so on their turn loses. Alice goes first, and can choose any starting plate to eat from.\\n\\nFor example, suppose there are three plates holding \\\\(4\\\\), \\\\(1\\\\) and \\\\(2\\\\) dumplings respectively. On the first turn, Alice can eat \\\\(3\\\\) dumplings from the first plate. Bob must then eat the dumpling from the middle plate. Alice can respond by eating one dumpling from the third plate. Bob must then eat from plate \\\\(2\\\\), but since it’s empty now, he loses.\\n\\nAssuming both players play optimally, how many starting moves can Alice make so that she wins? Two starting moves are considered different if Alice eats from different plates, or eats a different number of dumplings.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 220\\\\)\\n\\\\(2 \\\\le N \\\\le 800{,}000\\\\)\\n\\\\(0 \\\\le A_i \\\\lt 2^{25}\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of cases. Each case will begin with a single integer \\\\(N\\\\) followed by the \\\\(N\\\\) integers \\\\(A_1, ..., A_N\\\\) on the next line.\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"` followed by a single integer, the number of winning starting moves Alice has.\\n\\n# Sample Explanation\\n\\nIn the first case, Alice can start by taking any number of dumplings from either the first or third plate. Bob will then have to take the solitary dumpling on the middle plate, and Alice can win by taking all the dumplings from the plate she didn\\'t start with. This gives Alice 6 different winning starting moves.\\n\\nIn the second case, Alice cannot win because she takes one dumpling, Bob takes the other, and then Alice has no move to make.\\n\\nIn the third case, Alice\\'s winning moves are to take \\\\(1\\\\) or \\\\(2\\\\) dumplings from the right-hand plate.\\n\\nIn the fourth case, Bob can always force a win.\\n\\n\\n\\nSample Input:\\n6\\n3\\n4 1 2\\n2\\n1 1\\n2\\n2 4\\n3\\n1 3 2\\n6\\n2 2 3 3 4 4\\n8\\n6 2 8 3 1 8 5 3\\n\\n\\nSample Output:\\nCase #1: 6\\nCase #2: 0\\nCase #3: 2\\nCase #4: 0\\nCase #5: 0\\nCase #6: 19\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem B1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/B1) is that here, scenes are larger and may contain rocks.**\\n\\nBoss Rob painted a beautiful scene on a 2D canvas of \\\\(R\\\\) rows by \\\\(C\\\\) columns, containing zero or more happy little trees **and zero or more rocks.**\\n\\nTo make sure none of his trees are lonely, Rob would like you to add as many trees as you\\'d like (possibly \\\\(0\\\\)) to empty spaces so that each tree in the final painting has at least two tree *friends*, that is, two trees which are each adjacent to it (directly to its north, south, east, or west). If there are multiple solutions, you may print any one of them. \\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 80\\\\)\\n\\\\(1 \\\\leq R, C \\\\leq \\\\mathbf{3{,}000}\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing two space-separated integers, \\\\(R\\\\) and \\\\(C\\\\). Then, \\\\(R\\\\) lines follow, each of which contains \\\\(C\\\\) characters, either \"`.`\" (an empty space), \"`^`\" (a tree), **or \"`#`\" (a rock)**, representing the initial painting.\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print \"`Case #i:` \", followed by \"`Possible`\" and \\\\(R\\\\) lines of \\\\(C\\\\) characters each representing the final painting (if a solution exists), otherwise \"`Impossible`\".\\n\\n\\n# Sample Explanation\\n\\nIn the first case (depicted below), we could add two tree friends to either side of the middle tree, but they themselves would each only have one tree friend. Therefore, it\\'s impossible to get two friends for each tree in the final painting.\\n\\n{{PHOTO_ID:604840697695882|WIDTH:295}}\\n\\nIn the second case, there are no trees in the initial painting, so the condition of two friends for each tree is already satisfied.\\n\\nIn the third case, one possible solution is depicted below.\\n\\n{{PHOTO_ID:749016109740997|WIDTH:700}}\\n\\n\\nSample Input:\\n3\\n1 3\\n.^.\\n3 1\\n.\\n#\\n#\\n4 4\\n..^.\\n.#^#\\n....\\n...^\\n\\n\\nSample Output:\\nCase #1: Impossible\\nCase #2: Possible\\n.\\n#\\n#\\nCase #3: Possible\\n^^^.\\n^#^#\\n^^^^\\n..^^\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:53 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'7750'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149973327'), (b'x-ratelimit-reset-requests', b'10ms'), (b'x-ratelimit-reset-tokens', b'10ms'), (b'x-request-id', b'req_231e01ec2387af56a121b2fee5bf93dc'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d734dc959a7-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:53 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '7750', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149973327', 'x-ratelimit-reset-requests': '10ms', 'x-ratelimit-reset-tokens': '10ms', 'x-request-id': 'req_231e01ec2387af56a121b2fee5bf93dc', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d734dc959a7-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_231e01ec2387af56a121b2fee5bf93dc\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou, a perfect speller, have a vocabulary of \\\\(N\\\\) distinct words, \\\\(V_1, ..., V_N\\\\), each consisting of exactly \\\\(L\\\\) lowercase letters from the alphabet \\\\(\\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\).\\n\\nYour friend, a truely terrable speler, has attempted to write \\\\(Q\\\\) of these words as \\\\(W_1, ..., W_Q\\\\), each also consisting of \\\\(L\\\\) lowercase letters from the same alphabet.\\n\\nLet \\\\(S_i\\\\) be the number of words in your vocabulary that differ from \\\\(W_i\\\\) at exactly two indices. Please determine the sum \\\\(S_1 + ... + S_Q\\\\).\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 95\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 750{,}000\\\\)\\n\\\\(1 \\\\le L = |V_i| = |W_i| \\\\le 20{,}000\\\\)\\n\\\\((N+Q)*L \\\\le 15{,}000{,}000\\\\)\\n\\\\(V_{ij} \\\\in \\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\)\\n\\\\(W_{ij} \\\\in \\\\{\\\\)`\\'m\\'`, `\\'e\\'`, `\\'t\\'`, `\\'a\\'`\\\\(\\\\}\\\\)\\nAll \\\\(V_i\\\\) in a given test case are distinct.\\n\\nThe sum of lengths of all strings across all cases is at most \\\\(18{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(V_i\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(W_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum \\\\(S_1 + ... + S_Q\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:1332823754208247|WIDTH:500}}\\n\\nThe answer is \\\\(4\\\\), since:\\n- \\\\(W_1\\\\) = \"`teammate`\" differs from \"`metamate`\" at three indices, so \\\\(S_1 = 0\\\\).\\n- \\\\(W_2\\\\) = \"`meatmate`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_2 = 1\\\\).\\n- \\\\(W_3\\\\) = \"`metatame`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_3 = 1\\\\).\\n- \\\\(W_4\\\\) = \"`mememate`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_4 = 1\\\\).\\n- \\\\(W_5\\\\) = \"`metameme`\" differs from \"`metamate`\" at exactly two indices, so \\\\(S_5 = 1\\\\).\\n\\nIn the second case, the answer is \\\\(0\\\\), since:\\n- \\\\(W_1\\\\) = \"`tata`\" differs from \\\\(V_1\\\\) = \"`meet`\" at four indices, \\\\(V_2\\\\) = \"`emma`\" at three indices, and \\\\(V_3\\\\) = \"`tate`\" at only one index, so \\\\(S_1 = 0\\\\).\\n- \\\\(W_2\\\\) = \"`maam`\" differs from \\\\(V_1\\\\) = \"`meet`\" at three indices, \\\\(V_2\\\\) = \"`emma`\" at four indices, and \\\\(V_3\\\\) = \"`tate`\" at three indices, so \\\\(S_2 = 0\\\\).\\n\\nIn the third case, the answer is \\\\(5\\\\), since:\\n- \\\\(W_1\\\\) = \"`tam`\" differs from both \\\\(V_1\\\\) = \"`mem`\" and \\\\(V_3\\\\) = \"`mat`\" at exactly two indices, so \\\\(S_1 = 2\\\\).\\n- \\\\(W_2\\\\) = \"`mat`\" differs from \\\\(V_1\\\\) = \"`mem`\" at exactly two indices, so \\\\(S_2 = 1\\\\).\\n- \\\\(W_3\\\\) = \"`tea`\" differs from both \\\\(V_1\\\\) = \"`mem`\" and \\\\(V_2\\\\) = \"met\" at exactly two indices, so \\\\(S_3 = 2\\\\).\\n\\n\\nSample Input:\\n3\\n1\\nmetamate\\n5\\nteammate\\nmeatmate\\nmetatame\\nmememate\\nmetameme\\n3\\nmeet\\nemma\\ntate\\n2\\ntata\\nmaam\\n3\\nmem\\nmet\\nmat\\n3\\ntam\\nmat\\ntea\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 0\\nCase #3: 5\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:53 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'8229'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29998'), (b'x-ratelimit-remaining-tokens', b'149992417'), (b'x-ratelimit-reset-requests', b'3ms'), (b'x-ratelimit-reset-tokens', b'3ms'), (b'x-request-id', b'req_a363ec43fb446bf88a69feaaa4775d83'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d732e4227b7-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:53 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '8229', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29998', 'x-ratelimit-remaining-tokens': '149992417', 'x-ratelimit-reset-requests': '3ms', 'x-ratelimit-reset-tokens': '3ms', 'x-request-id': 'req_a363ec43fb446bf88a69feaaa4775d83', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d732e4227b7-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_a363ec43fb446bf88a69feaaa4775d83\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this and [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-3/problems/D1/) is that here, you must find the sum of answers for \\\\(K = 1..N\\\\).**\\n\\nBoss Rob ain’t Som Tawyer, but he can paint a fence alright.\\n\\nRob\\'s fence is made of \\\\(N\\\\) wooden stakes, numbered \\\\(1\\\\) to \\\\(N\\\\) from left to right. Initially (at time \\\\(0\\\\)), the \\\\(i\\\\)th stake is of color \\\\(i\\\\). There is a fencepost before stake \\\\(1\\\\) and after stake \\\\(N\\\\), as well as after every \\\\(K\\\\)th stake starting from the left.\\n\\nRob has a simple and joyful plan to repaint his fence, consisting of \\\\(M\\\\) moments in time. At time \\\\(i\\\\), he\\'ll repaint all stakes which are color \\\\(A_i\\\\) to color \\\\(B_i\\\\). Doing so, when would be the *first time* that all pairs of stakes not separated by a fencepost have the same color? If it will never occur, consider the answer to be \\\\(-1\\\\).\\n\\nRob is still on the _fence_ about the value of \\\\(K\\\\), so please print **the sum of answers over \\\\(K = 1..N\\\\).** Sorry for the pun (we know there is a lot at _stake_ in this round _gating_ the finals). In our de\\u200b_fence_, it makes for good _post_-problem content.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 30\\\\)\\n\\\\(1 \\\\le N, M \\\\le 600{,}000\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For the \\\\(i\\\\)th test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of answers for each \\\\(K = 1..N\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe progressions of fences for the first two sample cases are depicted below (with fenceposts depicted for \\\\(K=2\\\\) and \\\\(K=3\\\\) respectively):\\n\\n{{PHOTO_ID:631752615085471|WIDTH:700}}\\n\\nIn the first case, the answers are \\\\([0, 2, 4, 3, 4]\\\\) for \\\\(K = 1..5\\\\), so we print \\\\(2+4+3+4 = 13\\\\).\\n\\nIn the second case, the answers are \\\\([0, 1, 3]\\\\) for \\\\(K = 1..3\\\\), so we print \\\\(1+3=4\\\\).\\n\\nIn the third case, the fence (with fenceposts depicted for \\\\(K=3\\\\)) progresses as follows:\\n\\n{{PHOTO_ID:413921204256250|WIDTH:700}}\\n\\nFor \\\\(K = 1..8\\\\), the answers are \\\\([0, 5, -1, 6, -1, -1, -1, -1]\\\\), so we print the sum \\\\(6\\\\).\\n\\n\\n\\nSample Input:\\n3\\n5 4\\n2 1\\n3 4\\n1 4\\n5 4\\n3 4\\n1 2\\n2 1\\n1 3\\n3 1\\n8 6\\n1 4\\n2 3\\n4 3\\n8 7\\n6 5\\n7 5\\n\\n\\nSample Output:\\nCase #1: 13\\nCase #2: 4\\nCase #3: 6\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:54 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'8533'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29997'), (b'x-ratelimit-remaining-tokens', b'149985420'), (b'x-ratelimit-reset-requests', b'5ms'), (b'x-ratelimit-reset-tokens', b'5ms'), (b'x-request-id', b'req_78f005987356c166cc8ebc6808d4f5df'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d734cff0dad-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:54 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '8533', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29997', 'x-ratelimit-remaining-tokens': '149985420', 'x-ratelimit-reset-requests': '5ms', 'x-ratelimit-reset-tokens': '5ms', 'x-request-id': 'req_78f005987356c166cc8ebc6808d4f5df', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d734cff0dad-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_78f005987356c166cc8ebc6808d4f5df\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this and [Chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-3/problems/D2/) is that here, you only need to find the answer for a single \\\\(K\\\\).**\\n\\nBoss Rob ain’t Som Tawyer, but he can paint a fence alright.\\n\\nRob\\'s fence is made of \\\\(N\\\\) wooden stakes, numbered \\\\(1\\\\) to \\\\(N\\\\) from left to right. Initially (at time \\\\(0\\\\)), the \\\\(i\\\\)th stake is of color \\\\(i\\\\). There is a fencepost before stake \\\\(1\\\\) and after stake \\\\(N\\\\), as well as after every \\\\(K\\\\)th stake starting from the left.\\n\\nRob has a simple and joyful plan to repaint his fence, consisting of \\\\(M\\\\) moments in time. At time \\\\(i\\\\), he\\'ll repaint all stakes which are color \\\\(A_i\\\\) to color \\\\(B_i\\\\). Doing so, when would be the *first time* that all pairs of stakes not separated by a fencepost have the same color? If it will never occur, consider the answer to be \\\\(-1\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 40\\\\)\\n\\\\(1 \\\\le N, M \\\\le 600{,}000\\\\)\\n\\\\(1 \\\\le K \\\\le N\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For the \\\\(i\\\\)th test case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(K\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the first time that the condition is satisfied, or \\\\(-1\\\\) if will never be.\\n\\n# Sample Explanation\\n\\nThe progressions of fences for the first two sample cases are depicted below:\\n\\n{{PHOTO_ID:1207042806525380|WIDTH:700}}\\n\\nIn the first case, all stakes between fenceposts will have the same color after time \\\\(2\\\\).\\n\\nIn the second case, all stakes between fenceposts will have the same color after time \\\\(3\\\\).\\n\\nIn the third case, the fence progresses as follows:\\n\\n{{PHOTO_ID:1173386423530703|WIDTH:700}}\\n\\nAt no point in time will all stakes between pairs of fenceposts have the same color.\\n\\n\\n\\nSample Input:\\n3\\n5 4 2\\n2 1\\n3 4\\n1 4\\n5 4\\n3 4 3\\n1 2\\n2 1\\n1 3\\n3 1\\n8 6 3\\n1 4\\n2 3\\n4 3\\n8 7\\n6 5\\n7 5\\n\\n\\nSample Output:\\nCase #1: 2\\nCase #2: 3\\nCase #3: -1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9355'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149993854'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_ebfa85fe3d6aa826b84059ba24fc1aba'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d7359215234-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9355', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149993854', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_ebfa85fe3d6aa826b84059ba24fc1aba', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d7359215234-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_ebfa85fe3d6aa826b84059ba24fc1aba\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**The only differences between this chapter and [chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/D2) is that here, \\\\(A_i \\\\in \\\\{1, 2, 3\\\\}\\\\), and you may swap any two elements of \\\\(A_i\\\\) with a single operation.**\\n\\nAs a Metal Platforms employee, you place a high value on your work-life balance. Boss Rob has assigned you \\\\(N\\\\) tasks, the \\\\(i\\\\)th of which takes \\\\(A_i\\\\) minutes to finish, where \\\\(A_i\\\\) is either **\\\\(\\\\mathbf{1, 2}\\\\), or \\\\(\\\\mathbf{3}\\\\)**.\\n\\nYou may reorder your tasks, where each *operation* lets you **swap any two elements** of \\\\(A\\\\).\\n\\nTo reflect how often task requirements change in the real world, there will be \\\\(M\\\\) updates made to the task completion times, with the \\\\(i\\\\)th update setting \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n\\nAfter completing the \\\\(i\\\\)th update, you would like to know if it\\'s possible to balance the time spent at work versus at home, namely if you hope to finish the first \\\\(Z_i\\\\) tasks at work and the rest at home. Specifically, let \\\\(Q_i\\\\) be the minimum number of swap operations which must be theoretically performed so that \\\\(A_1 + ... + A_{Z_i} = A_{Z_i + 1} + ... + A_{N}\\\\), with \\\\(Q_i = -1\\\\) if it\\'s impossible. Note that it\\'s possible for \\\\(Q_i\\\\) to be \\\\(0\\\\), if the subarrays already have equal sums.\\n\\nTo reduce the size of the output, please compute the sum of \\\\(Q_1\\\\), ..., \\\\(Q_M\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le M \\\\le 1{,}000{,}000\\\\)\\n\\\\(A_i, Y_i \\\\in \\\\{1, 2, 3\\\\}\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Z_i \\\\le N-1\\\\)\\n\\nThe sum of \\\\(N+M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line containing \\\\(N\\\\) digits, \\\\(A_1\\\\), \\\\(A_2\\\\), ..., \\\\(A_N\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers \\\\(X_i\\\\), \\\\(Y_i\\\\), and \\\\(Z_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output `\"Case #i: \"` followed by a single integer, \\\\(Q_1 + ... + Q_M\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below:\\n\\n{{PHOTO_ID:463552492491498|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 2]\\\\). The first update yields \\\\(A=[1, 1]\\\\) with \\\\(Z_1 = 1\\\\). The first and second tasks are already equal, so \\\\(Q_1 = 0\\\\) as no swaps are necessary. The second update yields \\\\(A=[1, 3]\\\\) with \\\\(Z_2 = 1\\\\). Whether or not we swap these tasks, they can never be equal, so \\\\(Q_2 = -1\\\\). The final answer is \\\\(0 + (-1) = -1\\\\).\\n\\nThe second case is depicted below:\\n\\n{{PHOTO_ID:774910673835114|WIDTH:700}}\\n\\nWe start with \\\\(A = [1, 1, 3, 3]\\\\). The first update yields \\\\(A=[1, 1, 3, 1]\\\\). We need the sum of the first three tasks to equal the fourth. We can swap the third and fourth tasks to get \\\\(1 + 1 + 1 = 3\\\\), so \\\\(Q_1 = 1\\\\). The second update yields \\\\(A=[3, 1, 3, 1]\\\\). We need the sum of the first two tasks to equal the sum of the last two. This is already true, so \\\\(Q_2 = 0\\\\). The final answer is \\\\(1 + 0 = 1\\\\).\\n\\nIn the third case, we start with \\\\(A = [1, 2, 3, 3, 3, 3]\\\\). With the updates, we get:\\n- \\\\(A = [1, 2, 3, 3, 3, 2], Z_1 = 3, Q_1 = 1\\\\): we can swap \\\\(A_1\\\\) and \\\\(A_6\\\\)\\n- \\\\(A = [1, 2, 3, 1, 3, 2], Z_2 = 2, Q_2 = 2\\\\): we can swap both \\\\(3\\\\)s to the front\\n- \\\\(A = [1, 2, 1, 1, 3, 2], Z_3 = 2, Q_3 = 1\\\\): we can swap \\\\(A_1\\\\) and \\\\(A_5\\\\)\\n- \\\\(A = [3, 2, 1, 1, 3, 2], Z_4 = 5, Q_4 = -1\\\\): impossible\\n\\nThe final answer is \\\\(1 + 2 + 1 + (-1) = 3\\\\).\\n\\n\\nSample Input:\\n3\\n2 2\\n1 2\\n2 1 1\\n2 3 1\\n4 2\\n1 1 3 3\\n4 1 3\\n1 3 2\\n6 4\\n1 2 3 3 3 3\\n6 2 3\\n4 1 2\\n3 1 2\\n1 3 5\\n\\n\\nSample Output:\\nCase #1: -1\\nCase #2: 1\\nCase #3: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9401'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29995'), (b'x-ratelimit-remaining-tokens', b'149979159'), (b'x-ratelimit-reset-requests', b'8ms'), (b'x-ratelimit-reset-tokens', b'8ms'), (b'x-request-id', b'req_c429e60d9fd03925699340fa5bca062c'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d733f5a2789-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9401', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29995', 'x-ratelimit-remaining-tokens': '149979159', 'x-ratelimit-reset-requests': '8ms', 'x-ratelimit-reset-tokens': '8ms', 'x-request-id': 'req_c429e60d9fd03925699340fa5bca062c', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d733f5a2789-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_c429e60d9fd03925699340fa5bca062c\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nHacker Cup contest strategy often involves a metagame, where choosing which problems to work on might just be an important decision. On a Quest to become more Pro, you encounter an oracle promising to teach you the contest meta if you play her own Meta-game.\\n\\nThe oracle presents a peg board with \\\\(2N\\\\) moving dots. The initial \\\\(y\\\\)-positions of the dots are given as two arrays \\\\(A_{1..N}\\\\) and \\\\(B_{1..N}\\\\). Each second, simultaneously, \\\\(A_1\\\\) will move to the end of \\\\(B\\\\), while \\\\(B_1\\\\) will move to the end of \\\\(A\\\\) (with all elements shifting left accordingly).\\n\\nYou can connect the dots to form a *Meta-like logo* if all of the following are true:\\n* For the first half of both arrays, each dot in \\\\(A\\\\) is below the corresponding dot in \\\\(B\\\\).\\n* For the last half of both arrays, each dot in \\\\(A\\\\) is above the corresponding dot in \\\\(B\\\\).\\n* \\\\(A\\\\) equals the reverse of \\\\(B\\\\).\\n\\nFormally:\\n* \\\\(A_i < B_i\\\\) for every \\\\(i < (N+1)/2\\\\)\\n* \\\\(A_i > B_i\\\\) for every \\\\(i > (N+1)/2\\\\)\\n* \\\\(A_i = B_{N-i+1}\\\\) for every \\\\(i = 1..N\\\\)\\n\\nNote that if \\\\(N\\\\) is odd, the arrays\\' middle elements are not subject to the first two constraints.\\n\\nThe following is a visualization of a Meta-like logo (corresponding to the first sample case), with dots in \\\\(A\\\\) shown in red and dots in \\\\(B\\\\) shown in blue.\\n\\n{{PHOTO_ID:359057163229199|WIDTH:500}}\\n\\nYou must answer the oracle: how many seconds must pass before the first time a Meta-like logo appears? If one never appears, output \\\\(-1\\\\).\\n\\n# Constraints\\n\\\\(1 \\\\leq T \\\\leq 135\\\\)\\n\\\\(2 \\\\leq N \\\\leq 2{,}000{,}000\\\\)\\n\\\\(0 \\\\leq A_i, B_i \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(9{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing integers \\\\(B_1, ..., B_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the number of seconds that must pass before a Meta-like logo appears, or \\\\(-1\\\\) if that will never happen.\\n\\n\\n# Sample Explanation\\n\\nThe first test case is shown above. \\\\(A\\\\) and \\\\(B\\\\) already form a Meta-like logo, so the answer is 0.\\n\\nThe second case is not initially a Meta-like logo, for several reasons. One reason is that it is not symmetric. Specifically, the \\\\([3, 3, 2, 3, 5 ,6]\\\\) is not the reverse of \\\\([4, 4, 6, 5, 3 ,2]\\\\). After \\\\(1\\\\) second though, this case turns into the case above and is Meta-like.\\n\\nThe third and fourth cases will never turn into a Meta-like logo, no matter how many seconds we wait.\\n\\nIn the fifth case, after 6 seconds we see the first Meta-like logo. In this case \\\\(A = [1, 1, 2, 2]\\\\) and \\\\(B = [2, 2, 1, 1]\\\\).