BALTIC-WAY 2025 Problem 13

In a cyclic quadrilateral $ABCD$, the lines $AB$ and $CD$ intersect at $E$. A point $P$ lies inside $ABCD$ and satisfies

$$\angle BAP = \angle PCB \quad \text{and} \quad \angle CBP = \angle PDC.$$

Prove that $PE \perp BC$.