2024 AIME II Problem 8

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 inside of S, it is internally 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 \frac{m}{n}, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.