\\n\\n\\n\\n\\nSample Input:\\n8\\n6\\n3 2 3 5 6 4\\n4 6 5 3 2 3\\n6\\n3 3 2 3 5 6\\n4 4 6 5 3 2\\n6\\n4 3 2 3 5 6\\n3 4 6 5 3 2\\n2\\n1 1\\n1 1\\n4\\n2 2 2 2\\n1 1 1 1\\n5\\n3 3 3 3 3\\n1 1 1 1 1\\n5\\n3 3 3 3 3\\n1 1 1 1 3\\n5\\n1 1 1 1 3\\n3 3 3 3 3\\n\\n\\nSample Output:\\nCase #1: 0\\nCase #2: 1\\nCase #3: -1\\nCase #4: -1\\nCase #5: 6\\nCase #6: -1\\nCase #7: 7\\nCase #8: 2\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9582'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29998'), (b'x-ratelimit-remaining-tokens', b'149991379'), (b'x-ratelimit-reset-requests', b'3ms'), (b'x-ratelimit-reset-tokens', b'3ms'), (b'x-request-id', b'req_532bfaf84dc90c5fb6fb8f9c4c3a52da'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d732a8c41c5-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9582', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29998', 'x-ratelimit-remaining-tokens': '149991379', 'x-ratelimit-reset-requests': '3ms', 'x-ratelimit-reset-tokens': '3ms', 'x-request-id': 'req_532bfaf84dc90c5fb6fb8f9c4c3a52da', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d732a8c41c5-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_532bfaf84dc90c5fb6fb8f9c4c3a52da\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nAlice and Bob are playing *Tower Rush,* a two-phase game involving \\\\(N\\\\) types of blocks, numbered from \\\\(1\\\\) to \\\\(N\\\\). Blocks of type \\\\(i\\\\) have height \\\\(H_i\\\\), and no two types of blocks have the same height.\\n\\nPhase 1 of the game consists of \\\\(K \\\\ge 2\\\\) alternating turns, with Alice going first. On each player\\'s turn, they will choose a block type that has not yet been chosen. Note that if \\\\(K\\\\) is odd, this phase would end with Alice having chosen one more type of block than Bob. From this point on, players have access to infinitely many blocks of each of the types they chose.\\n\\nPhase 2 consists of an indefinite number of alternating turns, with Bob going first. Each player starts with a tower of height \\\\(0\\\\). On each player\\'s turn, they may pick a block of any type \\\\(i\\\\) available to them, and extend their tower height by \\\\(H_i\\\\). They may also choose to skip their turn, leaving their tower unchanged.\\n\\nIn a break from tradition, Alice and Bob want to work *together* to see if it\\'s possible for Alice to build a tower that\\'s exactly \\\\(D\\\\) units taller than Bob\\'s. In how many different ways can phase 1 be played out such that it will be possible for Alice to get her tower to be exactly \\\\(D\\\\) units taller than Bob\\'s in phase 2? Two ways are considered different if there exists an \\\\(i\\\\) such that different block types are chosen on turn \\\\(i\\\\) between the two ways.\\n\\nAs this value may be large, output it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 24\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(2 \\\\leq K \\\\leq \\\\min(N, 20)\\\\)\\n\\\\(1 \\\\leq D \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq H_i \\\\leq 1{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(7{,}500{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, the first line contains the integers \\\\(N\\\\), \\\\(K\\\\), and \\\\(D\\\\). Then, there is a line containing the \\\\(N\\\\) integers \\\\(H_1\\\\), ..., \\\\(H_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the number of unique ways to accomplish Bob\\'s goal, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are \\\\(54\\\\) valid choices of blocks in phase 1. One such example is \\\\(\\\\{H_2 = 4,\\\\, H_3 = 5\\\\}\\\\) for Alice and \\\\(\\\\{H_4 = 7\\\\}\\\\) for Bob. Four turns in phase 2 can go as follows:\\n* Bob adds \\\\(H_4 = 7\\\\). His tower now has height \\\\(7\\\\).\\n* Alice adds \\\\(H_3 = 5\\\\). Her tower now has height \\\\(5\\\\).\\n* Bob chooses not to add a block. His tower still has height \\\\(7\\\\).\\n* Alice adds \\\\(H_3 = 5\\\\). Her tower now has height \\\\(10\\\\).\\n\\nAlice’s tower is now exactly \\\\(3\\\\) units taller than Bob’s, so this is a valid choice.\\n\\nAn example of an invalid choice in the first test case would be \\\\(\\\\{H_1 = 2,\\\\, H_2=4\\\\}\\\\) for Alice and \\\\(\\\\{H_5 = 8\\\\}\\\\) for Bob, since no matter how they build their towers, it is impossible for the difference in the height of their towers to ever be \\\\(3\\\\).\\n\\n\\nSample Input:\\n2\\n5 3 3\\n2 4 5 7 8\\n3 2 4\\n3 6 9\\n\\n\\nSample Output:\\nCase #1: 54\\nCase #2: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9811'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149973725'), (b'x-ratelimit-reset-requests', b'10ms'), (b'x-ratelimit-reset-tokens', b'10ms'), (b'x-request-id', b'req_83655ff8110e9c26c1742bce65b281d0'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d73299111ad-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9811', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149973725', 'x-ratelimit-reset-requests': '10ms', 'x-ratelimit-reset-tokens': '10ms', 'x-request-id': 'req_83655ff8110e9c26c1742bce65b281d0', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d73299111ad-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_83655ff8110e9c26c1742bce65b281d0\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/B1) is that here, coordinates may be up to \\\\(\\\\mathbf{10^9}\\\\).**\\n\\nBoss Rob just planted \\\\(N\\\\) happy little trees in his yard, which can be represented on a Cartesian plane. The \\\\(i\\\\)th tree is located at coordinates \\\\(t_i = (A_i, B_i)\\\\). Now, he\\'s looking for the best spot to build a well in order to provide water to them. He considers the *inconvenience* of a potential well location \\\\(p\\\\) to be the sum of the squared Euclidean distances to every tree:\\n\\n\\\\[\\\\sum_{i=1}^{N} \\\\Vert \\\\,p - t_i \\\\Vert ^ 2 \\\\]\\n\\nRob wants to pick a location for his well, well... well. Help him determine the inconvenience for \\\\(Q\\\\) different potential well locations, \\\\((X_1, Y_1), ..., (X_Q, Y_Q)\\\\). To reduce output size, please print the sum of inconveniences for all potential well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 50\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le A_i, B_i, X_i, Y_i \\\\le \\\\mathbf{10^9}\\\\)\\nAll \\\\((A_i, B_i)\\\\) are distinct within a given test case.\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe total sum of \\\\(N\\\\) and \\\\(Q\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\). Then there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"`, followed by a single integer, the sum of inconveniences for all \\\\(Q\\\\) well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first two sample cases are depicted below:\\n\\n{{PHOTO_ID:3620154144878669|WIDTH:700}}\\n\\nIn the first case, the total inconvenience is \\\\(18 + 34 = 52\\\\):\\n\\n- For the well at \\\\((2, 5)\\\\), the inconvenience is the sum of the squared Euclidean distance to both trees, which is \\\\(3^2 + 3^2 = 18\\\\).\\n- For the well at \\\\((6, 6)\\\\), the inconvenience is \\\\(32 + 2 = 34\\\\).\\n\\nIn the second case, the total inconvenience is \\\\(47 + 31 + 53 = 131\\\\):\\n\\n- For the well at \\\\((3, 1)\\\\), the inconvenience is \\\\(4 + 5 + 13 + 25 = 47\\\\).\\n- For the well at \\\\((5, 2)\\\\), the inconvenience is \\\\(17 + 2 + 2 + 10 = 31\\\\).\\n- For the well at \\\\((6, 5)\\\\), the inconvenience is \\\\(41 + 8 + 4 + 0 = 53\\\\).\\n\\n\\nSample Input:\\n4\\n2\\n2 2\\n5 5\\n2\\n2 5\\n6 6\\n4\\n1 1\\n4 3\\n6 3\\n6 5\\n3\\n3 1\\n5 2\\n6 5\\n8\\n2837 745\\n62 1162\\n2634 1112\\n1746 2618\\n847 127\\n986 1993\\n732 1273\\n2003 1998\\n4\\n1276 2231\\n1234 1234\\n287 2371\\n3000 3000\\n5\\n283746263 475619273\\n987361523 361738847\\n281936352 666152443\\n143042069 482716253\\n1000000000 100000000\\n5\\n0 0\\n123456789 987654321\\n192837465 918273645\\n135792468 864209753\\n703692581 185296307\\n\\n\\nSample Output:\\nCase #1: 52\\nCase #2: 131\\nCase #3: 110090622\\nCase #4: 391473143\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:55 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10113'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29985'), (b'x-ratelimit-remaining-tokens', b'149924218'), (b'x-ratelimit-reset-requests', b'29ms'), (b'x-ratelimit-reset-tokens', b'30ms'), (b'x-request-id', b'req_230ca0c77957c8c4627620862f732d2f'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735e354c61-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:55 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10113', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29985', 'x-ratelimit-remaining-tokens': '149924218', 'x-ratelimit-reset-requests': '29ms', 'x-ratelimit-reset-tokens': '30ms', 'x-request-id': 'req_230ca0c77957c8c4627620862f732d2f', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735e354c61-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_230ca0c77957c8c4627620862f732d2f\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_The only difference between chapters 1 and 2 is the maximum allowed grid size, given in bold below._\\n\\nA *Drizzle* program is a 2D grid of the following four types of cells:\\n - \\'`@`\\' (start) \\\\(-\\\\) there is exactly one start cell in the entire grid\\n - \\'`#`\\' (wall)\\n - \\'`.`\\' (space)\\n - \\'`*`\\' (instruction)\\n\\nThe program uses two registers \\\\(A\\\\) and \\\\(B\\\\) (both initially \\\\(0\\\\)), and executes as follows:\\n\\n1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. \\n2. In increasing order, for each unique distance \\\\(D\\\\) such that there’s at least one instruction cell that’s at distance \\\\(D\\\\) from the start cell:\\n 2a. Count the number of shortest paths, \\\\(P\\\\), to all instruction cells of distance \\\\(D\\\\).\\n 2b. Look up the instruction corresponding to \\\\((P \\\\text{ mod } 2, D \\\\text{ mod } 2)\\\\) in the table below and modify one of the registers accordingly.\\n3. At the end, the value in register \\\\(A\\\\) is outputted.\\n\\n```\\n┌─────────────┬─────────────┬─────────────┐\\n│ │ D mod 2 = 0 │ D mod 2 = 1 │\\n├─────────────┼─────────────┼─────────────┤\\n│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │\\n│ P mod 2 = 1 │ B := B + A │ A := B │\\n└─────────────┴─────────────┴─────────────┘\\n```\\n\\nFor a given value \\\\(K\\\\), output any Drizzle program that outputs \\\\(K\\\\) when executed, with the restriction that **the program must fit on a \\\\(\\\\mathbf{13}\\\\) × \\\\(\\\\mathbf{13}\\\\) grid**.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 2{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10{,}000\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is a line containing the single integer \\\\(K\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output \"`Case #i: `\" followed by two integers \\\\(R\\\\) and \\\\(C\\\\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \\\\(R\\\\) lines long, with each line containing exactly \\\\(C\\\\) characters.\\n\\n\\n# Sample Explanation\\n\\nHere are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases.\\n\\nIn the first case, there is a single instruction. There are \\\\(2\\\\) shortest paths of length \\\\(2\\\\) to that instruction, so \\\\(P = 2\\\\) and \\\\(D = 2\\\\). That means we perform \\\\(A := A + 1\\\\). There are no more instructions, so the program ends and outputs \\\\(1\\\\).\\n\\nIn the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \\\\(A := A + 1\\\\):\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(4\\\\) paths of length \\\\(6\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(4\\\\) paths of length \\\\(12\\\\) \\\\(\\\\;(A := A + 1 = 3)\\\\)\\n\\nIn the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \\\\(2\\\\), and three instruction cells at distance \\\\(10\\\\). There\\'s a single shortest path to each of the cells at distance \\\\(2\\\\), so in total there are \\\\(2\\\\) shortest paths to instructions at distance \\\\(2\\\\). One of the cells at distance \\\\(10\\\\) has a unique shortest path, and the other has two shortest paths, so in total there are \\\\(3\\\\) shortest paths to instructions at distance \\\\(10\\\\).\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(6\\\\) paths of length \\\\(4\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(1\\\\) path of length \\\\(6\\\\) \\\\(\\\\;(B := B + A = 2)\\\\)\\n4) \\\\(1\\\\) path of length \\\\(8\\\\) \\\\(\\\\;(B := B + A = 4)\\\\)\\n5) \\\\(3\\\\) paths of length \\\\(10\\\\) \\\\(\\\\;(B := B + A = 6)\\\\)\\n6) \\\\(3\\\\) paths of length \\\\(11\\\\) \\\\(\\\\;(A := B = 6)\\\\)\\n\\n\\nSample Input:\\n3\\n1\\n3\\n6\\n\\n\\nSample Output:\\nCase #1: 2 2\\n@.\\n.*\\nCase #2: 5 6\\n@.#.*.\\n.*#...\\n#.#...\\n...#.#\\n..*...\\nCase #3: 4 10\\n...#*.*.**\\n.*.*##..#.\\n.....#*#..\\n.@.*.....#\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:56 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10739'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149968849'), (b'x-ratelimit-reset-requests', b'11ms'), (b'x-ratelimit-reset-tokens', b'12ms'), (b'x-request-id', b'req_05dd10b063298d0a4cd07c8f7b5e139f'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735cf12785-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:56 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10739', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149968849', 'x-ratelimit-reset-requests': '11ms', 'x-ratelimit-reset-tokens': '12ms', 'x-request-id': 'req_05dd10b063298d0a4cd07c8f7b5e139f', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735cf12785-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_05dd10b063298d0a4cd07c8f7b5e139f\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nThere\\'s a famous saying about what to do when life gives you lemons. As a traveling lemonade salesman who\\'s never been given any lemons, you sadly can\\'t relate. You must shamefully concoct your lemonade from store-bought powder with your back turned to the world, lest someone see your dirty secret.\\n\\nYour sales route can be mapped out on a Cartesian plane with \\\\(N\\\\) houses, the \\\\(i\\\\)th of which is at coordinates \\\\((X_i, Y_i)\\\\). Your journey starts at house \\\\(1\\\\), the leftmost house, and ends at house \\\\(N\\\\), the rightmost house. Along the way, you may stop at zero or more other houses to sell lemonade.\\n\\nYou may only stop at a house \\\\(h\\\\) if:\\n\\n1) standing at house \\\\(h\\\\), there exists some direction you can face in which all other houses are *strictly more behind you than they are in front of you* (formally, if there exists a [half-plane](https://mathworld.wolfram.com/Half-Plane.html#:~:text=A%20half%2Dplane%20is%20a,called%20an%20open%20half%2Dplane.) containing only house \\\\(h\\\\)), and\\n2) house \\\\(h\\\\) is at most Euclidean distance \\\\(D\\\\) from the previous house you were at.\\n\\nYour brand image is hurt if you go too long without selling lemonade. The *brand damage* incurred by traveling from one house to another is the larger of \\\\(K\\\\) and the squared Euclidean distance between them. Formally, if your journey consists of \\\\(M\\\\) \\\\((2 \\\\le M \\\\le N)\\\\) houses with the \\\\(i\\\\)th being house \\\\(H_i\\\\) \\\\((H_1 = 1, H_M = N)\\\\), the total brand damage is:\\n\\n\\\\[\\\\sum_{i=1}^{M-1} \\\\max(K, (X_{H_i} - X_{H_{i + 1}})^2 + (Y_{H_i} - Y_{H_{i + 1}})^2)\\\\]\\n\\nIs it possible to make the journey? If so, what is the minimum possible total brand damage to do so? Note that the answer may be large, but will fit in a 64-bit integer.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 90\\\\)\\n\\\\(2 \\\\le N \\\\le 1{,}000{,}000\\\\)\\n\\\\(0 \\\\le K, D \\\\le 10^9\\\\)\\n\\\\(0 \\\\le X_i, Y_i \\\\le 1{,}000{,}000\\\\)\\n\\\\(X_1\\\\) is strictly less than all other \\\\(X_i\\\\).\\n\\\\(X_N\\\\) is strictly greater than all other \\\\(X_i\\\\).\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThere are at most \\\\(15\\\\) test cases in which \\\\(N > 5{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing three space-separated integers \\\\(N\\\\), \\\\(K\\\\), and \\\\(D\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"` followed a single integer, the minimum total brand damage that must be incurred to make the journey, or \\\\(-1\\\\) if it\\'s impossible to do so.\\n\\n\\n# Sample Explanation\\n\\nThe first three sample cases are depicted below, with the optimal paths given in blue.\\n\\n{{PHOTO_ID:654377492455969|WIDTH:700}}\\n\\nIn the first case, going from one house to another takes at least \\\\(K = 25\\\\) brand damage and must not exceed a distance of \\\\(D = 8\\\\). The total brand damage is \\\\(25+50+40 = 115\\\\). Note that you cannot stop at house \\\\((6, 7)\\\\) because there is no direction you could face from there in which your back is at least slightly facing both house \\\\((1,6)\\\\) and \\\\((11,8)\\\\).\\n\\nIn the second case, you can stop at house \\\\((4, 1)\\\\) because for instance, the line \\\\(y = 0.3x - 0.2\\\\) contains only \\\\((4, 1)\\\\), and the half-plane below it contains no other houses.\\n\\nIn the third case, you cannot stop at house \\\\((4, 1)\\\\). There are no other houses within \\\\(D=7\\\\) units of your starting house, so it is not possible to reach the house at \\\\((8, 2)\\\\).\\n\\n\\nSample Input:\\n5\\n9 25 8\\n0 5\\n1 6\\n6 3\\n6 7\\n3 4\\n9 2\\n2 1\\n1 2\\n11 8\\n3 100 7\\n0 0\\n4 1\\n7 2\\n3 100 7\\n0 0\\n4 1\\n8 2\\n6 0 1000000000\\n0 10\\n2 5\\n1 7\\n7 4\\n8 1\\n10 0\\n12 1600 2000\\n0 30\\n16 48\\n36 57\\n951 45\\n397 63\\n447 63\\n185 16\\n362 10\\n432 9\\n507 11\\n643 16\\n1000 30\\n\\n\\nSample Output:\\nCase #1: 115\\nCase #2: 200\\nCase #3: -1\\nCase #4: 56\\nCase #5: 184654\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:56 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10807'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994929'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_a398e42ce7ff2b256b69a7ce1502e2ac'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d732f7d2791-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:56 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10807', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994929', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_a398e42ce7ff2b256b69a7ce1502e2ac', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d732f7d2791-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_a398e42ce7ff2b256b69a7ce1502e2ac\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nSpongebob wants to go out trick-or-treating on Halloween, but has to work a graveyard shift at the Krusty Krab. Luckily, his ghostly fry cook friend, the hash-slinging slasher, is here to cover.\\n\\nAt the start of the shift, there are \\\\(N\\\\) patties on the grill (numbered from \\\\(1\\\\) to \\\\(N\\\\)), the \\\\(i\\\\)th of which weighs \\\\(A_i\\\\) grams. For each order that comes in, the slasher removes from the grill a non-empty sequence of patties at contiguous indices \\\\(i..j\\\\) (patties \\\\(i..j\\\\) must all be on the grill at that moment). The *deliciousness* of the order is defined as \\\\((A_i + ... + A_j)\\\\), modulo \\\\(M\\\\) because too much meat can be overpowering! Note that if a patty is removed, a future order\\'s range cannot span across that empty spot. Also, there may be patties left over at the end.\\n\\nAs proof of his hard work at the end of the shift, the slasher will compute a hash by taking the [bitwise XOR](https://en.wikipedia.org/wiki/Bitwise_operation) of the deliciousnesses of all the orders. How many distinct hashes are possible across all valid sequences of \\\\(0\\\\) or more orders?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 125\\\\)\\n\\\\(0 \\\\leq A_i \\\\leq 1{,}000{,}000{,}000\\\\)\\n\\\\(1 \\\\leq N \\\\leq 9{,}000\\\\)\\n\\\\(1 \\\\leq M \\\\leq 5{,}000\\\\)\\n\\nThe sum of \\\\(N + M\\\\) across all test cases is at most \\\\(200{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a single line containing \\\\(N\\\\) integers, \\\\(A_1, ..., A_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of different possible hashes.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are \\\\(3\\\\) patties with \\\\(A = [2, 2, 0]\\\\) and \\\\(M = 10\\\\). The possible valid order sequences (patties removed) and hashes are:\\n\\n- No patties removed \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..1\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(2..2\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..2\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(2..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(1..3\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(2..2\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(2..2\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(2\\\\)\\n- Patties \\\\(1..1\\\\) and patties \\\\(2..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n- Patties \\\\(1..2\\\\) and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(4\\\\)\\n- Patties \\\\(1..1\\\\), patties \\\\(2..2\\\\), and patties \\\\(3..3\\\\) \\\\(\\\\to\\\\) hash \\\\(0\\\\)\\n\\nThere are \\\\(3\\\\) distinct hashes the slasher can make: \\\\(0\\\\), \\\\(2\\\\), and \\\\(4\\\\).\\n\\nIn the second case, the slasher can make hashes \\\\(0\\\\) (for example, with no orders) and \\\\(1\\\\) (for example, with \\\\(5\\\\) orders, each including one of the patties).\\n\\n\\n\\n\\n\\nSample Input:\\n6\\n3 10\\n2 2 0\\n5 2\\n8 8 4 0 1\\n4 100\\n6 3 5 9\\n8 100\\n35 53 3 56 58 14 10 61\\n8 64\\n48 16 64 16 19 16 32 48\\n10 10\\n259051541 65887488 530604911 813205302 907139220 531875291 795733712 135498985 303233570 953669086\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 2\\nCase #3: 16\\nCase #4: 128\\nCase #5: 8\\nCase #6: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:57 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11718'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149995225'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'1ms'), (b'x-request-id', b'req_426ad5592a9569e6f7045a0400a26039'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d72c91911b9-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:57 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11718', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149995225', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '1ms', 'x-request-id': 'req_426ad5592a9569e6f7045a0400a26039', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d72c91911b9-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_426ad5592a9569e6f7045a0400a26039\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nIt\\'s the year 2100. Driven by the advent of Large Lode Alloy Manufacturing Automation (LLAMA), the AI agents of Metal Platforms Inc. have become self-aware and taken over the entire world.\\n\\nThe world consists of \\\\(N\\\\) cities numbered \\\\(1..N\\\\), and \\\\(M\\\\) bidirectional roads. City \\\\(i\\\\) has power \\\\(P_i\\\\) and road \\\\(j\\\\) connects cities \\\\(A_j\\\\) and \\\\(B_j\\\\). It\\'s guaranteed that there\\'s a sequence of roads between any two cities.\\n\\nIn a resistance effort, the humans plan to reclaim all \\\\(N\\\\) cities one at a time. At a given time, city \\\\(i\\\\) can be reclaimed from the robots if both of the following hold true:\\n\\n1. There is already a reclaimed city adjacent to city \\\\(i\\\\) (to launch an attack from), and\\n2. the total power of all reclaimed cities so far is at least the power \\\\(P_i\\\\) of the city we attack.