{"task": {"agent_timeout": 3000, "task": "aime_88", "verifier_timeout": 3000, "instruction": "  Solve the following problem. Reason step by step, create the file `/app/answer.txt`, and put your final answer there.\n  Your answer should be a single integer from 1 to 999 inclusive.\n\n    Torus $T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane of the circle that is a distance $6$ from the center of the circle (so like a donut). Let $S$ be a sphere with a radius $11$. When $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radius $r_i$, and when $T$ rests on the outside of $S$, it is externally tangent to $S$ along a circle with radius $r_o$. The difference $r_i-r_o$ can be written as $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n  [asy] unitsize(0.3 inch); draw(ellipse((0,0), 3, 1.75)); draw((-1.2,0.1)..(-0.8,-0.03)..(-0.4,-0.11)..(0,-0.15)..(0.4,-0.11)..(0.8,-0.03)..(1.2,0.1)); draw((-1,0.04)..(-0.5,0.12)..(0,0.16)..(0.5,0.12)..(1,0.04)); draw((0,2.4)--(0,-0.15)); draw((0,-0.15)--(0,-1.75), dashed); draw((0,-1.75)--(0,-2.25)); draw(ellipse((2,0), 1, 0.9)); draw((2.03,-0.02)--(2.9,-0.4)); [/asy]\n", "memory": "", "runnable": false, "difficulty": "unknown", "language": "", "cpus": "", "instruction_truncated": false, "category": "reasoning", "compose": false, "has_solution": true, "oracle": null, "docker_image": "", "taskset": "aime", "tags": ["math", "AIME"]}, "runs": []}