HMMT November 2024 H10

Isabella the geologist discovers a diamond deep underground via an X-ray machine. The diamond has the shape of a convex cyclic pentagon $P A B C D$ with $A D \| B C$. Soon after the discovery, her X-ray breaks, and she only recovers partial information about its dimensions. She knows that $A D=70$, $B C=55, P A: P D=3: 4$, and $P B: P C=5: 6$. Compute $P B$.