HMMT 2021 U32

Let acute triangle $A B C$ have circumcenter $O$, and let $M$ be the midpoint of $B C$. Let $P$ be the unique point such that $\angle B A P=\angle C A M, \angle C A P=\angle B A M$, and $\angle A P O=90^{\circ}$. If $A O=53, O M=28$, and $A M=75$, compute the perimeter of $\triangle B P C$.