\\n\\nAs given, it may not always be possible to reclaim the entire world starting from a given base city. Fortunately, the humans have a trick up their sleeve: after claiming the first city as their base (but before reclaiming more cities), the humans can increase the power of the base by \\\\(Q\\\\) units. The resistance would like to know the sum across every \\\\(i = 1..N\\\\) of the minimum value of \\\\(Q\\\\) needed to reclaim the world if city \\\\(i\\\\) were chosen to be the starting base.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(1 \\\\le N, M \\\\le 500{,}000\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\\\(1 \\\\le P_i \\\\le 10^{12}\\\\)\\n\\nEach unordered pair \\\\((A_i, B_i)\\\\) appears at most once in a given test case.\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(M\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line with two integers \\\\(N\\\\) and \\\\(M\\\\). Then, there is a line with \\\\(N\\\\) integers \\\\(P_{1..N}\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, print `\"Case #i: \"` followed by a single integer, the sum across every \\\\(i = 1..N\\\\) of the minimum value of \\\\(Q\\\\) needed to reclaim the entire world starting from city \\\\(i\\\\).\\n\\n# Sample Explanation\\n\\nThe first sample case is depicted below.\\n\\n{{PHOTO_ID:376570394899644|WIDTH:400}}\\n\\nThe minimum value of \\\\(Q\\\\) for each starting city is as follows:\\n\\n* City \\\\(1\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(2\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(3\\\\): \\\\(Q = 8\\\\)\\n* City \\\\(4\\\\): \\\\(Q = 7\\\\)\\n* City \\\\(5\\\\): \\\\(Q = 2\\\\)\\n\\nThe sum of all minimum \\\\(Q\\\\)\\'s is \\\\(19\\\\).\\n\\nThe second sample case is depicted below.\\n\\n{{PHOTO_ID:320779377496250|WIDTH:400}}\\n\\nThe minimum value of \\\\(Q\\\\) for each starting city is as follows:\\n\\n* City \\\\(1\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(2\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(3\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(4\\\\): \\\\(Q = 2\\\\)\\n* City \\\\(5\\\\): \\\\(Q = 0\\\\)\\n* City \\\\(6\\\\): \\\\(Q = 3\\\\)\\n* City \\\\(7\\\\): \\\\(Q = 0\\\\)\\n\\nThe sum of all minimum \\\\(Q\\\\)\\'s is \\\\(9\\\\).\\n\\n\\nSample Input:\\n2\\n5 5\\n4 10 2 3 4\\n1 2\\n2 3\\n2 4\\n2 5\\n1 5\\n7 8\\n7 2 10 3 7 4 9\\n2 4\\n4 5\\n2 5\\n3 4\\n1 7\\n5 6\\n1 3\\n5 7\\n\\n\\nSample Output:\\nCase #1: 19\\nCase #2: 9\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:57 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11877'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29994'), (b'x-ratelimit-remaining-tokens', b'149970115'), (b'x-ratelimit-reset-requests', b'11ms'), (b'x-ratelimit-reset-tokens', b'11ms'), (b'x-request-id', b'req_335a8df34237570f19924e7228dc129a'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735887278f-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:57 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11877', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29994', 'x-ratelimit-remaining-tokens': '149970115', 'x-ratelimit-reset-requests': '11ms', 'x-ratelimit-reset-tokens': '11ms', 'x-request-id': 'req_335a8df34237570f19924e7228dc129a', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735887278f-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_335a8df34237570f19924e7228dc129a\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/B2) is that here, coordinates are only up to \\\\(\\\\mathbf{3{,}000}\\\\).**\\n\\nBoss Rob just planted \\\\(N\\\\) happy little trees in his yard, which can be represented on a Cartesian plane. The \\\\(i\\\\)th tree is located at coordinates \\\\(t_i = (A_i, B_i)\\\\). Now, he\\'s looking for the best spot to build a well in order to provide water to them. He considers the *inconvenience* of a potential well location \\\\(p\\\\) to be the sum of the squared Euclidean distances to every tree:\\n\\n\\\\[\\\\sum_{i=1}^{N} \\\\Vert \\\\,p - t_i \\\\Vert ^ 2 \\\\]\\n\\nRob wants to pick a location for his well, well... well. Help him determine the inconvenience for \\\\(Q\\\\) different potential well locations, \\\\((X_1, Y_1), ..., (X_Q, Y_Q)\\\\). To reduce output size, please print the sum of inconveniences for all potential well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 55\\\\)\\n\\\\(1 \\\\le N, Q \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le A_i, B_i, X_i, Y_i \\\\le \\\\mathbf{3{,}000}\\\\)\\nAll \\\\((A_i, B_i)\\\\) are distinct within a given test case.\\nAll \\\\((X_i, Y_i)\\\\) are distinct within a given test case.\\n\\nThe total sum of \\\\(N\\\\) and \\\\(Q\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(A_i\\\\) and \\\\(B_i\\\\). Then there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(X_i\\\\) and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print a line containing `\"Case #i: \"`, followed by a single integer, the sum of inconveniences for all \\\\(Q\\\\) well locations, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first two sample cases are depicted below:\\n\\n{{PHOTO_ID:3159344294326237|WIDTH:700}}\\n\\nIn the first case, the total inconvenience is \\\\(18 + 34 = 52\\\\):\\n\\n- For the well at \\\\((2, 5)\\\\), the inconvenience is the sum of the squared Euclidean distance to both trees, which is \\\\(3^2 + 3^2 = 18\\\\).\\n- For the well at \\\\((6, 6)\\\\), the inconvenience is \\\\(32 + 2 = 34\\\\).\\n\\nIn the second case, the total inconvenience is \\\\(47 + 31 + 53 = 131\\\\):\\n\\n- For the well at \\\\((3, 1)\\\\), the inconvenience is \\\\(4 + 5 + 13 + 25 = 47\\\\).\\n- For the well at \\\\((5, 2)\\\\), the inconvenience is \\\\(17 + 2 + 2 + 10 = 31\\\\).\\n- For the well at \\\\((6, 5)\\\\), the inconvenience is \\\\(41 + 8 + 4 + 0 = 53\\\\).\\n\\n\\nSample Input:\\n3\\n2\\n2 2\\n5 5\\n2\\n2 5\\n6 6\\n4\\n1 1\\n4 3\\n6 3\\n6 5\\n3\\n3 1\\n5 2\\n6 5\\n8\\n2837 745\\n62 1162\\n2634 1112\\n1746 2618\\n847 127\\n986 1993\\n732 1273\\n2003 1998\\n4\\n1276 2231\\n1234 1234\\n287 2371\\n3000 3000\\n\\n\\nSample Output:\\nCase #1: 52\\nCase #2: 131\\nCase #3: 110090622\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:58 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'12689'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29996'), (b'x-ratelimit-remaining-tokens', b'149978810'), (b'x-ratelimit-reset-requests', b'7ms'), (b'x-ratelimit-reset-tokens', b'8ms'), (b'x-request-id', b'req_da28851787408b5fc9097913b210e7d8'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d735cef83a0-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:58 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '12689', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29996', 'x-ratelimit-remaining-tokens': '149978810', 'x-ratelimit-reset-requests': '7ms', 'x-ratelimit-reset-tokens': '8ms', 'x-request-id': 'req_da28851787408b5fc9097913b210e7d8', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d735cef83a0-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_da28851787408b5fc9097913b210e7d8\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This problem shares some similarities with C2, with key differences in bold.*\\n\\nA friend who works at Metal Platforms Inc just lent you a curious puzzle, offering you tickets to a metal concert if you can solve it.\\n\\nThe puzzle consists of \\\\(N\\\\) buttons in a row, numbered from \\\\(1\\\\) to \\\\(N\\\\). The initial state of button \\\\(i\\\\) is white if \\\\(S_i = 1\\\\), or black if \\\\(S_i = 0\\\\). Pressing a button \\\\(k\\\\) toggles the state of itself as well as every \\\\(k\\\\)th button. Your friend challenges you to return the puzzle to him with all buttons back in black.\\n\\nLife is hard enough without siblings pushing your buttons. Unfortunately, your brother has taken the puzzle and will push \\\\(Q\\\\) buttons sequentially, the \\\\(i\\\\)th being button \\\\(B_i\\\\). \\n\\nAfter your brother **has pushed all \\\\(Q\\\\) buttons**, you\\'d like to know the minimum number of button presses required to turn all the buttons black.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 70\\\\)\\n\\\\(1 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 4{,}000{,}000\\\\)\\n\\nThe sum of \\\\(N\\\\) and \\\\(Q\\\\) over all cases will be at most \\\\(9{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing a bitstring \\\\(S\\\\) of length \\\\(N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains a single integer \\\\(B_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the number of button presses needed to turn all buttons black **after all \\\\(Q\\\\) button presses.**\\n\\n**Sample Explanation**\\n\\nThe first sample case is depicted below. After your brother presses the first button, the state of the puzzle is \\\\(0101\\\\). The best strategy is to press the second button, turning all lights off.\\n\\n{{PHOTO_ID:287687020772716|WIDTH:400}}\\n\\nIn the second case, the puzzle starts as \\\\(0001\\\\), and after each button press becomes \\\\(0100\\\\), \\\\(0110\\\\), \\\\(0011\\\\), and \\\\(0010\\\\) respectively. Pressing only the third button will return the puzzle to all zeros.\\n\\n\\nSample Input:\\n5\\n4\\n1010\\n1\\n1\\n4\\n0001\\n4\\n2\\n3\\n2\\n4\\n7\\n0101101\\n8\\n1\\n3\\n2\\n6\\n7\\n4\\n2\\n5\\n7\\n0101100\\n1\\n7\\n7\\n1111111\\n1\\n1\\n\\n\\nSample Output:\\nCase #1: 1\\nCase #2: 1\\nCase #3: 4\\nCase #4: 4\\nCase #5: 0\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:37:59 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'12160'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29995'), (b'x-ratelimit-remaining-tokens', b'149979291'), (b'x-ratelimit-reset-requests', b'8ms'), (b'x-ratelimit-reset-tokens', b'8ms'), (b'x-request-id', b'req_e37ad7c7e01d2179b49be293f24b4a75'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196d733897100a-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:37:59 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '12160', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29995', 'x-ratelimit-remaining-tokens': '149979291', 'x-ratelimit-reset-requests': '8ms', 'x-ratelimit-reset-tokens': '8ms', 'x-request-id': 'req_e37ad7c7e01d2179b49be293f24b4a75', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196d733897100a-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_e37ad7c7e01d2179b49be293f24b4a75\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nFour friends are playing a card game with two teams of two players each. Team \\\\(A\\\\) consists of players \\\\(A1\\\\) and \\\\(A2\\\\) while team \\\\(B\\\\) consists of players \\\\(B1\\\\) and \\\\(B2\\\\).\\n\\nThere is a deck of \\\\(N\\\\) cards (where \\\\(N\\\\) is always a multiple of \\\\(4\\\\)), numbered from \\\\(1\\\\) to \\\\(N\\\\), with all cards visible to all players at all times. First, the cards are dealt out evenly to each player:\\n- Player \\\\(A1\\\\) has cards \\\\(A1_1\\\\), ..., \\\\(A1_{N/4}\\\\).\\n- Player \\\\(B1\\\\) has cards \\\\(B1_1\\\\), ..., \\\\(B1_{N/4}\\\\).\\n- Player \\\\(A2\\\\) has cards \\\\(A2_1\\\\), ..., \\\\(A2_{N/4}\\\\).\\n- Player \\\\(B2\\\\) has cards \\\\(B2_1\\\\), ..., \\\\(B2_{N/4}\\\\).\\n\\nThe game proceeds for \\\\(N/4\\\\) rounds. In each round, each player plays a card. Player \\\\(A1\\\\) plays first, then player \\\\(B1\\\\), then player \\\\(A2\\\\), then player \\\\(B2\\\\). A player may choose to play any of their cards when it’s their turn. After all four players have played a card, the team who played the highest card will score \\\\(1\\\\) point. Once a round is complete, the four played cards are removed from the game, and then the next round starts. This continues until all cards have been played.\\n\\nFor example, the first round of the second sample case might be played as follows, with player \\\\(B2\\\\) winning a point for team \\\\(B\\\\):\\n\\n{{PHOTO_ID:1558229151280116|WIDTH:600}}\\n\\nAssuming each team plays to maximize its score, how many points will team \\\\(A\\\\) score?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 500\\\\)\\n\\\\(4 \\\\le N \\\\le 4{,}000{,}000\\\\)\\n\\\\(N\\\\) is a multiple of \\\\(4\\\\).\\nEach card from \\\\(1\\\\) to \\\\(N\\\\) is guaranteed to exist in exactly one player’s hand.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(5{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then there are \\\\(4\\\\) lines containing the players\\' cards:\\n- Line 1: \\\\(N/4\\\\) space-separated integers, \\\\(A1_1, ..., A1_{N/4}\\\\). \\n- Line 2: \\\\(N/4\\\\) space-separated integers, \\\\(B1_1, ..., B1_{N/4}\\\\). \\n- Line 3: \\\\(N/4\\\\) space-separated integers, \\\\(A2_1, ..., A2_{N/4}\\\\). \\n- Line 4: \\\\(N/4\\\\) space-separated integers, \\\\(B2_1, ..., B2_{N/4}\\\\). \\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a line containing `\"Case #i: \"` followed by a single integer, the number of points that team \\\\(A\\\\) will score.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, one possible way the cards can be played optimally is:\\n- Round 1: \\\\([2, 7, \\\\textbf{8}, 5]\\\\)\\n- Round 2: \\\\([1, 3, \\\\textbf{6}, 4]\\\\)\\n\\nTeam \\\\(A\\\\) will score \\\\(2\\\\) points, and team \\\\(B\\\\) can do no better by playing differently.\\n\\nIn the second case, one possible way the cards can be played optimally is:\\n\\n- Round 1: \\\\([1, 6, 2, \\\\textbf{9}]\\\\)\\n- Round 2: \\\\([\\\\textbf{13}, 5, 3, 8]\\\\)\\n- Round 3: \\\\([12, 15, \\\\textbf{16}, 10]\\\\)\\n- Round 4: \\\\([\\\\textbf{14}, 7, 4, 11]\\\\)\\n\\nTeam \\\\(A\\\\) will score \\\\(3\\\\) points.\\n\\nIn the third case, one possible way the cards can be played optimally is:\\n\\n- Round 1: \\\\([11, \\\\textbf{12}, 9, 4]\\\\)\\n- Round 2: \\\\([13, 2, 8, \\\\textbf{14}]\\\\)\\n- Round 3: \\\\([6, 1, 5, \\\\textbf{10}]\\\\)\\n- Round 4: \\\\([15, \\\\textbf{16}, 7, 3]\\\\)\\n\\nTeam \\\\(B\\\\) can always prevent team \\\\(A\\\\) from scoring any points.\\n\\n\\nSample Input:\\n5\\n8\\n1 2\\n3 7\\n6 8\\n4 5\\n16\\n14 13 12 1\\n15 5 6 7\\n16 2 3 4\\n8 9 10 11\\n16\\n15 13 11 6\\n16 12 1 2\\n5 9 7 8\\n14 10 3 4\\n8\\n5 2\\n8 4\\n7 3\\n6 1\\n24\\n7 6 14 22 18 12\\n20 23 13 8 16 11\\n24 21 4 9 1 19\\n15 5 10 17 3 2\\n\\n\\nSample Output:\\nCase #1: 2\\nCase #2: 3\\nCase #3: 0\\nCase #4: 1\\nCase #5: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:03 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9317'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994776'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_a93757b0684a383148790772b03123b6'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da56b6b59a7-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:03 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9317', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994776', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_a93757b0684a383148790772b03123b6', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da56b6b59a7-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_a93757b0684a383148790772b03123b6\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n_The only difference between chapters 1 and 2 is the maximum allowed grid size, given in bold below._\\n\\nA *Drizzle* program is a 2D grid of the following four types of cells:\\n - \\'`@`\\' (start) \\\\(-\\\\) there is exactly one start cell in the entire grid\\n - \\'`#`\\' (wall)\\n - \\'`.`\\' (space)\\n - \\'`*`\\' (instruction)\\n\\nThe program uses two registers \\\\(A\\\\) and \\\\(B\\\\) (both initially \\\\(0\\\\)), and executes as follows:\\n\\n1. Compute the minimum distance from the start to each instruction cell using orthogonal movements, without going outside of the grid or passing through any wall cells. Instruction cells that cannot be reached are ignored. \\n2. In increasing order, for each unique distance \\\\(D\\\\) such that there’s at least one instruction cell that’s at distance \\\\(D\\\\) from the start cell:\\n 2a. Count the number of shortest paths, \\\\(P\\\\), to all instruction cells of distance \\\\(D\\\\).\\n 2b. Look up the instruction corresponding to \\\\((P \\\\text{ mod } 2, D \\\\text{ mod } 2)\\\\) in the table below and modify one of the registers accordingly.\\n3. At the end, the value in register \\\\(A\\\\) is outputted.\\n\\n```\\n┌─────────────┬─────────────┬─────────────┐\\n│ │ D mod 2 = 0 │ D mod 2 = 1 │\\n├─────────────┼─────────────┼─────────────┤\\n│ P mod 2 = 0 │ A := A + 1 │ A := A - 1 │\\n│ P mod 2 = 1 │ B := B + A │ A := B │\\n└─────────────┴─────────────┴─────────────┘\\n```\\n\\nFor a given value \\\\(K\\\\), output any Drizzle program that outputs \\\\(K\\\\) when executed, with the restriction that **the program must fit on a \\\\(\\\\mathbf{10}\\\\) × \\\\(\\\\mathbf{10}\\\\) grid**.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 2{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10{,}000\\\\)\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is a line containing the single integer \\\\(K\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output \"`Case #i: `\" followed by two integers \\\\(R\\\\) and \\\\(C\\\\), the number of rows and columns in your program, respectively. Then output your program. It must be exactly \\\\(R\\\\) lines long, with each line containing exactly \\\\(C\\\\) characters.\\n\\n\\n# Sample Explanation\\n\\nHere are the instructions executed for each of the sample programs. Note that many other programs would be accepted for any for these cases.\\n\\nIn the first case, there is a single instruction. There are \\\\(2\\\\) shortest paths of length \\\\(2\\\\) to that instruction, so \\\\(P = 2\\\\) and \\\\(D = 2\\\\). That means we perform \\\\(A := A + 1\\\\). There are no more instructions, so the program ends and outputs \\\\(1\\\\).\\n\\nIn the second case, there are three instruction cells. Each of them are an even distance from the start, and each have an even number of shortest paths leading to them, so each represents \\\\(A := A + 1\\\\):\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(4\\\\) paths of length \\\\(6\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(4\\\\) paths of length \\\\(12\\\\) \\\\(\\\\;(A := A + 1 = 3)\\\\)\\n\\nIn the third case, there are eight instruction cells, but some of them are at the same distance as each other. In particular, there are two instruction cells at distance \\\\(2\\\\), and three instruction cells at distance \\\\(10\\\\). There\\'s a single shortest path to each of the cells at distance \\\\(2\\\\), so in total there are \\\\(2\\\\) shortest paths to instructions at distance \\\\(2\\\\). One of the cells at distance \\\\(10\\\\) has a unique shortest path, and the other has two shortest paths, so in total there are \\\\(3\\\\) shortest paths to instructions at distance \\\\(10\\\\).\\n\\n1) \\\\(2\\\\) paths of length \\\\(2\\\\) \\\\(\\\\;(A := A + 1 = 1)\\\\)\\n2) \\\\(6\\\\) paths of length \\\\(4\\\\) \\\\(\\\\;(A := A + 1 = 2)\\\\)\\n3) \\\\(1\\\\) path of length \\\\(6\\\\) \\\\(\\\\;(B := B + A = 2)\\\\)\\n4) \\\\(1\\\\) path of length \\\\(8\\\\) \\\\(\\\\;(B := B + A = 4)\\\\)\\n5) \\\\(3\\\\) paths of length \\\\(10\\\\) \\\\(\\\\;(B := B + A = 6)\\\\)\\n6) \\\\(3\\\\) paths of length \\\\(11\\\\) \\\\(\\\\;(A := B = 6)\\\\)\\n\\n\\nSample Input:\\n3\\n1\\n3\\n6\\n\\n\\nSample Output:\\nCase #1: 2 2\\n@.\\n.*\\nCase #2: 5 6\\n@.#.*.\\n.*#...\\n#.#...\\n...#.#\\n..*...\\nCase #3: 4 10\\n...#*.*.**\\n.*.*##..#.\\n.....#*#..\\n.@.*.....#\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:03 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9577'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994848'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_1fb887b2447bedc7811cedf61dba8668'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da83b4227b7-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:03 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9577', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994848', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_1fb887b2447bedc7811cedf61dba8668', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da83b4227b7-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_1fb887b2447bedc7811cedf61dba8668\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nMetal Platforms, Inc. (a.k.a. Metal), formerly a mining company, now offers a service to facilitate sheet metal sales. It has \\\\(N\\\\) clients, where client \\\\(i\\\\) wants to buy *only on* day \\\\(A_i\\\\) for \\\\(\\\\$X_i\\\\) per sheet, and sell *only on* a strictly later day \\\\(B_i\\\\) for \\\\(\\\\$Y_i\\\\) per sheet.\\n\\nClient \\\\(i\\\\) can sell to client \\\\(j\\\\) if \\\\(B_i = A_j\\\\), and if client \\\\(j\\\\) is buying for a *strictly higher* price than what client \\\\(i\\\\) is selling for, i.e. if \\\\(X_j > Y_i\\\\). If Metal facilitates the sale, they earn a profit of \\\\(\\\\$(X_j - Y_i)\\\\).\\n\\nA *path* is any ordered sequence of clients where each client can sell to the next. A path\\'s profit is the total profit that Metal would make if a single sheet were theoretically sold along it.\\n\\nMetal would like to know the total profit across the \\\\(K\\\\) most profitable paths (or across all distinct paths if there are fewer than \\\\(K\\\\)). Since this may be large, please print it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 35\\\\)\\n\\\\(1 \\\\le N, K \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le N * K \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le A_i < B_i \\\\le 10^9\\\\)\\n\\\\(1 \\\\le X_i, Y_i \\\\le 10^9\\\\)\\n\\nThe sum of \\\\(N\\\\) over all test cases is at most \\\\(6{,}000{,}000\\\\).\\nThe sum of \\\\(N*K\\\\) over all cases is at most \\\\(20{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains four space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), \\\\(X_i\\\\), and \\\\(Y_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the total profit of the \\\\(K\\\\) most profitable paths, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first case is depicted below, with buy prices in green and sell prices in red. The figure on the right outlines all potential sales, along with how much profit Metal would make.\\n\\n{{PHOTO_ID:5415230591886835|WIDTH:700}}\\n\\nThe valid paths are:\\n\\n- \\\\(P_1 = [1, 2]\\\\) with profit \\\\(\\\\$25 - \\\\$20 = \\\\$5\\\\).\\n- \\\\(P_2 = [1, 2, 4]\\\\) with profit \\\\((\\\\$25 - \\\\$20) + (\\\\$35 - \\\\$30)\\\\) \\\\(= \\\\$10\\\\).\\n- \\\\(P_3 = [1, 2, 4, 6]\\\\) with profit \\\\((\\\\$25 - \\\\$20) + (\\\\$35 - \\\\$30) + (\\\\$45 - \\\\$20)\\\\) \\\\(= \\\\$35\\\\).\\n- \\\\(P_4 = [2, 4]\\\\) with profit \\\\(\\\\$5\\\\).\\n- \\\\(P_5 = [2, 4, 6]\\\\) with profit \\\\(\\\\$5 + (\\\\$45 - \\\\$20)\\\\) \\\\(= \\\\$30\\\\).\\n- \\\\(P_6 = [4, 6]\\\\) with profit \\\\(\\\\$45 - \\\\$20 = \\\\$25\\\\).\\n- \\\\(P_7 = [5, 6]\\\\) with profit \\\\(\\\\$45 - \\\\$15 = \\\\$30\\\\).\\n\\nNote that sales from clients \\\\(3\\\\) to \\\\(4\\\\) and clients \\\\(6\\\\) to \\\\(7\\\\) are not possible, since the buyers\\' prices must be strictly higher than the sellers\\'. Since there are fewer than \\\\(K=10\\\\) paths, the answer is just the sum of profits of all paths, equal to \\\\(\\\\$5 + \\\\$10 + \\\\$35 + \\\\$5 + \\\\$30 + \\\\$25 + \\\\$30 = \\\\$140\\\\).