HMMT November 2022 U28

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Pick points $Q$ and $R$ on $A C$ and $A B$ such that $\angle C B Q=\angle B C R=90^{\circ}$. There exist two points $P_{1} \neq P_{2}$ in the plane of $A B C$ such that $\triangle P_{1} Q R, \triangle P_{2} Q R$, and $\triangle A B C$ are similar (with vertices in order). Compute the sum of the distances from $P_{1}$ to $B C$ and $P_{2}$ to $B C$.