HMMT 2017 G5

Let $A B C D$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $A C$ and $B D$. Let $R_{1}, R_{2}, R_{3}, R_{4}$ be the circumradii of triangles $A P B, B P C, C P D, D P A$ respectively. If $R_{1}=31$ and $R_{2}=24$ and $R_{3}=12$, find $R_{4}$.