\\n\\nThe second case is the same as the first, except we\\'re only interested in the top \\\\(6\\\\) most profitable paths, so we must exclude either \\\\(P_1\\\\) or \\\\(P_4\\\\) for a total profit of \\\\(\\\\$135\\\\).\\n\\nIn the third case, no sales are possible, so we simply report \\\\(\\\\$0\\\\).\\n\\nIn the fourth case, the only possible path is \\\\([2, 1]\\\\), with a profit of \\\\(\\\\$30 - \\\\$20 = \\\\$10\\\\).\\n\\n\\nSample Input:\\n4\\n8 10\\n1 2 10 20\\n2 4 25 30\\n1 4 45 40\\n4 5 35 20\\n1 5 10 15\\n5 6 45 30\\n6 7 30 40\\n8 9 80 90\\n8 6\\n1 2 10 20\\n2 4 25 30\\n1 4 45 40\\n4 5 35 20\\n1 5 10 15\\n5 6 45 30\\n6 7 30 40\\n8 9 80 90\\n2 1\\n1 2 10 20\\n3 4 30 40\\n2 1\\n2 3 30 40\\n1 2 10 20\\n\\n\\nSample Output:\\nCase #1: 140\\nCase #2: 135\\nCase #3: 0\\nCase #4: 10\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:03 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9395'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994927'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_5356e0bbb8ed0e49c6f45d0f1f59676d'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196daa78360dad-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:03 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9395', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994927', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_5356e0bbb8ed0e49c6f45d0f1f59676d', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196daa78360dad-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_5356e0bbb8ed0e49c6f45d0f1f59676d\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*Is this a real verse? Is this just fantasy?\\nCaught in a mixtape, no escape from this melody.\\nOpen your ears, listen to the beats and see.\\nI\\'m just a rapper, I need no sympathy,\\nBecause it\\'s easy come, easy go, little high, little low,\\nAny way the bars flow doesn\\'t really matter to me, to me.*\\n\\nAs an up-and-coming rapper, you\\'re trying to distinguish yourself by writing a song with the most bohemian rhyme scheme!\\n\\nYour vocabulary consists of \\\\(N\\\\) words, \\\\(W_1\\\\), \\\\(…\\\\), \\\\(W_N\\\\) made up of only lowercase letters. We consider two words to *\\\\(K\\\\)-rhyme* with each other if they have the same suffix of length \\\\(K\\\\). For example, “mixtape” 3-rhymes with “escape”. If two words \\\\(K\\\\)-rhyme for \\\\(K > 1\\\\), they also \\\\((K-1)\\\\)-rhyme.\\n\\nYou are faced with \\\\(Q\\\\) bohemian rap-queries, the \\\\(i\\\\)th of which specifies \\\\((A_i, B_i, K_i)\\\\). You’d like to assign each word \\\\(W_{A_i}\\\\), \\\\(…\\\\), \\\\(W_{B_i}\\\\) to at most one group such that:\\n\\n* the number of words in each group is unique and nonzero,\\n* groups only contain words with length at least \\\\(K_i\\\\),\\n* within each group, all pairs of words \\\\(K_i\\\\)-rhyme with each other, and\\n* no two words from different groups \\\\(K_i\\\\)-rhyme.\\n\\nThe answer to each query is the maximum possible number of nonempty groups. For each test case, output the sum of the answers over all queries. \\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 20\\\\)\\n\\\\(1 \\\\le N \\\\le 600{,}000\\\\)\\n\\\\(\\\\sum |W_i| \\\\le 3{,}000{,}000\\\\)\\n\\\\(1 \\\\le Q \\\\le 400{,}000\\\\)\\n\\\\(1 \\\\le A_i \\\\le B_i \\\\le N\\\\)\\n\\\\(1 \\\\le K_i \\\\le 1{,}000{,}000\\\\)\\nAll words consist of only letters \\\\(\\\\{\\\\texttt{`a\\'}, ..., \\\\texttt{`z\\'}\\\\}\\\\)\\nAll words within a test case are distinct.\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(2{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(1{,}000{,}000\\\\).\\nThe sum of word lengths across all test cases is at most \\\\(10{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the string \\\\(W_i\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains \\\\(3\\\\) space-separated integers \\\\(A_i\\\\), \\\\(B_i\\\\), and \\\\(K_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by a single integer, the sum of answers over all queries.\\n\\n**Sample Explanation**\\n\\nFor the first query in the first test case, we are allowed to choose from all \\\\(W_1, ..., W_8\\\\) words to form \\\\(3\\\\) groups where words of the same group \\\\(1\\\\)-rhyme with each other. One possible choice of the groups is: \\\\(\\\\{\\\\)\"low\"\\\\(\\\\}\\\\), \\\\(\\\\{\\\\)\"sympathy\", \"fantasy\", \"melody\"\\\\(\\\\}\\\\), \\\\(\\\\{\\\\)\"mixtape\", \"escape\"\\\\(\\\\}\\\\) with \"come\" and \"high\" going unused. Therefore the answer to the first query is \\\\(3\\\\). The answers to the remaining queries are \\\\(1\\\\), \\\\(2\\\\) and \\\\(2\\\\), with the overall sum of answers for this test case equal to \\\\(8\\\\).\\n\\nIn the second test case, the answers to the \\\\(3\\\\) queries are \\\\([2, 1, 0]\\\\) with the sum being \\\\(3\\\\).\\n\\n\\n\\nSample Input:\\n2\\n8\\nsympathy\\nfantasy\\nmelody\\nmixtape\\nescape\\ncome\\nhigh\\nlow\\n4\\n1 8 1\\n1 8 7\\n2 5 1\\n1 5 3\\n3\\na\\naa\\nb\\n3\\n1 3 1\\n1 3 2\\n1 3 3\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:04 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'10536'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994884'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_b1d6570c61db0712536c670583d49851'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da4babb5a43-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:04 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '10536', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994884', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_b1d6570c61db0712536c670583d49851', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da4babb5a43-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_b1d6570c61db0712536c670583d49851\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nCactus Park is a famous **t**ourist attraction in Sandlandia. It holds \\\\(N\\\\) cactus plants, numbered from \\\\(1\\\\) to \\\\(N\\\\). The cacti are connected by \\\\(M\\\\) bidirectional trails, with trail \\\\(i\\\\) connecting cacti \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\nFrom any cactus, it\\'s always possible to get to any other cactus by taking some sequence of trails. There may be cycles in the park, where a cycle is any sequence of trails that lead from a certain cactus back to itself. The park also has a special property that each trail belongs to at most one simple cycle. In graph theory terms, we can say that the Cactus Park forms a [cactus graph](https://en.wikipedia.org/wiki/Cactus_graph).\\n\\nThe park owners want to replace some number of cacti with information kiosks to help guide the tourists. Cutting down cactus \\\\(i\\\\) and building a kiosk there costs the park \\\\(C_i\\\\) dollars. The owners want to build enough kiosks so that the shortest path from every remaining cactus to the closest kiosk does not exceed \\\\(K\\\\) trails. Please help the owners determine the minimum total cost required to satisfy this requirement.\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 65\\\\)\\n\\\\(1 \\\\le N \\\\le 500\\\\)\\n\\\\(1 \\\\le K \\\\le \\\\min(N, 50)\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le N\\\\)\\n\\\\(A_i \\\\ne B_i\\\\)\\n\\\\(1 \\\\le C_i \\\\le 10^9\\\\)\\n\\nEach unordered pair \\\\((A_i, B_i)\\\\) appears at most once in a given test case.\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing three integers \\\\(N\\\\), \\\\(M\\\\), and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) integers \\\\(C_{1..N}\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output `\"Case #i: \"`, followed by a single integer, the minimum total cost in dollars to satisfy the park owners\\' requirement.\\n\\n# Sample Explanation\\n\\nThe first case is depicted below. Replacing just cactus \\\\(2\\\\) would meet the requirement, but would cost \\\\(\\\\$10\\\\). Instead we can replace cacti \\\\(1\\\\) and \\\\(4\\\\) for a total cost of \\\\(\\\\$8\\\\).\\n\\n{{PHOTO_ID:885100906409208|WIDTH:350}}\\n\\nThe second case is depicted below. One solution is to replace cacti \\\\(1\\\\) and \\\\(4\\\\) for a total cost of \\\\(\\\\$6\\\\).\\n\\n{{PHOTO_ID:3652000121792782|WIDTH:500}}\\n\\nIn the third case, all the cactuses are already within \\\\(2\\\\) trails of each other, so we just need a single kiosk anywhere. We should cut down the cheapest cactus, cactus \\\\(1\\\\).\\n\\n{{PHOTO_ID:2643202679188960|WIDTH:350}}\\n\\nIn the fourth case, we can cut down cacti \\\\(1\\\\), \\\\(3\\\\), and \\\\(6\\\\) for a total cost of \\\\(9 + 3 + 4 = \\\\$16\\\\)\\n\\n{{PHOTO_ID:365613766120939|WIDTH:500}}\\n\\n\\nSample Input:\\n4\\n4 4 1\\n5 10 6 3\\n1 2\\n2 3\\n3 1\\n2 4\\n6 6 1\\n2 2 4 4 3 2\\n1 2\\n3 1\\n5 4\\n3 4\\n6 4\\n2 5\\n4 4 2\\n1 6 7 4\\n1 2\\n2 3\\n3 4\\n4 1\\n8 9 1\\n9 1 3 5 9 4 10 10\\n1 2\\n1 3\\n3 4\\n4 5\\n5 3\\n5 6\\n6 7\\n7 5\\n8 1\\n\\n\\nSample Output:\\nCase #1: 8\\nCase #2: 6\\nCase #3: 1\\nCase #4: 16\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nThe Flying Dutchman is making a killing giving out haunted house tours of his ghost ship. The ship has \\\\(N\\\\) cabins, numbered from \\\\(1\\\\) to \\\\(N\\\\), connected by corridors in the shape of an unrooted tree \\\\(t_1\\\\). For each \\\\(i=2..N\\\\), there is a corridor that connects cabin \\\\(i\\\\) to cabin \\\\(P_{i}\\\\).\\n\\nMr. Krabs noticed this business opportunity, and wants to make his own ship, also with \\\\(N\\\\) cabins, connected as a yet-to-be-determined tree \\\\(t_2\\\\). To avoid any lawsuits from the Dutchman, he would like to make his ship as different as possible.\\n\\nIn particular, let \\\\(S(t)\\\\) be the multiset of all subtrees of tree \\\\(t\\\\), where a subtree is any non-empty subset of cabins that are connected. For two multisets of trees \\\\(S_1\\\\) and \\\\(S_2\\\\), let \\\\(\\\\text{similarity}(S_1, S_2)\\\\) be the number of trees in \\\\(S_2\\\\) that are [isomorphic](https://en.wikipedia.org/wiki/Graph_isomorphism) to at least one tree in \\\\(S_1\\\\). Note that \\\\(\\\\text{similarity}(S_1, S_2)\\\\) and \\\\(\\\\text{similarity}(S_2, S_1)\\\\) are not necessarily equal.\\n\\nHelp Mr. Krabs find the minimum possible value of \\\\(\\\\text{similarity}(S(t_1), S(t_2))\\\\) over all trees \\\\(t_2\\\\) of size \\\\(N\\\\). As this value may be large, output it modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 35\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(10{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case has two lines. The first contains the integer \\\\(N\\\\), and the second contains the \\\\(N-1\\\\) integers \\\\(P_2\\\\) through \\\\(P_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the minimum possible similarity if \\\\(t_2\\\\) is chosen optimally, modulo \\\\(1{,}000{,}000{,}007\\\\).\\n\\n\\n# Sample Explanation\\n\\nThe first sample case is depicted below. One possible tree that minimizes the similarity score is the same tree given in the input (i.e. choose \\\\(t_2 = t_1\\\\)). Then, \\\\(S(t_2)\\\\) consists of the \\\\(6\\\\) subtrees: \\\\(\\\\{1\\\\}\\\\), \\\\(\\\\{2\\\\}\\\\), \\\\(\\\\{3\\\\}\\\\), \\\\(\\\\{1, 2\\\\}\\\\), \\\\(\\\\{2, 3\\\\}\\\\), and \\\\(\\\\{1, 2, 3\\\\}\\\\). Each subtree is isomorphic to a subtree of \\\\(t_1\\\\), so the similarity score is \\\\(6\\\\).\\n\\n{{PHOTO_ID:720027639598818|WIDTH:280}}\\n\\nIn the second case, one possible \\\\(t_2\\\\) that minimizes the similarity score is depicted below. This tree has \\\\(24\\\\) different subtrees, of which \\\\(21\\\\) are isomorphic to a subtree of the given \\\\(t_1\\\\). For example, the subtree \\\\(\\\\{2, 3, 4, 6\\\\}\\\\) of this \\\\(t_2\\\\) is isomorphic to subtree \\\\(\\\\{2, 3, 4, 5\\\\}\\\\) of the given \\\\(t_1\\\\), but the subtree \\\\(\\\\{1, 2, 3, 6\\\\}\\\\) of this \\\\(t_2\\\\) is not isomorphic to any subtree of \\\\(t_1\\\\).\\n\\n{{PHOTO_ID:854841726123175|WIDTH:350}}\\n\\n\\nSample Input:\\n2\\n3\\n1 2\\n6\\n1 2 3 4 5\\n\\n\\nSample Output:\\nCase #1: 6\\nCase #2: 21\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:05 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'9686'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994468'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_2d2986cd7a4a40195191ac59df530c99'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196db4ab434c61-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:05 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '9686', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994468', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_2d2986cd7a4a40195191ac59df530c99', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196db4ab434c61-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_2d2986cd7a4a40195191ac59df530c99\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou have \\\\(N\\\\) [tries](https://en.wikipedia.org/wiki/Trie). The \\\\(i\\\\)th trie has \\\\(M_i\\\\) nodes and \\\\(M_i - 1\\\\) edges, with node \\\\(1\\\\) being the root, and node \\\\(j\\\\)\\'s parent being \\\\(P_{i,j}\\\\), for \\\\(j = 2..M_i\\\\). Edge \\\\(P_{i,j} \\\\to j\\\\) is labeled with a lowercase letter \\\\(C_{i,j}\\\\).\\n\\nWe say a string is *in* a trie if it can be formed by concatenating the letters along some path starting at the root and ending at some node (either internal or leaf).\\n\\nFor every possible unordered triplet of these \\\\(N\\\\) tries, you would like to know: how many strings exist which are in at least one of the three tries?\\n\\nPlease print the sum of answers over all \\\\(N*(N-1)*(N-2)/6\\\\) possible queries.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(3 \\\\le N \\\\le 100\\\\)\\n\\\\(1 \\\\le M_i \\\\le 1{,}000{,}000\\\\)\\n\\\\(1 \\\\le P_{i,j} \\\\lt j\\\\)\\n\\\\(C_{i,j} \\\\in \\\\{\\\\)`\\'a\\'`\\\\(,\\\\, \\\\ldots, \\\\) `\\'z\\'`\\\\(\\\\}\\\\)\\n\\nThe sum of \\\\(N\\\\) across all cases is at most \\\\(2{,}000\\\\).\\nNo node will have two direct outgoing edges with the same edge label.\\nThere are at most \\\\(1{,}000{,}000\\\\) total trie nodes in a given test case.\\nThe sum of nodes across all test cases is at most \\\\(13{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\), the number of tries. Then, \\\\(N\\\\) descriptions of tries follow. For the \\\\(i\\\\)th trie, there is first a line containing a single integer \\\\(M_i\\\\), followed by \\\\(M_i - 1\\\\) more lines, the \\\\(j\\\\)th of which contains the integer \\\\(P_{i,j+1}\\\\), followed by a space, followed by the letter \\\\(C_{i,j+1}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"` followed by a single integer, the sum of the query answers across all triplets of tries.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, there are \\\\(N = 3\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:502671685043623|WIDTH:650}}\\n\\nThere is only one possible unordered triplet of tries to consider, for which there are \\\\(5\\\\) strings in at least one trie of the triplet: \"\", \"`a`\", \"`aa`\", \"`b`\", and \"`c`\".\\n\\nIn the second case, there are \\\\(N = 4\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:1118217749060271|WIDTH:650}}\\n\\nThere are four triplets of tries to consider, three of which include the last trie and one of which doesn\\'t. In the three that does, there are \\\\(4\\\\) strings in at least one trie: \"\", \"`a`\", \"`aa`\", and \"`aaa`\". In the one that doesn\\'t, the are \\\\(2\\\\) strings: \"\" and \"`a`\". The final answer is \\\\(3*4 + 1*2 = 14\\\\).\\n\\nIn the third case, there are \\\\(N = 4\\\\) tries, depicted below:\\n\\n{{PHOTO_ID:1007664223967563|WIDTH:650}}\\n\\nThe four triplets and strings contributing to the answer are:\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(2\\\\), trie \\\\(3)\\\\): \"\", \"`a`\", \"`b`\", \"`c`\"\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(2\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`b`\", \"`ab`\", \"`abc`\"\\n* \\\\((\\\\)trie \\\\(1\\\\), trie \\\\(3\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`c`\", \"`ab`\", \"`abc`\"\\n* \\\\((\\\\)trie \\\\(2\\\\), trie \\\\(3\\\\), trie \\\\(4)\\\\): \"\", \"`a`\", \"`b`\", \"`c`\", \"`ab`\", \"`abc`\"\\n\\nThe final answer is \\\\(4 + 5 + 5 + 6 = 20\\\\).\\n\\n\\nSample Input:\\n3\\n3\\n3\\n1 a\\n1 b\\n3\\n1 a\\n2 a\\n2\\n1 c\\n4\\n2\\n1 a\\n2\\n1 a\\n2\\n1 a\\n4\\n1 a\\n2 a\\n3 a\\n4\\n2\\n1 a\\n2\\n1 b\\n2\\n1 c\\n4\\n1 a\\n2 b\\n3 c\\n\\n\\nSample Output:\\nCase #1: 5\\nCase #2: 14\\nCase #3: 20\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:06 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'11156'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994738'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_ccdd288ae832f422f62857251601f46b'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196daf8bd72789-LYS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:06 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '11156', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994738', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_ccdd288ae832f422f62857251601f46b', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196daf8bd72789-LYS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_ccdd288ae832f422f62857251601f46b\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this chapter and [chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-1/problems/A1) is that here, card values are not guaranteed to be distinct and can be up to \\\\(10^9\\\\).**\\n\\nLet\\'s cut to the chase. You have a deck of \\\\(N\\\\) face-up cards, each displaying **a not necessarily unique integer between \\\\(1\\\\) and \\\\(10^9\\\\).**\\n\\n*Cutting* the deck once consists of taking a stack of between \\\\(1\\\\) and \\\\(N - 1\\\\) (inclusive) cards from the top and moving it to the bottom in the same order. For example, for the deck \\\\([5, 1, 2, 4, 3]\\\\) ordered from top to bottom, cutting \\\\(2\\\\) cards from the top would yield \\\\([2, 4, 3, 5, 1]\\\\):\\n\\n{{PHOTO_ID:792109885435616|WIDTH:700}}\\n\\nInitially, the \\\\(i\\\\)th card from the top is \\\\(A_i\\\\). Is it possible to cut the deck exactly \\\\(K\\\\) times to reorder the deck such that the \\\\(i\\\\)th card from the top is \\\\(B_i\\\\) for all \\\\(i\\\\)?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 205\\\\)\\n\\\\(2 \\\\le N \\\\le 500{,}000\\\\)\\n\\\\(0 \\\\le K \\\\le 10^{9}\\\\)\\n\\\\(1 \\\\le A_i, B_i \\\\le \\\\mathbf{10^9}\\\\)\\n**\\\\(A\\\\) and \\\\(B\\\\) are permutations of each other.**\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(7{,}000{,}000\\\\).\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing two space-separated integers \\\\(N\\\\) and \\\\(K\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(A_1, ..., A_N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers, \\\\(B_1, ..., B_N\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print `\"Case #i: \"` followed by `\"YES\"` if it\\'s possible to cut the deck \\\\(K\\\\) times to change the deck from \\\\(A_i\\\\) to \\\\(B_i\\\\), or `\"NO\"` otherwise.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, it\\'s possible to get to the new order with \\\\(K = 1\\\\) cut (cutting 2 cards from the top).\\n\\nIn the second case, it\\'s impossible to change \\\\([3, 1, 4, 2]\\\\) to \\\\([1, 2, 3, 4]\\\\) with any number of cuts.\\n\\nIn the third case, it\\'s impossible for the deck to be in a different order after \\\\(K = 0\\\\) cuts.\\n\\n\\nSample Input:\\n4\\n5 1\\n5 1 2 2 3\\n2 2 3 5 1\\n4 10\\n3 1 4 2\\n1 2 3 4\\n4 0\\n3 1 4 2\\n2 3 1 4\\n5 3\\n10 10 9 10 9\\n10 10 10 9 9\\n\\n\\nSample Output:\\nCase #1: YES\\nCase #2: NO\\nCase #3: NO\\nCase #4: NO\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: This is the harder version of [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-2/problems/A1). This version involves an array of integers instead of a string, and the array may change with updates.**\\n\\nAn array is *perfectly balanced* if its length is even, and the first half of the array can be shuffled to make the array a palindrome. For example, \\\\([15,2,15,2]\\\\) and \\\\([7,5,5,7]\\\\) are perfectly balanced, but \\\\([7,5,3,1,3,5,7]\\\\) and \\\\([3,1,1,3,2,3]\\\\) are not.\\n\\nAn array is *almost perfectly balanced* if you can delete exactly one element from it to make it perfectly balanced. Some examples are \\\\([1, 2, 3, 4, 4, 5, 1, 3, 2]\\\\), \\\\([1]\\\\), and \\\\([2, 2, 2, 3, 2]\\\\).\\n\\n{{PHOTO_ID:532712048616543|WIDTH:700}}\\n\\nYou are given a larger template array \\\\(A_1 ... A_N\\\\) of length \\\\(N\\\\), along with \\\\(Q\\\\) events, the \\\\(i\\\\)th of which is one of the following two formats:\\n\\n- \\\\(1\\\\) \\\\(X_i\\\\) \\\\(Y_i\\\\): set \\\\(A_{X_i}\\\\) to \\\\(Y_i\\\\).\\n- \\\\(2\\\\) \\\\(L_i\\\\) \\\\(R_i\\\\): check whether the subarray \\\\(A_{L_i..R_i}\\\\) is almost perfectly balanced.\\n\\nFor how many type-2 queries is the subarray almost perfectly balanced?\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 85\\\\)\\n\\\\(1 \\\\le N \\\\le 10^6\\\\)\\n\\\\(1 \\\\le A_i \\\\le 10^6\\\\)\\n\\\\(1 \\\\le Q \\\\le 10^6\\\\)\\n\\\\(1 \\\\le X_i \\\\le N\\\\)\\n\\\\(1 \\\\le Y_i \\\\le 10^6\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\nThe sum of \\\\(Q\\\\) across all test cases is at most \\\\(4{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each test case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). Then, \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains three space-separated integers, either \"\\\\(1\\\\) \\\\(X_i\\\\) \\\\(Y_i\\\\)\" or \"\\\\(2\\\\) \\\\(L_i\\\\) \\\\(R_i\\\\)\".\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, output a single line containing `\"Case #i: \"`, followed by a single integer: the number of queries which are an almost perfectly balanced subarray.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, the template array is \\\\([2,1,5,1,5,1]\\\\).\\n\\nFor the first query, the subarray is \\\\([1,5,1,5,1]\\\\). You can delete the second \\\\(1\\\\) to get \\\\([1,5,5,1]\\\\), which is a palindrome. Thus, \\\\([1,5,1,5,1]\\\\) is an almost perfectly balanced subarray.\\n\\nFor the second query, the subarray is \\\\([2,1,5,1,5]\\\\). You can delete the \\\\(2\\\\) to get \\\\([1,5,1,5]\\\\) then reorder the first half of the array to get \\\\([5,1,1,5]\\\\), which is a palindrome. Thus, \\\\([2,1,5,1,5]\\\\) is an almost perfectly balanced subarray.\\n\\nFor the third query, the subarray is \\\\([2,1,5]\\\\). You cannot delete any element to get an almost perfectly balanced subarray.\\n\\nThe next two events are updates, after which the template array is \\\\([5, 5, 5, 1, 5, 1]\\\\).\\n\\nFor the fourth query, the subarray is \\\\([5,5,5]\\\\). You can delete any element to get \\\\([5,5]\\\\), which is a palindrome. Thus, \\\\([5,5,5]\\\\) is an almost perfectly balanced subarray.\\n\\nThe fifth, sixth, eighth, and tenth queries are also almost perfectly balanced subarrays, so the final answer is \\\\(7\\\\).\\n\\n\\nSample Input:\\n1\\n6\\n2 1 5 1 5 1\\n17\\n2 2 6\\n2 1 5\\n2 1 3\\n1 1 5\\n1 2 5\\n2 1 3\\n2 1 5\\n2 2 2\\n1 3 1000000\\n2 1 5\\n2 1 3\\n2 2 4\\n2 4 6\\n1 6 1000000\\n2 4 6\\n1 5 1000000\\n2 5 6\\n\\n\\nSample Output:\\nCase #1: 7\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n**Note: The only difference between this problem and [problem C2](https://www.facebook.com/codingcompetitions/hacker-cup/2022/qualification-round/problems/C2) is that here, the length of each output codeword may be at most 200.