AIME 2024 / AIME 2024 train 179961cd-80dd-5415-8772-a133e05732d5

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 fracmn, where m and n are relatively prime positive integers. Find m+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 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 $m+n$.