2024 AIME II Problems/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 outside of S, it is externally tangent to S along a circle with radius , and when T rests on the outside of S, it is externally tangent to S along a circle with radius . The difference can be written as , where and are relatively prime positive integers. Find m+n.