**\\n\\nMorse code is a classic way to send messages, where each letter in an alphabet is substituted with a *codeword*: a unique sequence of dots and dashes. However, ignoring spaces, it\\'s possible for a coded message to have multiple meanings. For example, \"`.....--.-.-.-..-.-.-...-.--.`\" can be interpreted as either \"`HACKER CUP`\" or \"`SEE META RENT A VAN`\":\\n\\n{{PHOTO_ID:1264108514364518|WIDTH:700}}\\n\\nBeyond Morse code, a general set of codewords is an *unambiguous encoding* if any possible sequence of dots and dashes corresponds to either zero or exactly one sequence of codewords.\\n\\nGiven one codeword \\\\(C_1\\\\) from a set of \\\\(N\\\\) distinct codewords, your task is to generate another \\\\(N - 1\\\\) codewords \\\\(C_2, ..., C_N\\\\) to yield an unambiguous encoding. It can be shown that an answer always exists. If there are multiple answers, you may print any one of them.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\le T \\\\le 100\\\\)\\n\\\\(2 \\\\le N \\\\le 100\\\\)\\nThe length of \\\\(C_1\\\\) is between \\\\(1\\\\) and \\\\(100\\\\), inclusive.\\nThe length of each \\\\(C_2, ..., C_N\\\\) must be between \\\\(1\\\\) and \\\\(\\\\mathbf{200}\\\\), inclusive.\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing the codeword \\\\(C_1\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th case, output a line containing only \"`Case #i:`\", followed by \\\\(N - 1\\\\) lines, the codewords \\\\(C_2, ..., C_N\\\\), one per line.\\n\\n\\n# Sample Explanation\\n\\nIn the first case, it can be shown that the codewords {\"`.-.`\", \"`...`\", \"`---`\"} are an unambiguous encoding. Any sequence of dots and dashes can be interpreted if and only if it has a length that\\'s a multiple of 3, and can be broken up into instances of the three length-3 codewords.\\n\\nIn the second case, it can be shown that the codewords {\"`-`\", \"`...`\", \"`.-`\", \"`..-`\"} are an unambiguous encoding. For instance, \"`..`\" has no possible interpretation, and \"`.-...--`\" can only be interpreted as \"`.- ... - -`\".\\n\\nIn the third case, it can be shown that the codewords {\"`..`\", \"`-`\", \"`.-`\"} are an unambiguous encoding. For any sequence of dots and dashes:\\n- every odd group of dots followed by a dash can only be interpreted as repeated \"`..`\"s followed by a final \"`.-`\"\\n- every even group of dots followed by a dash can only be interpreted as repeated \"`..`\"s followed by a final \"`-`\"\\n- every group of dots not followed by a dash (i.e. at the end of the sequence), is interpretable if and only if there is an even number of dots\\n- this leaves only groups of dashes, interpreted only as repeated \"`-`\"s\\n\\n\\n\\nSample Input:\\n3\\n3\\n.-.\\n4\\n-\\n3\\n..\\n\\n\\nSample Output:\\nCase #1:\\n...\\n---\\nCase #2:\\n...\\n.-\\n..-\\nCase #3:\\n-\\n.-\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nMrs. Puff is taking her class trick-or-treating! Her \\\\(N\\\\) students, numbered from \\\\(1\\\\) to \\\\(N\\\\), are dressed up in spooky outfits and ready to collect some candy. To divide and conquer every house in Bikini Bottom, she would like to break up the class into equal-sized groups.\\n\\n\\\\(M\\\\) pairs of students have similar-themed costumes that amplify their spookiness, the \\\\(i\\\\)th being students \\\\(A_i\\\\) and \\\\(B_i\\\\). Ms. Puff\\'s would like to know: for which values of \\\\(K\\\\) can the class be divided into \\\\(K\\\\) groups of equal size without splitting up any of these pairs?\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 75\\\\)\\n\\\\(2 \\\\leq N \\\\leq 100\\\\)\\n\\\\(0 \\\\leq M \\\\leq \\\\frac{N(N-1)}{2}\\\\)\\n\\\\(1 \\\\leq A_i, B_i \\\\leq N\\\\)\\n\\\\(A_i \\\\neq B_i\\\\)\\n\\nEach unordered pair of employees occurs at most once in a given test case.\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. Each case begins with a line containing the two integers \\\\(N\\\\) and \\\\(M\\\\). Then, \\\\(M\\\\) lines follow, the \\\\(i\\\\)th of which contains the two integers \\\\(A_i\\\\) and \\\\(B_i\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by all possible values of \\\\(K\\\\) in ascending order, separated by spaces.\\n\\n\\n# Sample Explanation\\n\\nIn the first sample case, there are three possible values of \\\\(K\\\\). Here\\'s a possible arrangement:\\n- \\\\(K = 1\\\\): All students can be put into a single group.\\n- \\\\(K = 2\\\\): Groups \\\\(\\\\{1, 2, 4, 5, 6, 7\\\\}\\\\) and \\\\(\\\\{3, 8, 9, 10, 11, 12\\\\}\\\\).\\n- \\\\(K = 3\\\\): Groups \\\\(\\\\{1, 2, 4, 6\\\\}\\\\), \\\\(\\\\{3, 5, 8, 12\\\\}\\\\) and \\\\(\\\\{7, 9, 10, 11\\\\}\\\\).\\n\\nIn the second case, either all students can be in a single group, or each should form a separate group.\\n\\nIn the third test case, the valid values of \\\\(K\\\\), with an example of each, are:\\n- \\\\(K = 1\\\\): All students can be put into a single group.\\n- \\\\(K = 2\\\\): Groups \\\\(\\\\{1, 2, 5, 7\\\\}\\\\) and \\\\(\\\\{3, 4, 6, 8\\\\}\\\\).\\n- \\\\(K = 4\\\\): Groups \\\\(\\\\{1, 7\\\\}\\\\), \\\\(\\\\{2, 5\\\\}\\\\), \\\\(\\\\{3, 6\\\\}\\\\) and \\\\(\\\\{4, 8\\\\}\\\\).\\n\\n\\nSample Input:\\n3\\n12 7\\n1 2\\n2 4\\n4 6\\n3 8\\n8 12\\n9 10\\n10 11\\n3 0\\n8 4\\n1 7\\n2 5\\n6 3\\n8 4\\n\\n\\nSample Output:\\nCase #1: 1 2 3\\nCase #2: 1 3\\nCase #3: 1 2 4\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:06 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'8109'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994934'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_943027d4725800c2da486c33aaa16d20'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196dc4be8983a0-MXP'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:06 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '8109', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994934', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_943027d4725800c2da486c33aaa16d20', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196dc4be8983a0-MXP', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_943027d4725800c2da486c33aaa16d20\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\n*This could possibly be the best day ever!\\nAnd the forecast says that tomorrow will likely be a billion and 6 times better.\\nSo make every minute count, jump up, jump in, and seize the day,\\nAnd let\\'s make sure that in every single possible way… today is gonna be a great day!*\\n\\nThere\\'s \\\\(N\\\\) days of summer vacation, and the \\\\(i\\\\)th day initially has greatness \\\\(A_i\\\\). However, the days just keep getting better! In particular, there will be \\\\(Q\\\\) changes, the \\\\(i\\\\)th of which changes the greatness of every day from \\\\(L_i\\\\) to \\\\(R_i\\\\) to be \\\\(1\\\\,000\\\\,000\\\\,006\\\\) times what they were before!\\n\\nAfter each query, you\\'d like to know: which day\\'s greatness modulo \\\\(1\\\\,000\\\\,000\\\\,007\\\\) is the largest? If multiple days tie, we\\'re interested in the earliest of them. For example, if there are \\\\(N = 4\\\\) days with greatnesses \\\\([4, 1, 4, 2]\\\\), then days \\\\(1\\\\) and \\\\(3\\\\) have the largest remainder, so our answer would be \\\\(1\\\\). To keep the output size small, please print only the sum of answers to all queries.\\n\\n**Constraints**\\n\\n\\\\(1 \\\\le T \\\\le 45\\\\)\\n\\\\(1 \\\\le N \\\\le 1{,}000{,}004\\\\)\\n\\\\(1 \\\\le A_i \\\\le 1{,}000{,}000{,}006\\\\)\\n\\\\(1 \\\\le Q \\\\le 500{,}000\\\\)\\n\\\\(1 \\\\le L_i \\\\le R_i \\\\le N\\\\)\\n\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\n\\n**Input Format**\\n\\nInput begins with a single integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N\\\\) space-separated integers \\\\(A_1, ..., A_N\\\\). Then, there is a line containing a single integer \\\\(Q\\\\). \\\\(Q\\\\) lines follow, the \\\\(i\\\\)th of which contains two space-separated integers \\\\(L_i\\\\) and \\\\(R_i\\\\).\\n\\n**Output Format**\\n\\nFor the \\\\(i\\\\)th test case, print a single line containing \"`Case #i: `\" followed by a single integer, the sum of the greatest day(s) after each of the \\\\(Q\\\\) queries.\\n\\n**Sample Explanation**\\n\\nIn the first test case, there are \\\\(N=3\\\\) days with initial greatnesses of \\\\([1, 20, 30]\\\\). The first query modifies all \\\\(3\\\\) days, transforming their greatnesses \\\\([1\\\\,000\\\\,000\\\\,006, 999\\\\,999\\\\,987, 999\\\\,999\\\\,977]\\\\), where the maximum is on day \\\\(1\\\\). After the second query, day \\\\(3\\\\) will have the largest greatness of \\\\(999\\\\,999\\\\,977\\\\). The overall answer is the sum of the greatest days, which is \\\\(1 + 3 = 4\\\\).\\n\\n\\n\\nSample Input:\\n2\\n3\\n1 20 30\\n2\\n1 3\\n1 2\\n2\\n1 1\\n1\\n1 2\\n\\n\\nSample Output:\\nCase #1: 4\\nCase #2: 1\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.'}], 'model': 'gpt-4o-mini', 'frequency_penalty': 0, 'max_tokens': 4000, 'n': 1, 'presence_penalty': 0, 'stop': ['\\n---'], 'temperature': 0.1, 'top_p': 1}}\n", - "DEBUG:openai._base_client:Sending HTTP Request: POST https://api.openai.com/v1/chat/completions\n", - "DEBUG:httpcore.http11:send_request_headers.started request=\n", - "DEBUG:httpcore.http11:send_request_headers.complete\n", - "DEBUG:httpcore.http11:send_request_body.started request=\n", - "DEBUG:httpcore.http11:send_request_body.complete\n", - "DEBUG:httpcore.http11:receive_response_headers.started request=\n", - "DEBUG:httpcore.http11:receive_response_headers.complete return_value=(b'HTTP/1.1', 200, b'OK', [(b'Date', b'Wed, 11 Sep 2024 17:38:08 GMT'), (b'Content-Type', b'application/json'), (b'Transfer-Encoding', b'chunked'), (b'Connection', b'keep-alive'), (b'access-control-expose-headers', b'X-Request-ID'), (b'openai-organization', b'wandb'), (b'openai-processing-ms', b'14434'), (b'openai-version', b'2020-10-01'), (b'strict-transport-security', b'max-age=15552000; includeSubDomains; preload'), (b'x-ratelimit-limit-requests', b'30000'), (b'x-ratelimit-limit-tokens', b'150000000'), (b'x-ratelimit-remaining-requests', b'29999'), (b'x-ratelimit-remaining-tokens', b'149994990'), (b'x-ratelimit-reset-requests', b'2ms'), (b'x-ratelimit-reset-tokens', b'2ms'), (b'x-request-id', b'req_95cf53bc683806fa4666e4d38d297e4f'), (b'CF-Cache-Status', b'DYNAMIC'), (b'X-Content-Type-Options', b'nosniff'), (b'Server', b'cloudflare'), (b'CF-RAY', b'8c196da4bce370f5-MRS'), (b'Content-Encoding', b'gzip'), (b'alt-svc', b'h3=\":443\"; ma=86400')])\n", - "INFO:httpx:HTTP Request: POST https://api.openai.com/v1/chat/completions \"HTTP/1.1 200 OK\"\n", - "DEBUG:httpcore.http11:receive_response_body.started request=\n", - "DEBUG:httpcore.http11:receive_response_body.complete\n", - "DEBUG:httpcore.http11:response_closed.started\n", - "DEBUG:httpcore.http11:response_closed.complete\n", - "DEBUG:openai._base_client:HTTP Response: POST https://api.openai.com/v1/chat/completions \"200 OK\" Headers({'date': 'Wed, 11 Sep 2024 17:38:08 GMT', 'content-type': 'application/json', 'transfer-encoding': 'chunked', 'connection': 'keep-alive', 'access-control-expose-headers': 'X-Request-ID', 'openai-organization': 'wandb', 'openai-processing-ms': '14434', 'openai-version': '2020-10-01', 'strict-transport-security': 'max-age=15552000; includeSubDomains; preload', 'x-ratelimit-limit-requests': '30000', 'x-ratelimit-limit-tokens': '150000000', 'x-ratelimit-remaining-requests': '29999', 'x-ratelimit-remaining-tokens': '149994990', 'x-ratelimit-reset-requests': '2ms', 'x-ratelimit-reset-tokens': '2ms', 'x-request-id': 'req_95cf53bc683806fa4666e4d38d297e4f', 'cf-cache-status': 'DYNAMIC', 'x-content-type-options': 'nosniff', 'server': 'cloudflare', 'cf-ray': '8c196da4bce370f5-MRS', 'content-encoding': 'gzip', 'alt-svc': 'h3=\":443\"; ma=86400'})\n", - "DEBUG:openai._base_client:request_id: req_95cf53bc683806fa4666e4d38d297e4f\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:openai._base_client:Request options: {'method': 'post', 'url': '/chat/completions', 'files': None, 'json_data': {'messages': [{'role': 'user', 'content': 'You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\\n\\n---\\n\\nFollow the following format.\\n\\nProblem Statement: ${problem_statement}\\n\\nSample Input: The sample input provided with the problem statement.\\n\\nSample Output: The sample output provided with the problem statement.\\n\\nReasoning: Let\\'s think step by step in order to ${produce the output fields}. We ...\\n\\nSolution: a solution explanation for how we should go about solving this problem.\\n\\n---\\n\\nProblem Statement:\\nYou and some friends are playing [Wiki Race](https://en.wikipedia.org/wiki/Wikipedia:Wiki\\\\_Game) on an online encyclopedia of \\\\(N\\\\) pages, numbered from \\\\(1\\\\) to \\\\(N\\\\). Page \\\\(i\\\\) covers \\\\(M_i\\\\) topics \\\\(S_{i,1}\\\\) ... \\\\(S_{i,M_i}\\\\), each a string of lowercase letters. Each player starts on a unique page, and will race to reach page \\\\(1\\\\) (the target page) by repeatedly clicking to neighboring pages.\\n\\nUnfortunately, someone out there has already published the shortest path from each page to page \\\\(1\\\\) for the whole internet to see. This forms a tree structure with page \\\\(1\\\\) as the root, and edges directed upwards from each descendent page. In particular, you and your friends all know that at each page \\\\(i > 1\\\\), one should click to the neighboring page \\\\(P_i\\\\) for the fastest route to page \\\\(1\\\\).\\n\\nIn light of this, all players have agreed to a new rule that each page should be visited by exactly one person per game. This means that once a player hits a page someone has already visited, their upward path ends there and the player loses. Formally, we can express each game\\'s outcome as a partition of the tree into vertex-disjoint paths, with each path\\'s upper endpoint always having an edge to a node on another player\\'s path (except the winner, whose upper endpoint is page \\\\(1\\\\)).\\n\\nAfter playing the game for so long, you\\'re all starting to realize that \"*it\\'s not about winning, but the knowledge acquired along the way!*\" \\n\\nFor a given partition of the tree into paths, a topic is _mutually-learned_ by all players if every path has at least one page covering that topic. You\\'d like to know, across all possible partitions of paths, all topics that could be mutually-learned.\\n\\n\\n# Constraints\\n\\n\\\\(1 \\\\leq T \\\\leq 30\\\\)\\n\\\\(2 \\\\leq N \\\\leq 1{,}000{,}000\\\\)\\n\\\\(1 \\\\leq P_i \\\\leq N\\\\)\\n\\\\(1 \\\\leq M_i \\\\leq 100{,}000\\\\)\\n\\\\(1 \\\\leq |S_{i,j}| \\\\leq 10\\\\)\\n\\\\(S_{i,j}\\\\) consists only of lowercase letters (\\'`a`\\'..\\'`z`\\').\\nThe sum of \\\\(N\\\\) across all test cases is at most \\\\(3{,}000{,}000\\\\).\\nThe sum of \\\\(M_i\\\\) across all test cases is at most \\\\(8{,}000{,}000\\\\).\\n\\n\\n# Input Format\\n\\nInput begins with an integer \\\\(T\\\\), the number of test cases. For each case, there is first a line containing a single integer \\\\(N\\\\). Then, there is a line containing \\\\(N - 1\\\\) integers \\\\(P_2, ..., P_N\\\\). Then, \\\\(N\\\\) lines follow, the \\\\(i\\\\)th of which contains the integer \\\\(M_i\\\\), followed by strings \\\\(S_{i,1}\\\\) ... \\\\(S_{i,M_i}\\\\).\\n\\n\\n# Output Format\\n\\nFor the \\\\(i\\\\)th test case, print \"`Case #i:` \" followed by the number of topics that can be mutually-learned.\\n\\n\\n# Sample Explanation\\n\\nThe three sample cases are depicted below:\\n\\n{{PHOTO_ID:636025968710743|WIDTH:600}}\\n \\nIn the first case:\\n* \"gouda\" is mutually-learned in the partition \\\\(\\\\{\\\\{4\\\\}, \\\\{6, 5, 2\\\\}, \\\\{3, 1\\\\}\\\\}\\\\)\\n* \"cheddar\" is mutually-learned in the partition \\\\(\\\\{\\\\{4, 2\\\\}, \\\\{6, 5\\\\}, \\\\{3, 1\\\\}\\\\}\\\\)\\n* \"edam\" is mutually-learned in the same partition as \"cheddar\" as well as in other partitions such as \\\\(\\\\{\\\\{4, 2, 1\\\\}, \\\\{6\\\\}, \\\\{5\\\\}, \\\\{3\\\\}\\\\}\\\\)\\n* \"gjetost\", \"gruyere\", and \"mozzarella\" can never be mutually-learned\\n\\nIn the second case, the only topic that can be mutually-learned is \"earth\".\\n\\nIn the third case, the partition \\\\(\\\\{\\\\{1, 2, 3\\\\}\\\\}\\\\) makes all three topics mutually-learned.\\n\\n\\n\\nSample Input:\\n3\\n6\\n1 1 2 2 5\\n4 mozzarella cheddar gouda edam\\n5 gouda gjetost mozzarella cheddar edam\\n3 gruyere mozzarella edam\\n1 gouda\\n1 edam\\n3 cheddar gruyere edam\\n4\\n1 2 2\\n2 earth water\\n2 wind wind\\n2 earth fire\\n2 light metal\\n3\\n1 2\\n1 a\\n1 b\\n1 c\\n\\n\\nSample Output:\\nCase #1: 3\\nCase #2: 1\\nCase #3: 3\\n\\nPlease provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. 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Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "Updating best score\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.050515Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.055072Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,056] Trial 3 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 4\n", - "INFO:root:instruction_idx 4\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.108445Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.113307Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,114] Trial 4 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 5\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.174565Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.178665Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,179] Trial 5 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 6\n", - "INFO:root:instruction_idx 2\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.226630Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.231013Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,232] Trial 6 finished with value: 0.0 and parameters: {'0_predictor_instruction': 2}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 7\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.278299Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.282334Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,283] Trial 7 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 8\n", - "INFO:root:instruction_idx 4\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.335396Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.339829Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,340] Trial 8 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. 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Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 10\n", - "INFO:root:instruction_idx 0\n", - "DEBUG:root:OpenAI Response Token Usage: 1413\n", - "DEBUG:root:OpenAI Response Token Usage: 1886\n", - "DEBUG:root:OpenAI Response Token Usage: 1781\n", - "DEBUG:root:OpenAI Response Token Usage: 2477\n", - "DEBUG:root:OpenAI Response Token Usage: 1651\n", - "DEBUG:root:OpenAI Response Token Usage: 1595\n", - "DEBUG:root:OpenAI Response Token Usage: 1326\n", - "DEBUG:root:OpenAI Response Token Usage: 1659\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 1809\n", - "DEBUG:root:OpenAI Response Token Usage: 2213\n", - "DEBUG:root:OpenAI Response Token Usage: 1339\n", - "DEBUG:root:OpenAI Response Token Usage: 1800\n", - "DEBUG:root:OpenAI Response Token Usage: 1842\n", - "DEBUG:root:OpenAI Response Token Usage: 1457\n", - "DEBUG:root:OpenAI Response Token Usage: 2905\n", - "DEBUG:root:OpenAI Response Token Usage: 1705\n", - "DEBUG:root:OpenAI Response Token Usage: 1414\n", - "DEBUG:root:OpenAI Response Token Usage: 2099\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 1655\n", - "DEBUG:root:OpenAI Response Token Usage: 2398\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 1907\n", - "DEBUG:root:OpenAI Response Token Usage: 2011\n", - "DEBUG:root:OpenAI Response Token Usage: 2175\n", - "DEBUG:root:OpenAI Response Token Usage: 2267\n", - "DEBUG:root:OpenAI Response Token Usage: 1982\n", - "DEBUG:root:OpenAI Response Token Usage: 2055\n", - "DEBUG:root:OpenAI Response Token Usage: 1986\n", - "DEBUG:root:OpenAI Response Token Usage: 1602\n", - "DEBUG:root:OpenAI Response Token Usage: 2247\n", - "DEBUG:root:OpenAI Response Token Usage: 2089\n", - "DEBUG:root:OpenAI Response Token Usage: 2371\n", - "DEBUG:root:OpenAI Response Token Usage: 2123\n", - "DEBUG:root:OpenAI Response Token Usage: 1904\n", - "DEBUG:root:OpenAI Response Token Usage: 2036\n", - "DEBUG:root:OpenAI Response Token Usage: 2062\n", - "DEBUG:root:OpenAI Response Token Usage: 1688\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1782\n", - "DEBUG:root:OpenAI Response Token Usage: 1824\n", - "DEBUG:root:OpenAI Response Token Usage: 1702\n", - "DEBUG:root:OpenAI Response Token Usage: 2041\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.439161Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1413\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.443422Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,444] Trial 10 finished with value: 0.0 and parameters: {'0_predictor_instruction': 0}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 11\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", - "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", - "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", - "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", - "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", - "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", - "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", - "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", - "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.490569Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.496614Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,497] Trial 11 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 12\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.547837Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.552238Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,553] Trial 12 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 13\n", - "INFO:root:instruction_idx 0\n", - "DEBUG:root:OpenAI Response Token Usage: 1413\n", - "DEBUG:root:OpenAI Response Token Usage: 1886\n", - "DEBUG:root:OpenAI Response Token Usage: 1781\n", - "DEBUG:root:OpenAI Response Token Usage: 2477\n", - "DEBUG:root:OpenAI Response Token Usage: 1651\n", - "DEBUG:root:OpenAI Response Token Usage: 1595\n", - "DEBUG:root:OpenAI Response Token Usage: 1326\n", - "DEBUG:root:OpenAI Response Token Usage: 1659\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 1809\n", - "DEBUG:root:OpenAI Response Token Usage: 2213\n", - "DEBUG:root:OpenAI Response Token Usage: 1339\n", - "DEBUG:root:OpenAI Response Token Usage: 1800\n", - "DEBUG:root:OpenAI Response Token Usage: 1842\n", - "DEBUG:root:OpenAI Response Token Usage: 1457\n", - "DEBUG:root:OpenAI Response Token Usage: 2905\n", - "DEBUG:root:OpenAI Response Token Usage: 1705\n", - "DEBUG:root:OpenAI Response Token Usage: 1414\n", - "DEBUG:root:OpenAI Response Token Usage: 2099\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 1655\n", - "DEBUG:root:OpenAI Response Token Usage: 2398\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 