{"task": {"agent_timeout": 600, "task": "omnimath_7", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nA table tennis club hosts a series of doubles matches following several rules:\n(i)  each player belongs to two pairs at most;\n(ii) every two distinct pairs play one game against each other at most;\n(iii) players in the same pair do not play against each other when they pair with others respectively.\nEvery player plays a certain number of games in this series. All these distinct numbers make up a set called the \u201c[i]set of games[/i]\u201d. Consider a set $A=\\{a_1,a_2,\\ldots ,a_k\\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding [i]set of games [/i] is equal to set $A$ exists.\n\n## Instructions\n\nSolve the mathematical problem above and write your final answer to `/workspace/answer.txt`.\n\n**Important**: Write only your final answer to the file, not the full solution process.\n\n### Guidelines\n\n- Provide your final numerical answer or mathematical expression\n- Write the answer as plain text (no special formatting needed)\n- Be precise and clear in your answer\n- The answer should directly respond to what the problem asks for\n\n### Example\n\nIf the problem asks \"What is 2 + 2?\", your answer file should contain:\n\n```\n4\n```\n\nYour answer will be evaluated against the correct solution using an automated grading system.\n", "memory": "", "runnable": false, "difficulty": "hard", "language": "", "cpus": "", "instruction_truncated": false, "category": "math", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "omnimath", "tags": ["omni-math"]}, "runs": []}