OSN 2021 Problem 7

Given $\triangle ABC$ with circumcircle $\ell$ . Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$ . Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$ . Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$ . Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$ .