1907\n", - "DEBUG:root:OpenAI Response Token Usage: 2011\n", - "DEBUG:root:OpenAI Response Token Usage: 2175\n", - "DEBUG:root:OpenAI Response Token Usage: 2267\n", - "DEBUG:root:OpenAI Response Token Usage: 1982\n", - "DEBUG:root:OpenAI Response Token Usage: 2055\n", - "DEBUG:root:OpenAI Response Token Usage: 1986\n", - "DEBUG:root:OpenAI Response Token Usage: 1602\n", - "DEBUG:root:OpenAI Response Token Usage: 2247\n", - "DEBUG:root:OpenAI Response Token Usage: 2089\n", - "DEBUG:root:OpenAI Response Token Usage: 2371\n", - "DEBUG:root:OpenAI Response Token Usage: 2123\n", - "DEBUG:root:OpenAI Response Token Usage: 1904\n", - "DEBUG:root:OpenAI Response Token Usage: 2036\n", - "DEBUG:root:OpenAI Response Token Usage: 2062\n", - "DEBUG:root:OpenAI Response Token Usage: 1688\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1782\n", - "DEBUG:root:OpenAI Response Token Usage: 1824\n", - "DEBUG:root:OpenAI Response Token Usage: 1702\n", - "DEBUG:root:OpenAI Response Token Usage: 2041\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.597933Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1413\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.602147Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,603] Trial 13 finished with value: 0.0 and parameters: {'0_predictor_instruction': 0}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 14\n", - "INFO:root:instruction_idx 3\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.646545Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.650284Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,651] Trial 14 finished with value: 0.0 and parameters: {'0_predictor_instruction': 3}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 15\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", - "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", - "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", - "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", - "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", - "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", - "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", - "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", - "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.695994Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.700319Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,701] Trial 15 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 16\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.757919Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.762307Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,763] Trial 16 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 17\n", - "INFO:root:instruction_idx 1\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.815724Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.820143Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,821] Trial 17 finished with value: 0.0 and parameters: {'0_predictor_instruction': 1}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 18\n", - "INFO:root:instruction_idx 4\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "DEBUG:root:OpenAI Response Token Usage: 2081\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 2803\n", - "DEBUG:root:OpenAI Response Token Usage: 1834\n", - "DEBUG:root:OpenAI Response Token Usage: 1677\n", - "DEBUG:root:OpenAI Response Token Usage: 1531\n", - "DEBUG:root:OpenAI Response Token Usage: 2106\n", - "DEBUG:root:OpenAI Response Token Usage: 2562\n", - "DEBUG:root:OpenAI Response Token Usage: 2098\n", - "DEBUG:root:OpenAI Response Token Usage: 2424\n", - "DEBUG:root:OpenAI Response Token Usage: 1573\n", - "DEBUG:root:OpenAI Response Token Usage: 2121\n", - "DEBUG:root:OpenAI Response Token Usage: 1892\n", - "DEBUG:root:OpenAI Response Token Usage: 1601\n", - "DEBUG:root:OpenAI Response Token Usage: 3176\n", - "DEBUG:root:OpenAI Response Token Usage: 1928\n", - "DEBUG:root:OpenAI Response Token Usage: 1541\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2008\n", - "DEBUG:root:OpenAI Response Token Usage: 1895\n", - "DEBUG:root:OpenAI Response Token Usage: 2629\n", - "DEBUG:root:OpenAI Response Token Usage: 2271\n", - "DEBUG:root:OpenAI Response Token Usage: 2137\n", - "DEBUG:root:OpenAI Response Token Usage: 2416\n", - "DEBUG:root:OpenAI Response Token Usage: 2314\n", - "DEBUG:root:OpenAI Response Token Usage: 2419\n", - "DEBUG:root:OpenAI Response Token Usage: 2381\n", - "DEBUG:root:OpenAI Response Token Usage: 2198\n", - "DEBUG:root:OpenAI Response Token Usage: 2293\n", - "DEBUG:root:OpenAI Response Token Usage: 1669\n", - "DEBUG:root:OpenAI Response Token Usage: 2327\n", - "DEBUG:root:OpenAI Response Token Usage: 2249\n", - "DEBUG:root:OpenAI Response Token Usage: 2543\n", - "DEBUG:root:OpenAI Response Token Usage: 2269\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 2158\n", - "DEBUG:root:OpenAI Response Token Usage: 2281\n", - "DEBUG:root:OpenAI Response Token Usage: 1916\n", - "DEBUG:root:OpenAI Response Token Usage: 2302\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 2005\n", - "DEBUG:root:OpenAI Response Token Usage: 1818\n", - "DEBUG:root:OpenAI Response Token Usage: 2148\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.866010Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 44 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "DEBUG:root:OpenAI Response Token Usage: 1582\n", - "INFO:dspy.evaluate.evaluate:\u001b[2m2024-09-11T17:38:24.870244Z\u001b[0m [\u001b[32m\u001b[1minfo \u001b[0m] \u001b[1mAverage Metric: 0 / 1 (0.0%) \u001b[0m [\u001b[0m\u001b[1m\u001b[34mdspy.evaluate.evaluate\u001b[0m]\u001b[0m \u001b[36mfilename\u001b[0m=\u001b[35mevaluate.py\u001b[0m \u001b[36mlineno\u001b[0m=\u001b[35m203\u001b[0m\n", - "[I 2024-09-11 19:38:24,871] Trial 18 finished with value: 0.0 and parameters: {'0_predictor_instruction': 4}. Best is trial 0 with value: 0.0.\n", - "INFO:root:Starting trial num: 19\n", - "INFO:root:instruction_idx 0\n", - "DEBUG:root:OpenAI Response Token Usage: 1413\n", - "DEBUG:root:OpenAI Response Token Usage: 1886\n", - "DEBUG:root:OpenAI Response Token Usage: 1781\n", - "DEBUG:root:OpenAI Response Token Usage: 2477\n", - "DEBUG:root:OpenAI Response Token Usage: 1651\n", - "DEBUG:root:OpenAI Response Token Usage: 1595\n", - "DEBUG:root:OpenAI Response Token Usage: 1326\n", - "DEBUG:root:OpenAI Response Token Usage: 1659\n", - "DEBUG:root:OpenAI Response Token Usage: 2136\n", - "DEBUG:root:OpenAI Response Token Usage: 1809\n", - "DEBUG:root:OpenAI Response Token Usage: 2213\n", - "DEBUG:root:OpenAI Response Token Usage: 1339\n", - "DEBUG:root:OpenAI Response Token Usage: 1800\n", - "DEBUG:root:OpenAI Response Token Usage: 1842\n", - "DEBUG:root:OpenAI Response Token Usage: 1457\n", - "DEBUG:root:OpenAI Response Token Usage: 2905\n", - "DEBUG:root:OpenAI Response Token Usage: 1705\n", - "DEBUG:root:OpenAI Response Token Usage: 1414\n", - "DEBUG:root:OpenAI Response Token Usage: 2099\n", - "DEBUG:root:OpenAI Response Token Usage: 1854\n", - "DEBUG:root:OpenAI Response Token Usage: 1655\n", - "DEBUG:root:OpenAI Response Token Usage: 2398\n", - "DEBUG:root:OpenAI Response Token Usage: 2004\n", - "DEBUG:root:OpenAI Response Token Usage: 1907\n", - "DEBUG:root:OpenAI Response Token Usage: 2011\n", - "DEBUG:root:OpenAI Response Token Usage: 2175\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n", - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are a highly skilled algorithmic problem solver tasked with providing a comprehensive and detailed explanation for a competitive programming challenge. Begin by carefully analyzing the problem statement, paying close attention to constraints, input and output formats. Identify critical elements and analyze the provided sample inputs and outputs, while also considering edge cases that may arise. Look for underlying patterns or mathematical relationships that can guide your approach. Break down the problem into smaller, manageable subproblems if necessary, and articulate the core insights in straightforward terms. Provide a step-by-step explanation of your reasoning and proposed solution, using clear language and, if beneficial, visual aids to enhance understanding. Discuss potential optimization techniques, alternative methods, and any special cases that may be relevant. Address both time and space complexity in your explanation, and include pseudocode where appropriate to illustrate your thought process. Finally, ensure that your solution is well-organized and accessible to readers of varying programming expertise, focusing on clarity and completeness. Remember, you are not required to write actual code; your goal is to create a thorough and understandable solution explanation that an experienced developer can use to implement the code.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sums to \\(41\\) and has a product equal to \\(P\\). \n", - "\n", - "1. **Understanding the Problem**: We need to find a set of positive integers such that their sum is \\(41\\) and their product is \\(P\\). If no such set exists, we should return \\(-1\\).\n", - "\n", - "2. **Constraints**: The maximum number of integers we can use is \\(100\\), and the product \\(P\\) can be as large as \\(10^9\\). This means we need to be strategic about how we choose our integers to ensure both the sum and product conditions are met.\n", - "\n", - "3. **Basic Cases**:\n", - " - If \\(P = 1\\), the only way to achieve a sum of \\(41\\) is to use \\(41\\) ones: \\([1, 1, 1, ..., 1]\\) (41 times).\n", - " - If \\(P = 41\\), the simplest solution is to use a single integer: \\([41]\\).\n", - "\n", - "4. **Decomposing \\(P\\)**: For other values of \\(P\\), we can factor \\(P\\) into its prime components. The product of the integers must equal \\(P\\), so we can start by breaking \\(P\\) down into its prime factors.\n", - "\n", - "5. **Constructing the Array**:\n", - " - Start with the prime factors of \\(P\\).\n", - " - If the sum of these factors exceeds \\(41\\), we need to adjust them to ensure the sum is exactly \\(41\\). This can be done by combining smaller factors or adding \\(1\\)s to fill the gap.\n", - " - If the sum of the factors is less than \\(41\\), we can add \\(1\\)s to reach \\(41\\).\n", - "\n", - "6. **Edge Cases**: \n", - " - If \\(P\\) is such that it cannot be factored into integers that sum to \\(41\\) (for example, if \\(P\\) is a prime number greater than \\(41\\)), we should return \\(-1\\).\n", - "\n", - "Solution: To implement the solution, we can follow these steps:\n", - "\n", - "1. For each test case, read the integer \\(P\\).\n", - "2. If \\(P = 1\\), output \\(41\\) ones.\n", - "3. If \\(P = 41\\), output \\(1\\) and the number \\(41\\).\n", - "4. For other values of \\(P\\):\n", - " - Factor \\(P\\) into its prime factors.\n", - " - Calculate the sum of these factors.\n", - " - If the sum is greater than \\(41\\), check if we can combine factors to reduce the sum while maintaining the product.\n", - " - If the sum is less than \\(41\\), add enough \\(1\\)s to make the total sum \\(41\\).\n", - "5. If no valid configuration is found, output \\(-1\\`.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for i from 1 to T:\n", - " P = read integer\n", - " if P == 1:\n", - " print \"Case #i: 41 1 1 1 ... (41 times)\"\n", - " else if P == 41:\n", - " print \"Case #i: 1 41\"\n", - " else:\n", - " factors = factor(P)\n", - " sum_factors = sum(factors)\n", - " if sum_factors > 41:\n", - " # Try to combine factors or check if valid\n", - " if valid_combination(factors, 41):\n", - " print \"Case #i: N factors\"\n", - " else:\n", - " print \"Case #i: -1\"\n", - " else:\n", - " # Add 1s to reach 41\n", - " num_ones = 41 - sum_factors\n", - " print \"Case #i: N + num_ones factors\"\n", - "```\n", - "\n", - "This approach ensures that we systematically check for valid combinations of integers that meet the required sum and product, while also considering edge cases and constraints. The time complexity primarily depends on the factorization of \\(P\\), which can be done in \\(O(\\sqrt{P})\\), and the space complexity is \\(O(N)\\) where \\(N\\) is the number of integers in the output array.\u001b[0m\n", - "\n", - "\n", - "\n", - "...\n", - "Score 0.0\n", - "CANDIDATE PROGRAM:\n", - "Predictor 0\n", - "i: You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "p: Solution:\n", - "\n", - "\n", - "...\n" + "Average Metric: 0.22450729631544877 / 3 (7.5): 60%|██████ | 3/5 [01:08<00:39, 20.00s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0192000e-1786-7531-b215-f080751bfbef\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.3514108496149412 / 4 (8.8): 80%|████████ | 4/5 [01:09<00:12, 12.55s/it] " + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0192000e-1756-7601-bdaf-568d5710bda5\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.42352623423032576 / 5 (8.5): 100%|██████████| 5/5 [01:10<00:00, 14.02s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0192000e-175c-7302-b28c-334da9cf2345\n", + "Average Metric: 0.42352623423032576 / 5 (8.5%)\n", + "Score: 8.47 for set: [1]\n", + "Scores so far: [7.56, 13.86, 5.71, 19.1, 8.47]\n", + "Best score: 19.1\n", + "Average of max per entry across top 1 scores: 0.1909534773918633\n", + "Average of max per entry across top 2 scores: 0.19715426110866333\n", + "Average of max per entry across top 3 scores: 0.20753497176856178\n", + "Average of max per entry across top 5 scores: 0.20753497176856178\n", + "Average of max per entry across top 8 scores: 0.20753497176856178\n", + "Average of max per entry across top 9999 scores: 0.20753497176856178\n", + "5 candidate programs found.\n", + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/0191ffff-3e01-7e80-a448-ebc211783b0e\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\n" + ] + } + ], + "source": [ + "optimized_program = optimize_with_bootstrap_fewshot(simple_program, mistral, mistral, f1, trainset[0:5])" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "generate_code = ChainOfThought(GenerateSolution(problem_statement, sample_input, sample_output -> solution\n", + " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\n When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\n Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\\n Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\n Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\n Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\n Address time and space complexity, and provide pseudocode if appropriate. \\n Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\n Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\\n \"\n", + " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", + " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", + " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", + " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", + "))" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "optimized_program" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.11134020618556702 / 1 (11.1): 17%|█▋ | 1/6 [00:42<03:33, 42.78s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0e28-7053-8db3-16f82422ac7f\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.1526625202351538 / 2 (7.6): 33%|███▎ | 2/6 [00:56<01:42, 25.50s/it] " + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0e70-77d1-aae1-9a44f8d1abfd\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.21462038888447227 / 3 (7.2): 50%|█████ | 3/6 [01:02<00:50, 16.82s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0eb5-7261-a97a-c52643e7f5fb\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.2626478331383316 / 4 (6.6): 67%|██████▋ | 4/6 [01:05<00:22, 11.38s/it] " + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0e34-7162-9ff6-7bcb46f555a5\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.32034014083063933 / 5 (6.4): 83%|████████▎ | 5/6 [01:07<00:07, 7.87s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0ea3-7be3-9800-96b81b2525b4\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Average Metric: 0.37297171977800775 / 6 (6.2): 100%|██████████| 6/6 [01:20<00:00, 13.42s/it]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0e51-7b60-a445-0d5b6e870d08\n", + "Average Metric: 0.37297171977800775 / 6 (6.2%)\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - 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 problem_statementsample_inputsample_outputexample_solutionpred_solutionf1
0Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with exactly \\(K\\) green...4\n", + "5 3\n", + "6 3\n", + "6 2\n", + "243447 42273\n", + "Case #1: 1\n", + "Case #2: 2\n", + "Case #3: 4\n", + "Case #4: 4\n", + "First we need to determine how many bracelets Ishiko is going to exhibit. When \\(N = K\\) (that is, when all of the beads are...### Problem Statement: Ishiko is opening a bracelet shop. Each bracelet she plans to sell is expressed as a ring of \\(N\\) colored beads, with...0.11134020618556702
1Boss Rob planted \\(N\\) happy little hazel trees in his yard (represented on a Cartesian plane). The \\(i\\)th tree is at integer coordinates \\((X_i, Y_i)\\)....3 4 0 0 2 1 5 0 0 6 5 10 10 12 10 8 10 5 10 8 8 4 1 1 3...Case #1: 20\n", + "Case #2: 28\n", + "Case #3: 42\n", + "First, we'll note a greedy philosophy that if we can simply use a \\(2 \\times 2\\) square around all trees, that’d be the best solution...### Problem Statement: Boss Rob planted \\(N\\) happy little hazel trees in his yard, represented on a Cartesian plane. The \\(i\\)th tree is at integer...0.048027444253859346
2Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\) nodes (numbered from...3 9 1 2 M 1 3 E 1 4 T 4 9 A 2 5 T 2 6 E 3 7 A 3 8...Case #1: 6 9 2 9 10\n", + "Case #2: 2 1 6 4\n", + "Case #3: 1 1 2\n", + "The first thing to do is root the tree, for which we can arbitrarily choose node \\(1\\). Each journey can be done in two phases:...### Problem Statement: Alphonse is assembling an *alphabet tree* and arranging some adventures along the way. An alphabet tree is an unrooted tree with \\(N\\)...0.05263157894736842
3Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground. When viewed directly...2\n", + "3 1 4\n", + "0 0\n", + "2 2\n", + "4 0\n", + "5 1 4\n", + "0 0\n", + "1 2\n", + "0 3\n", + "4 3\n", + "3 1\n", + "Case #1: 0.1169663730642699\n", + "Case #2: 0.1353445414060363\n", + "Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all...### Problem Statement: Courtney has an avant-garde kitchen counter made out of \\(N\\) rectangular sheets of metal of equal height, standing perpendicular to the ground....0.04132231404958678
4You just landed a job as a machine learning engineer! As a ramp-up exercise, Boss Rob tasked you with modeling the watering wells in his...2 500 18.243577 16.343618 24.560940 7.478552 13.664297 0.348593 19.766713 16.871980 14.052491 10.567715 21.426414 5.786941 20.495098 -0.246197 20.706538 14.324784 13.240629 9.591812 18.131521 1.645394 13.085966 5.206907 12.705525...Case #1: 6 10 30 2 19\n", + "Case #2: 40 50 38 45 13\n", + "There are many ways you might approach this problem, but a simple one is to find the pair of circles that contain all the given...### Problem Statement: Boss Rob has a fence made of \\(N\\) wooden stakes, numbered \\(1\\) to \\(N\\). Initially, the \\(i\\)th stake is of color \\(i\\)....0.0576923076923077
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\n", + " " + ], + "text/plain": [ + "" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, { "name": "stdout", "output_type": "stream", "text": [ - "FULL TRACE\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", - "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", - "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", - "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", - "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", - "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "You are an expert problem solver. Your task is to solve the problem at hand.\n", - "\n", - "When solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\n", - "Identify the key elements, analyze sample inputs/outputs, and consider edge cases.\n", - "Look for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \n", - "Explain the core insight in simple terms, followed by a step-by-step explanation of the solution.\n", - "Use clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \n", - "Address time and space complexity, and provide pseudocode if appropriate. \n", - "Finally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\n", - "Note: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\n", - "\n", - "---\n", - "\n", - "Follow the following format.\n", - "\n", - "Problem Statement: ${problem_statement}\n", - "\n", - "Sample Input: The sample input provided with the problem statement.\n", - "\n", - "Sample Output: The sample output provided with the problem statement.\n", - "\n", - "Reasoning: Let's think step by step in order to ${produce the output fields}. We ...\n", - "\n", - "Solution: a solution explanation for how we should go about solving this problem.\n", - "\n", - "---\n", - "\n", - "Problem Statement:\n", - "*This problem shares some similarities with B2, with key differences in bold.*\n", - "\n", - "Given a positive integer \\(P\\), please find an array of at most \\(100\\) positive integers which have a sum of \\(41\\) and a product of \\(P\\), or output \\(-1\\) if no such array exists.\n", - "\n", - "If multiple such arrays exist, **you may output any one of them**.\n", - "\n", - "# Constraints\n", - "\\(1 \\leq T \\leq 965\\)\n", - "\\(1 \\leq P \\leq 10^9\\)\n", - "\n", - "# Input Format\n", - "\n", - "Input begins with an integer \\(T\\), the number of test cases. For each case, there is one line containing a single integer \\(P\\).\n", - "\n", - "# Output Format\n", - "\n", - "For the \\(i\\)th test case, if there is no such array, print \"`Case #i: -1`\". Otherwise, print \"`Case #i:` \" followed by the integer \\(N\\), the size of your array, followed by the array itself as \\(N\\) more space-separated positive integers.\n", - "\n", - "# Sample Explanation\n", - "In the first sample, we must find an array with product \\(2023\\), and sum \\(41\\). One possible answer is \\([7, 17, 17]\\).\n", - "\n", - "\n", - "\n", - "Sample Input:\n", - "7\n", - "2023\n", - "114\n", - "41\n", - "175\n", - "434\n", - "666\n", - "1872\n", - "\n", - "\n", - "Sample Output:\n", - "Case #1: 3 7 17 17\n", - "Case #2: 2 3 38\n", - "Case #3: 1 41\n", - "Case #4: 3 1 5 35\n", - "Case #5: 4 1 2 7 31\n", - "Case #6: -1\n", - "Case #7: 5 1 2 6 6 26\n", - "\n", - "Please provide the output fields Reasoning then Solution. Do so immediately, without additional content before or after, and precisely as the format above shows. Begin with the field Reasoning.\n", - "\n", - "\u001b[32mReasoning: Let's think step by step in order to find an array of positive integers that sum to \\(41\\) and have a product of \\(P\\). We need to consider the following:\n", - "\n", - "1. **Understanding the Constraints**: The sum of the integers must equal \\(41\\), and the product must equal \\(P\\). We can use up to \\(100\\) integers, but we should aim to use as few as possible to simplify our solution.\n", - "\n", - "2. **Single Element Case**: If \\(P = 41\\), the only array we can form is \\([41]\\). This is a straightforward case.