{"task": {"agent_timeout": 600, "task": "omnimath_1595", "verifier_timeout": 60, "instruction": "# Mathematical Problem\n\nLet $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \\times 3$ grid, and then removing the center copy. Let $a_{n}$ be the expected value of $\\left|x-x^{\\prime}\\right|+\\left|y-y^{\\prime}\\right|$, where $(x, y)$ and $\\left(x^{\\prime}, y^{\\prime}\\right)$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that $$\\lim _{n \\rightarrow \\infty} \\frac{a_{n}}{3^{n}}=\\frac{a}{b}$$ Compute $100 a+b$.\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": []}