2021 AIME II Problem 10

Two spheres with radii 36 and one sphere with radius 13 are each externally tangent to the other two spheres and to two different planes \mathcal{P} and \mathcal{Q}. The intersection of planes \mathcal{P} and \mathcal{Q} is the line \ell. The distance from line \ell to the point where the sphere with radius 13 is tangent to plane \mathcal{P} is \tfrac{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.