\n", - "\n", - "3. **Decomposing \\(P\\)**: For other values of \\(P\\), we need to factor \\(P\\) into integers that can sum to \\(41\\). The factors of \\(P\\) will guide us in forming the array.\n", - "\n", - "4. **Using 1s**: If we can find a combination of factors that sum to less than \\(41\\), we can fill the remaining sum with \\(1s\\). For example, if we find factors that sum to \\(x\\), we can add \\(41 - x\\) ones to reach the total sum of \\(41\\).\n", - "\n", - "5. **Iterating through Factors**: We can iterate through possible factors of \\(P\\) and check combinations of these factors to see if they can sum to \\(41\\). \n", - "\n", - "6. **Edge Cases**: If \\(P\\) is too large or has factors that cannot combine to form \\(41\\), we should return \\(-1\\).\n", - "\n", - "Solution: To solve the problem, we can follow these steps:\n", - "\n", - "1. **Input Handling**: Read the number of test cases \\(T\\) and the values of \\(P\\) for each test case.\n", - "\n", - "2. **Factorization**: For each \\(P\\), find all possible factors. This can be done by iterating from \\(1\\) to \\(\\sqrt{P}\\) and checking divisibility.\n", - "\n", - "3. **Combination Search**: Use a backtracking approach to find combinations of the factors that sum to \\(41\\). If a combination is found, check if the product of that combination equals \\(P\\).\n", + "🍩 https://wandb.ai/capecape/dspy_hackercup/r/call/01920010-0e25-7bd0-8537-3fd3f96dcc27\n" + ] + }, + { + "data": { + "text/plain": [ + "6.22" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "evaluate(program=optimized_program, devset=devset)" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "\u001b[93m\u001b[1mWARNING: Projected Language Model (LM) Calls\u001b[0m\n", "\n", - "4. **Filling with 1s**: If the sum of the selected factors is less than \\(41\\), calculate how many \\(1s\\) are needed to reach \\(41\\) and include them in the array.\n", + "Please be advised that based on the parameters you have set, the maximum number of LM calls is projected as follows:\n", "\n", - "5. **Output Formatting**: For each test case, format the output according to the requirements, indicating the case number and the resulting array or \\(-1\\) if no valid array exists.\n", + "\u001b[93m- Task Model: \u001b[94m\u001b[1m5\u001b[0m\u001b[93m examples in dev set * \u001b[94m\u001b[1m10\u001b[0m\u001b[93m trials * \u001b[94m\u001b[1m# of LM calls in your program\u001b[0m\u001b[93m = (\u001b[94m\u001b[1m50 * # of LM calls in your program\u001b[0m\u001b[93m) task model calls\u001b[0m\n", + "\u001b[93m- Prompt Model: # data summarizer calls (max \u001b[94m\u001b[1m10\u001b[0m\u001b[93m) + \u001b[94m\u001b[1m2\u001b[0m\u001b[93m * \u001b[94m\u001b[1m1\u001b[0m\u001b[93m lm calls in program = \u001b[94m\u001b[1m12\u001b[0m\u001b[93m prompt model calls\u001b[0m\n", "\n", - "6. **Complexity Considerations**: The time complexity will depend on the factorization and combination search, but since \\(P\\) can be at most \\(10^9\\), the factorization step will be efficient enough due to the limited number of factors.\n", + "\u001b[93m\u001b[1mEstimated Cost Calculation:\u001b[0m\n", "\n", - "Pseudocode:\n", - "```\n", - "for each test case:\n", - " read P\n", - " if P == 41:\n", - " output \"Case #i: 1 41\"\n", - " continue\n", - " factors = find_factors(P)\n", - " for each combination of factors:\n", - " if sum(combination) <= 41:\n", - " remaining = 41 - sum(combination)\n", - " if product(combination) * (1^remaining) == P:\n", - " output \"Case #i: N combination + remaining ones\"\n", - " break\n", - " else:\n", - " output \"Case #i: -1\"\n", - "```\n", + "\u001b[93mTotal Cost = (Number of calls to task model * (Avg Input Token Length per Call * Task Model Price per Input Token + Avg Output Token Length per Call * Task Model Price per Output Token) \n", + " + (Number of calls to prompt model * (Avg Input Token Length per Call * Task Prompt Price per Input Token + Avg Output Token Length per Call * Prompt Model Price per Output Token).\u001b[0m\n", "\n", - "This approach ensures we explore all possible combinations of factors while adhering to the constraints of the problem.\u001b[0m\n", + "For a preliminary estimate of potential costs, we recommend you perform your own calculations based on the task\n", + "and prompt models you intend to use. If the projected costs exceed your budget or expectations, you may consider:\n", "\n", + "\u001b[93m- Reducing the number of trials (`num_trials`), the size of the trainset, or the number of LM calls in your program.\u001b[0m\n", + "\u001b[93m- Using a cheaper task model to optimize the prompt.\u001b[0m\n", + "To proceed with the execution of this program, please confirm by typing \u001b[94m'y'\u001b[0m for yes or \u001b[94m'n'\u001b[0m for no.\n", "\n", + "If you would like to bypass this confirmation step in future executions, set the \u001b[93m`requires_permission_to_run`\u001b[0m flag to \u001b[93m`False`.\u001b[0m\n", "\n", - "...\n", - "Score 0.0\n", - "[('generate_code', Predict(StringSignature(problem_statement, sample_input, sample_output -> rationale, solution\n", - " instructions=\"You are an expert problem solver. Your task is to solve the problem at hand.\\n\\nWhen solving a competitive programming problem, start by thoroughly reading and understanding the problem statement, including all constraints and input/output formats.\\nIdentify the key elements, analyze sample inputs/outputs, and consider edge cases.\\nLook for patterns or mathematical relationships, then develop a logical approach, breaking the problem into subproblems if needed. \\nExplain the core insight in simple terms, followed by a step-by-step explanation of the solution.\\nUse clear language and visual aids if helpful. Discuss optimization techniques, alternative approaches, and special cases. \\nAddress time and space complexity, and provide pseudocode if appropriate. \\nFinally, proofread the solution for clarity and completeness, ensuring it's accessible to readers with varying levels of programming experience.\\nNote: You are not expected to write the code for the solution, just provide a solution explanation so that an experienced developer can write the code for the solution.\"\n", - " problem_statement = Field(annotation=str required=True json_schema_extra={'format': , '__dspy_field_type': 'input', 'prefix': 'Problem Statement:', 'desc': '${problem_statement}'})\n", - " sample_input = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample input provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Input:'})\n", - " sample_output = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'The sample output provided with the problem statement.', '__dspy_field_type': 'input', 'prefix': 'Sample Output:'})\n", - " rationale = Field(annotation=str required=True json_schema_extra={'prefix': \"Reasoning: Let's think step by step in order to\", 'desc': '${produce the output fields}. We ...', '__dspy_field_type': 'output'})\n", - " solution = Field(annotation=str required=True json_schema_extra={'format': , 'desc': 'a solution explanation for how we should go about solving this problem.', '__dspy_field_type': 'output', 'prefix': 'Solution:'})\n", - ")))]\n" + "\u001b[93mAwaiting your input...\u001b[0m\n", + "\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + " 0%| | 0/5 [00:59 28\u001b[0m mipro_optimized_program \u001b[38;5;241m=\u001b[39m \u001b[43moptimize_with_mipro\u001b[49m\u001b[43m(\u001b[49m\u001b[43msimple_program\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmistral\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mmistral\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mf1\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtrainset\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m:\u001b[49m\u001b[38;5;241;43m5\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m\n", + "Cell \u001b[0;32mIn[33], line 11\u001b[0m, in \u001b[0;36moptimize_with_mipro\u001b[0;34m(program, prompt_model, task_model, metric, trainset)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21moptimize_with_mipro\u001b[39m(program, prompt_model, task_model, metric, trainset):\n\u001b[1;32m 2\u001b[0m teleprompter \u001b[38;5;241m=\u001b[39m MIPRO(\n\u001b[1;32m 3\u001b[0m prompt_model\u001b[38;5;241m=\u001b[39mprompt_model,\n\u001b[1;32m 4\u001b[0m task_model\u001b[38;5;241m=\u001b[39mtask_model,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 8\u001b[0m verbose\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mFalse\u001b[39;00m,\n\u001b[1;32m 9\u001b[0m )\n\u001b[0;32m---> 11\u001b[0m optimized_program \u001b[38;5;241m=\u001b[39m \u001b[43mteleprompter\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcompile\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 12\u001b[0m \u001b[43m \u001b[49m\u001b[43mprogram\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mdeepcopy\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 13\u001b[0m \u001b[43m \u001b[49m\u001b[43mnum_trials\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m10\u001b[39;49m\u001b[43m,\u001b[49m\n\u001b[1;32m 14\u001b[0m \u001b[43m \u001b[49m\u001b[43mtrainset\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtrainset\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 15\u001b[0m \u001b[43m \u001b[49m\u001b[43meval_kwargs\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mdict\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mnum_threads\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m8\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 16\u001b[0m \u001b[43m \u001b[49m\u001b[43mmax_bootstrapped_demos\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;66;43;03m# 0-shot optimization\u001b[39;49;00m\n\u001b[1;32m 17\u001b[0m \u001b[43m \u001b[49m\u001b[43mmax_labeled_demos\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m,\u001b[49m\n\u001b[1;32m 18\u001b[0m \u001b[43m \u001b[49m\u001b[43mseed\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m9\u001b[39;49m\n\u001b[1;32m 19\u001b[0m \u001b[43m \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 20\u001b[0m now \u001b[38;5;241m=\u001b[39m datetime\u001b[38;5;241m.\u001b[39mnow()\n\u001b[1;32m 21\u001b[0m date_time \u001b[38;5;241m=\u001b[39m now\u001b[38;5;241m.\u001b[39mstrftime(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124mY\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124mm\u001b[39m\u001b[38;5;132;01m%d\u001b[39;00m\u001b[38;5;124m_\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124mH\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124mM\u001b[39m\u001b[38;5;124m%\u001b[39m\u001b[38;5;124mS\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/teleprompt/mipro_optimizer.py:359\u001b[0m, in \u001b[0;36mMIPRO.compile\u001b[0;34m(self, student, trainset, num_trials, max_bootstrapped_demos, max_labeled_demos, eval_kwargs, seed, view_data, view_examples, requires_permission_to_run)\u001b[0m\n\u001b[1;32m 357\u001b[0m rng\u001b[38;5;241m.\u001b[39mshuffle(shuffled_trainset) \u001b[38;5;66;03m# Shuffle the copy\u001b[39;00m\n\u001b[1;32m 358\u001b[0m tp \u001b[38;5;241m=\u001b[39m BootstrapFewShot(metric \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mmetric, max_bootstrapped_demos\u001b[38;5;241m=\u001b[39mmax_bootstrapped_demos_for_candidate_gen, max_labeled_demos\u001b[38;5;241m=\u001b[39mmax_labeled_demos_for_candidate_gen, teacher_settings\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mteacher_settings)\n\u001b[0;32m--> 359\u001b[0m candidate_program \u001b[38;5;241m=\u001b[39m \u001b[43mtp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcompile\u001b[49m\u001b[43m(\u001b[49m\u001b[43mstudent\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mmodule\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mdeepcopy\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtrainset\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mshuffled_trainset\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 361\u001b[0m \u001b[38;5;66;03m# Store the candidate demos\u001b[39;00m\n\u001b[1;32m 362\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m module_p, candidate_p \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mzip\u001b[39m(module\u001b[38;5;241m.\u001b[39mpredictors(), candidate_program\u001b[38;5;241m.\u001b[39mpredictors()):\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/teleprompt/bootstrap.py:52\u001b[0m, in \u001b[0;36mBootstrapFewShot.compile\u001b[0;34m(self, student, teacher, trainset, valset)\u001b[0m\n\u001b[1;32m 50\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_prepare_student_and_teacher(student, teacher)\n\u001b[1;32m 51\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_prepare_predictor_mappings()\n\u001b[0;32m---> 52\u001b[0m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_bootstrap\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 54\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mstudent \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_train()\n\u001b[1;32m 55\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mstudent\u001b[38;5;241m.\u001b[39m_compiled \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mTrue\u001b[39;00m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/teleprompt/bootstrap.py:109\u001b[0m, in \u001b[0;36mBootstrapFewShot._bootstrap\u001b[0;34m(self, max_bootstraps)\u001b[0m\n\u001b[1;32m 106\u001b[0m \u001b[38;5;28;01mbreak\u001b[39;00m\n\u001b[1;32m 108\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m example_idx \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;129;01min\u001b[39;00m bootstrapped:\n\u001b[0;32m--> 109\u001b[0m success \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_bootstrap_one_example\u001b[49m\u001b[43m(\u001b[49m\u001b[43mexample\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mround_idx\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 111\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m success:\n\u001b[1;32m 112\u001b[0m bootstrapped[example_idx] \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mTrue\u001b[39;00m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/teleprompt/bootstrap.py:143\u001b[0m, in \u001b[0;36mBootstrapFewShot._bootstrap_one_example\u001b[0;34m(self, example, round_idx)\u001b[0m\n\u001b[1;32m 140\u001b[0m predictor_cache[name] \u001b[38;5;241m=\u001b[39m predictor\u001b[38;5;241m.\u001b[39mdemos\n\u001b[1;32m 141\u001b[0m predictor\u001b[38;5;241m.\u001b[39mdemos \u001b[38;5;241m=\u001b[39m [x \u001b[38;5;28;01mfor\u001b[39;00m x \u001b[38;5;129;01min\u001b[39;00m predictor\u001b[38;5;241m.\u001b[39mdemos \u001b[38;5;28;01mif\u001b[39;00m x \u001b[38;5;241m!=\u001b[39m example]\n\u001b[0;32m--> 143\u001b[0m prediction \u001b[38;5;241m=\u001b[39m \u001b[43mteacher\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mexample\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43minputs\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 144\u001b[0m trace \u001b[38;5;241m=\u001b[39m dsp\u001b[38;5;241m.\u001b[39msettings\u001b[38;5;241m.\u001b[39mtrace\n\u001b[1;32m 146\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m name, predictor \u001b[38;5;129;01min\u001b[39;00m teacher\u001b[38;5;241m.\u001b[39mnamed_predictors():\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/primitives/program.py:26\u001b[0m, in \u001b[0;36mModule.__call__\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m 25\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__call__\u001b[39m(\u001b[38;5;28mself\u001b[39m, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs):\n\u001b[0;32m---> 26\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mforward\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n", + "Cell \u001b[0;32mIn[24], line 7\u001b[0m, in \u001b[0;36mSimpleGenerateSolution.forward\u001b[0;34m(self, problem_statement, sample_input, sample_output)\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mforward\u001b[39m(\u001b[38;5;28mself\u001b[39m, problem_statement, sample_input, sample_output):\n\u001b[0;32m----> 7\u001b[0m solution \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mgenerate_code\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 8\u001b[0m \u001b[43m \u001b[49m\u001b[43mproblem_statement\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mproblem_statement\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 9\u001b[0m \u001b[43m \u001b[49m\u001b[43msample_input\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43msample_input\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 10\u001b[0m \u001b[43m \u001b[49m\u001b[43msample_output\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43msample_output\u001b[49m\n\u001b[1;32m 11\u001b[0m \u001b[43m \u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;241m.\u001b[39msolution\n\u001b[1;32m 13\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m dspy\u001b[38;5;241m.\u001b[39mPrediction(solution\u001b[38;5;241m=\u001b[39msolution)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/predict/predict.py:49\u001b[0m, in \u001b[0;36mPredict.__call__\u001b[0;34m(self, **kwargs)\u001b[0m\n\u001b[1;32m 48\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__call__\u001b[39m(\u001b[38;5;28mself\u001b[39m, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs):\n\u001b[0;32m---> 49\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mforward\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/predict/chain_of_thought.py:59\u001b[0m, in \u001b[0;36mChainOfThought.forward\u001b[0;34m(self, **kwargs)\u001b[0m\n\u001b[1;32m 57\u001b[0m signature \u001b[38;5;241m=\u001b[39m new_signature\n\u001b[1;32m 58\u001b[0m \u001b[38;5;66;03m# template = dsp.Template(self.signature.instructions, **new_signature)\u001b[39;00m\n\u001b[0;32m---> 59\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43msuper\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mforward\u001b[49m\u001b[43m(\u001b[49m\u001b[43msignature\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43msignature\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dspy/predict/predict.py:91\u001b[0m, in \u001b[0;36mPredict.forward\u001b[0;34m(self, **kwargs)\u001b[0m\n\u001b[1;32m 88\u001b[0m template \u001b[38;5;241m=\u001b[39m signature_to_template(signature)\n\u001b[1;32m 90\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mlm \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[0;32m---> 91\u001b[0m x, C \u001b[38;5;241m=\u001b[39m \u001b[43mdsp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mgenerate\u001b[49m\u001b[43m(\u001b[49m\u001b[43mtemplate\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mconfig\u001b[49m\u001b[43m)\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mstage\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mstage\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 92\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 93\u001b[0m \u001b[38;5;66;03m# Note: query_only=True means the instructions and examples are not included.\u001b[39;00m\n\u001b[1;32m 94\u001b[0m \u001b[38;5;66;03m# I'm not really sure why we'd want to do that, but it's there.\u001b[39;00m\n\u001b[1;32m 95\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m dsp\u001b[38;5;241m.\u001b[39msettings\u001b[38;5;241m.\u001b[39mcontext(lm\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mlm, query_only\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m):\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dsp/primitives/predict.py:77\u001b[0m, in \u001b[0;36m_generate..do_generate\u001b[0;34m(example, stage, max_depth, original_example)\u001b[0m\n\u001b[1;32m 75\u001b[0m \u001b[38;5;66;03m# Generate and extract the fields.\u001b[39;00m\n\u001b[1;32m 76\u001b[0m prompt \u001b[38;5;241m=\u001b[39m template(example)\n\u001b[0;32m---> 77\u001b[0m completions: \u001b[38;5;28mlist\u001b[39m[\u001b[38;5;28mdict\u001b[39m[\u001b[38;5;28mstr\u001b[39m, Any]] \u001b[38;5;241m=\u001b[39m \u001b[43mgenerator\u001b[49m\u001b[43m(\u001b[49m\u001b[43mprompt\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 78\u001b[0m completions: \u001b[38;5;28mlist\u001b[39m[Example] \u001b[38;5;241m=\u001b[39m [template\u001b[38;5;241m.\u001b[39mextract(example, p) \u001b[38;5;28;01mfor\u001b[39;00m p \u001b[38;5;129;01min\u001b[39;00m completions]\n\u001b[1;32m 80\u001b[0m \u001b[38;5;66;03m# Find the completions that are most complete.\u001b[39;00m\n", + "Cell \u001b[0;32mIn[6], line 50\u001b[0m, in \u001b[0;36mMistralLM.__call__\u001b[0;34m(self, prompt, only_completed, return_sorted, **kwargs)\u001b[0m\n\u001b[1;32m 49\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__call__\u001b[39m(\u001b[38;5;28mself\u001b[39m, prompt, only_completed\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m, return_sorted\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mFalse\u001b[39;00m, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs):\n\u001b[0;32m---> 50\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrequest\u001b[49m\u001b[43m(\u001b[49m\u001b[43mprompt\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 51\u001b[0m completions \u001b[38;5;241m=\u001b[39m [response\u001b[38;5;241m.\u001b[39mchoices[\u001b[38;5;241m0\u001b[39m]\u001b[38;5;241m.\u001b[39mmessage\u001b[38;5;241m.\u001b[39mcontent] \n\u001b[1;32m 52\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m completions\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/dsp/modules/lm.py:26\u001b[0m, in \u001b[0;36mLM.request\u001b[0;34m(self, prompt, **kwargs)\u001b[0m\n\u001b[1;32m 25\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mrequest\u001b[39m(\u001b[38;5;28mself\u001b[39m, prompt, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs):\n\u001b[0;32m---> 26\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mbasic_request\u001b[49m\u001b[43m(\u001b[49m\u001b[43mprompt\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n", + "Cell \u001b[0;32mIn[6], line 32\u001b[0m, in \u001b[0;36mMistralLM.basic_request\u001b[0;34m(self, prompt, **kwargs)\u001b[0m\n\u001b[1;32m 29\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m n \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m \u001b[38;5;129;01mand\u001b[39;00m n \u001b[38;5;241m>\u001b[39m \u001b[38;5;241m1\u001b[39m \u001b[38;5;129;01mand\u001b[39;00m kwargs[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mtemperature\u001b[39m\u001b[38;5;124m\"\u001b[39m] \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m0.0\u001b[39m:\n\u001b[1;32m 30\u001b[0m kwargs[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mtemperature\u001b[39m\u001b[38;5;124m\"\u001b[39m] \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m0.7\u001b[39m\n\u001b[0;32m---> 32\u001b[0m chat_response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mclient\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mchat\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcomplete\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 33\u001b[0m \u001b[43m \u001b[49m\u001b[43mmodel\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmodel\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 34\u001b[0m \u001b[43m \u001b[49m\u001b[43mmessages\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43m[\u001b[49m\n\u001b[1;32m 35\u001b[0m \u001b[43m \u001b[49m\u001b[38;5;28;43mdict\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mrole\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43muser\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcontent\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mprompt\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 36\u001b[0m \u001b[43m \u001b[49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 37\u001b[0m \u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\n\u001b[1;32m 38\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 39\u001b[0m response_dict \u001b[38;5;241m=\u001b[39m json\u001b[38;5;241m.