AIME 2026a Problem 3

A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200$, and the hemisphere and disk have the same center. Let $\mathcal T$ be the region of points $P$ in the disk such that a sphere of radius 42 can be placed on top of the disk at $P$ and lie completely inside the hemisphere. The area of $\mathcal T$ divided by the area of the disk is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.