{"task": {"agent_timeout": 600, "task": "omnimath_1136", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nLet $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50 \\%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and $2^{n}$ face each other in the last round? Express your answer in terms of $n$.\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": []}