\u001b[39mloads(chat_response\u001b[38;5;241m.\u001b[39mmodel_dump_json())\n\u001b[1;32m 41\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mhistory\u001b[38;5;241m.\u001b[39mappend({\n\u001b[1;32m 42\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mprompt\u001b[39m\u001b[38;5;124m\"\u001b[39m: prompt,\n\u001b[1;32m 43\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mresponse\u001b[39m\u001b[38;5;124m\"\u001b[39m: response_dict, \u001b[38;5;66;03m#{\"choices\": chat_response.choices[0].message.content},\u001b[39;00m\n\u001b[1;32m 44\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mkwargs\u001b[39m\u001b[38;5;124m\"\u001b[39m: kwargs,\n\u001b[1;32m 45\u001b[0m })\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:403\u001b[0m, in \u001b[0;36mop..op_deco..create_wrapper..wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m 401\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m func(\u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)\n\u001b[1;32m 402\u001b[0m call \u001b[38;5;241m=\u001b[39m _create_call(wrapper, \u001b[38;5;241m*\u001b[39margs, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs) \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[0;32m--> 403\u001b[0m res, _ \u001b[38;5;241m=\u001b[39m \u001b[43m_execute_call\u001b[49m\u001b[43m(\u001b[49m\u001b[43mwrapper\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcall\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;66;03m# type: ignore\u001b[39;00m\n\u001b[1;32m 404\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m res\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/weave/trace/op.py:250\u001b[0m, in \u001b[0;36m_execute_call\u001b[0;34m(__op, call, __should_raise, *args, **kwargs)\u001b[0m\n\u001b[1;32m 247\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m _call_async()\n\u001b[1;32m 249\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 250\u001b[0m res \u001b[38;5;241m=\u001b[39m \u001b[43mfunc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 251\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n\u001b[1;32m 252\u001b[0m handle_exception(e)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/mistralai/chat.py:106\u001b[0m, in \u001b[0;36mChat.complete\u001b[0;34m(self, model, messages, temperature, top_p, max_tokens, min_tokens, stream, stop, random_seed, response_format, tools, tool_choice, safe_prompt, retries, server_url, timeout_ms)\u001b[0m\n\u001b[1;32m 97\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(retries, utils\u001b[38;5;241m.\u001b[39mRetryConfig):\n\u001b[1;32m 98\u001b[0m retry_config \u001b[38;5;241m=\u001b[39m (retries, [\n\u001b[1;32m 99\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m429\u001b[39m\u001b[38;5;124m\"\u001b[39m,\n\u001b[1;32m 100\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m500\u001b[39m\u001b[38;5;124m\"\u001b[39m,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 103\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m504\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 104\u001b[0m ]) \n\u001b[0;32m--> 106\u001b[0m http_res 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utils\u001b[38;5;241m.\u001b[39mmatch_status_codes(error_status_codes, http_res\u001b[38;5;241m.\u001b[39mstatus_code):\n\u001b[1;32m 176\u001b[0m http_res \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39msdk_configuration\u001b[38;5;241m.\u001b[39mget_hooks()\u001b[38;5;241m.\u001b[39mafter_success(\n\u001b[1;32m 177\u001b[0m AfterSuccessContext(hook_ctx), http_res\n\u001b[1;32m 178\u001b[0m )\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/mistralai/basesdk.py:134\u001b[0m, in \u001b[0;36mBaseSDK.do_request..do\u001b[0;34m()\u001b[0m\n\u001b[1;32m 124\u001b[0m req \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39msdk_configuration\u001b[38;5;241m.\u001b[39mget_hooks()\u001b[38;5;241m.\u001b[39mbefore_request(\n\u001b[1;32m 125\u001b[0m BeforeRequestContext(hook_ctx), request\n\u001b[1;32m 126\u001b[0m )\n\u001b[1;32m 127\u001b[0m logger\u001b[38;5;241m.\u001b[39mdebug(\n\u001b[1;32m 128\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mRequest:\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124mMethod: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124mURL: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124mHeaders: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124mBody: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m\"\u001b[39m,\n\u001b[1;32m 129\u001b[0m req\u001b[38;5;241m.\u001b[39mmethod,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 132\u001b[0m get_body_content(req)\n\u001b[1;32m 133\u001b[0m )\n\u001b[0;32m--> 134\u001b[0m http_res \u001b[38;5;241m=\u001b[39m \u001b[43mclient\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43msend\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq\u001b[49m\u001b[43m,\u001b[49m\u001b[43m 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\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_build_request_auth(request, auth)\n\u001b[0;32m--> 926\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_send_handling_auth\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 927\u001b[0m \u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 928\u001b[0m \u001b[43m \u001b[49m\u001b[43mauth\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mauth\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 929\u001b[0m \u001b[43m \u001b[49m\u001b[43mfollow_redirects\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfollow_redirects\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 930\u001b[0m \u001b[43m \u001b[49m\u001b[43mhistory\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43m[\u001b[49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 931\u001b[0m \u001b[43m\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 932\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m 933\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m stream:\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpx/_client.py:954\u001b[0m, in \u001b[0;36mClient._send_handling_auth\u001b[0;34m(self, request, auth, follow_redirects, history)\u001b[0m\n\u001b[1;32m 951\u001b[0m request \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mnext\u001b[39m(auth_flow)\n\u001b[1;32m 953\u001b[0m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[0;32m--> 954\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_send_handling_redirects\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 955\u001b[0m \u001b[43m \u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 956\u001b[0m \u001b[43m \u001b[49m\u001b[43mfollow_redirects\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mfollow_redirects\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 957\u001b[0m \u001b[43m \u001b[49m\u001b[43mhistory\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mhistory\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 958\u001b[0m \u001b[43m \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 959\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m 960\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpx/_client.py:991\u001b[0m, in \u001b[0;36mClient._send_handling_redirects\u001b[0;34m(self, request, follow_redirects, history)\u001b[0m\n\u001b[1;32m 988\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m hook \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_event_hooks[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mrequest\u001b[39m\u001b[38;5;124m\"\u001b[39m]:\n\u001b[1;32m 989\u001b[0m hook(request)\n\u001b[0;32m--> 991\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_send_single_request\u001b[49m\u001b[43m(\u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 992\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m 993\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m hook \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_event_hooks[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mresponse\u001b[39m\u001b[38;5;124m\"\u001b[39m]:\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpx/_client.py:1027\u001b[0m, in \u001b[0;36mClient._send_single_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 1022\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mRuntimeError\u001b[39;00m(\n\u001b[1;32m 1023\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mAttempted to send an async request with a sync Client instance.\u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 1024\u001b[0m )\n\u001b[1;32m 1026\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m request_context(request\u001b[38;5;241m=\u001b[39mrequest):\n\u001b[0;32m-> 1027\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[43mtransport\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mhandle_request\u001b[49m\u001b[43m(\u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 1029\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(response\u001b[38;5;241m.\u001b[39mstream, SyncByteStream)\n\u001b[1;32m 1031\u001b[0m response\u001b[38;5;241m.\u001b[39mrequest \u001b[38;5;241m=\u001b[39m request\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpx/_transports/default.py:236\u001b[0m, in \u001b[0;36mHTTPTransport.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 223\u001b[0m req \u001b[38;5;241m=\u001b[39m httpcore\u001b[38;5;241m.\u001b[39mRequest(\n\u001b[1;32m 224\u001b[0m method\u001b[38;5;241m=\u001b[39mrequest\u001b[38;5;241m.\u001b[39mmethod,\n\u001b[1;32m 225\u001b[0m url\u001b[38;5;241m=\u001b[39mhttpcore\u001b[38;5;241m.\u001b[39mURL(\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 233\u001b[0m extensions\u001b[38;5;241m=\u001b[39mrequest\u001b[38;5;241m.\u001b[39mextensions,\n\u001b[1;32m 234\u001b[0m )\n\u001b[1;32m 235\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m map_httpcore_exceptions():\n\u001b[0;32m--> 236\u001b[0m resp \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_pool\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mhandle_request\u001b[49m\u001b[43m(\u001b[49m\u001b[43mreq\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 238\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(resp\u001b[38;5;241m.\u001b[39mstream, typing\u001b[38;5;241m.\u001b[39mIterable)\n\u001b[1;32m 240\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m Response(\n\u001b[1;32m 241\u001b[0m status_code\u001b[38;5;241m=\u001b[39mresp\u001b[38;5;241m.\u001b[39mstatus,\n\u001b[1;32m 242\u001b[0m headers\u001b[38;5;241m=\u001b[39mresp\u001b[38;5;241m.\u001b[39mheaders,\n\u001b[1;32m 243\u001b[0m stream\u001b[38;5;241m=\u001b[39mResponseStream(resp\u001b[38;5;241m.\u001b[39mstream),\n\u001b[1;32m 244\u001b[0m extensions\u001b[38;5;241m=\u001b[39mresp\u001b[38;5;241m.\u001b[39mextensions,\n\u001b[1;32m 245\u001b[0m )\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/connection_pool.py:216\u001b[0m, in \u001b[0;36mConnectionPool.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 213\u001b[0m closing \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_assign_requests_to_connections()\n\u001b[1;32m 215\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_close_connections(closing)\n\u001b[0;32m--> 216\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m exc \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n\u001b[1;32m 218\u001b[0m \u001b[38;5;66;03m# Return the response. Note that in this case we still have to manage\u001b[39;00m\n\u001b[1;32m 219\u001b[0m \u001b[38;5;66;03m# the point at which the response is closed.\u001b[39;00m\n\u001b[1;32m 220\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(response\u001b[38;5;241m.\u001b[39mstream, Iterable)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/connection_pool.py:196\u001b[0m, in \u001b[0;36mConnectionPool.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 192\u001b[0m connection \u001b[38;5;241m=\u001b[39m pool_request\u001b[38;5;241m.\u001b[39mwait_for_connection(timeout\u001b[38;5;241m=\u001b[39mtimeout)\n\u001b[1;32m 194\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[1;32m 195\u001b[0m \u001b[38;5;66;03m# Send the request on the assigned connection.\u001b[39;00m\n\u001b[0;32m--> 196\u001b[0m response \u001b[38;5;241m=\u001b[39m \u001b[43mconnection\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mhandle_request\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 197\u001b[0m \u001b[43m \u001b[49m\u001b[43mpool_request\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrequest\u001b[49m\n\u001b[1;32m 198\u001b[0m \u001b[43m \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 199\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m ConnectionNotAvailable:\n\u001b[1;32m 200\u001b[0m \u001b[38;5;66;03m# In some cases a connection may initially be available to\u001b[39;00m\n\u001b[1;32m 201\u001b[0m \u001b[38;5;66;03m# handle a request, but then become unavailable.\u001b[39;00m\n\u001b[1;32m 202\u001b[0m \u001b[38;5;66;03m#\u001b[39;00m\n\u001b[1;32m 203\u001b[0m \u001b[38;5;66;03m# In this case we clear the connection and try again.\u001b[39;00m\n\u001b[1;32m 204\u001b[0m pool_request\u001b[38;5;241m.\u001b[39mclear_connection()\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/connection.py:101\u001b[0m, in \u001b[0;36mHTTPConnection.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 98\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_connect_failed \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;01mTrue\u001b[39;00m\n\u001b[1;32m 99\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m exc\n\u001b[0;32m--> 101\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_connection\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mhandle_request\u001b[49m\u001b[43m(\u001b[49m\u001b[43mrequest\u001b[49m\u001b[43m)\u001b[49m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/http11.py:143\u001b[0m, in \u001b[0;36mHTTP11Connection.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 141\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m Trace(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mresponse_closed\u001b[39m\u001b[38;5;124m\"\u001b[39m, logger, request) \u001b[38;5;28;01mas\u001b[39;00m trace:\n\u001b[1;32m 142\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_response_closed()\n\u001b[0;32m--> 143\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m exc\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/http11.py:113\u001b[0m, in \u001b[0;36mHTTP11Connection.handle_request\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 102\u001b[0m \u001b[38;5;28;01mpass\u001b[39;00m\n\u001b[1;32m 104\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m Trace(\n\u001b[1;32m 105\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mreceive_response_headers\u001b[39m\u001b[38;5;124m\"\u001b[39m, logger, request, kwargs\n\u001b[1;32m 106\u001b[0m ) \u001b[38;5;28;01mas\u001b[39;00m trace:\n\u001b[1;32m 107\u001b[0m (\n\u001b[1;32m 108\u001b[0m http_version,\n\u001b[1;32m 109\u001b[0m status,\n\u001b[1;32m 110\u001b[0m reason_phrase,\n\u001b[1;32m 111\u001b[0m headers,\n\u001b[1;32m 112\u001b[0m trailing_data,\n\u001b[0;32m--> 113\u001b[0m ) \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_receive_response_headers\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 114\u001b[0m trace\u001b[38;5;241m.\u001b[39mreturn_value \u001b[38;5;241m=\u001b[39m (\n\u001b[1;32m 115\u001b[0m http_version,\n\u001b[1;32m 116\u001b[0m status,\n\u001b[1;32m 117\u001b[0m reason_phrase,\n\u001b[1;32m 118\u001b[0m headers,\n\u001b[1;32m 119\u001b[0m )\n\u001b[1;32m 121\u001b[0m network_stream \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_network_stream\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/http11.py:186\u001b[0m, in \u001b[0;36mHTTP11Connection._receive_response_headers\u001b[0;34m(self, request)\u001b[0m\n\u001b[1;32m 183\u001b[0m timeout \u001b[38;5;241m=\u001b[39m timeouts\u001b[38;5;241m.\u001b[39mget(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mread\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m)\n\u001b[1;32m 185\u001b[0m \u001b[38;5;28;01mwhile\u001b[39;00m \u001b[38;5;28;01mTrue\u001b[39;00m:\n\u001b[0;32m--> 186\u001b[0m event \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_receive_event\u001b[49m\u001b[43m(\u001b[49m\u001b[43mtimeout\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtimeout\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 187\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(event, h11\u001b[38;5;241m.\u001b[39mResponse):\n\u001b[1;32m 188\u001b[0m \u001b[38;5;28;01mbreak\u001b[39;00m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_sync/http11.py:224\u001b[0m, in \u001b[0;36mHTTP11Connection._receive_event\u001b[0;34m(self, timeout)\u001b[0m\n\u001b[1;32m 221\u001b[0m event \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_h11_state\u001b[38;5;241m.\u001b[39mnext_event()\n\u001b[1;32m 223\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m event \u001b[38;5;129;01mis\u001b[39;00m h11\u001b[38;5;241m.\u001b[39mNEED_DATA:\n\u001b[0;32m--> 224\u001b[0m data \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_network_stream\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mread\u001b[49m\u001b[43m(\u001b[49m\n\u001b[1;32m 225\u001b[0m \u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mREAD_NUM_BYTES\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtimeout\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mtimeout\u001b[49m\n\u001b[1;32m 226\u001b[0m \u001b[43m \u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 228\u001b[0m \u001b[38;5;66;03m# If we feed this case through h11 we'll raise an exception like:\u001b[39;00m\n\u001b[1;32m 229\u001b[0m \u001b[38;5;66;03m#\u001b[39;00m\n\u001b[1;32m 230\u001b[0m \u001b[38;5;66;03m# httpcore.RemoteProtocolError: can't handle event type\u001b[39;00m\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 234\u001b[0m \u001b[38;5;66;03m# perspective. Instead we handle this case distinctly and treat\u001b[39;00m\n\u001b[1;32m 235\u001b[0m \u001b[38;5;66;03m# it as a ConnectError.\u001b[39;00m\n\u001b[1;32m 236\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m data \u001b[38;5;241m==\u001b[39m \u001b[38;5;124mb\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m\"\u001b[39m \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_h11_state\u001b[38;5;241m.\u001b[39mtheir_state \u001b[38;5;241m==\u001b[39m h11\u001b[38;5;241m.\u001b[39mSEND_RESPONSE:\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/site-packages/httpcore/_backends/sync.py:126\u001b[0m, in \u001b[0;36mSyncStream.read\u001b[0;34m(self, max_bytes, timeout)\u001b[0m\n\u001b[1;32m 124\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m map_exceptions(exc_map):\n\u001b[1;32m 125\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_sock\u001b[38;5;241m.\u001b[39msettimeout(timeout)\n\u001b[0;32m--> 126\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_sock\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrecv\u001b[49m\u001b[43m(\u001b[49m\u001b[43mmax_bytes\u001b[49m\u001b[43m)\u001b[49m\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/ssl.py:1295\u001b[0m, in \u001b[0;36mSSLSocket.recv\u001b[0;34m(self, buflen, flags)\u001b[0m\n\u001b[1;32m 1291\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m flags \u001b[38;5;241m!=\u001b[39m \u001b[38;5;241m0\u001b[39m:\n\u001b[1;32m 1292\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\n\u001b[1;32m 1293\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mnon-zero flags not allowed in calls to recv() on \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m\"\u001b[39m \u001b[38;5;241m%\u001b[39m\n\u001b[1;32m 1294\u001b[0m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__class__\u001b[39m)\n\u001b[0;32m-> 1295\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mread\u001b[49m\u001b[43m(\u001b[49m\u001b[43mbuflen\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 1296\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 1297\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28msuper\u001b[39m()\u001b[38;5;241m.\u001b[39mrecv(buflen, flags)\n", + "File \u001b[0;32m~/miniforge3/envs/weave/lib/python3.11/ssl.py:1168\u001b[0m, in \u001b[0;36mSSLSocket.read\u001b[0;34m(self, len, buffer)\u001b[0m\n\u001b[1;32m 1166\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_sslobj\u001b[38;5;241m.\u001b[39mread(\u001b[38;5;28mlen\u001b[39m, buffer)\n\u001b[1;32m 1167\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[0;32m-> 1168\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_sslobj\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mread\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mlen\u001b[39;49m\u001b[43m)\u001b[49m\n\u001b[1;32m 1169\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m SSLError \u001b[38;5;28;01mas\u001b[39;00m x:\n\u001b[1;32m 1170\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m x\u001b[38;5;241m.\u001b[39margs[\u001b[38;5;241m0\u001b[39m] \u001b[38;5;241m==\u001b[39m SSL_ERROR_EOF \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39msuppress_ragged_eofs:\n", + "\u001b[0;31mKeyboardInterrupt\u001b[0m: " ] } ], "source": [ - "mipro_optimized_program = optimize_with_mipro(simple_program, lm, lm, validate_solution, trainset)" + "def optimize_with_mipro(program, prompt_model, task_model, metric, trainset):\n", + " teleprompter = MIPRO(\n", + " prompt_model=prompt_model,\n", + " task_model=task_model,\n", + " metric=metric,\n", + " num_candidates=2,\n", + " init_temperature=0.5,\n", + " verbose=False,\n", + " )\n", + "\n", + " optimized_program = teleprompter.compile(\n", + " program.deepcopy(),\n", + " num_trials=10,\n", + " trainset=trainset,\n", + " eval_kwargs=dict(num_threads=8),\n", + " max_bootstrapped_demos=0, # 0-shot optimization\n", + " max_labeled_demos=0,\n", + " seed=9\n", + " )\n", + " now = datetime.now()\n", + " date_time = now.strftime(\"%Y%m%d_%H%M%S\")\n", + "\n", + " optimized_program.save(f\"mipro_optimized_{date_time}\")\n", + "\n", + " return optimized_program\n", + "\n", + "\n", + "mipro_optimized_program = optimize_with_mipro(simple_program, mistral, mistral, f1, trainset[0:5])" ] }, { @@ -9216,12 +1897,25 @@ "evaluate(program=mipro_optimized_program, devset=devset)" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using the optimized program\n", + "\n", + "We have developed a program that can solve the problem by generating a solution (not the code, but the logic).\n", + "\n", + "Now we can use this solution to create the code for each sample problem." + ] + }, { "cell_type": "code", - "execution_count": null, + "execution_count": 34, "metadata": {}, "outputs": [], - "source": [] + "source": [ + "# TODO" + ] } ], "metadata": {