OSN 2019 Problem 3

Given that $ABCD$ is a rectangle such that $AD > AB$ , where $E$ is on $AD$ such that $BE \perp AC$ . Let $M$ be the intersection of $AC$ and $BE$ . Let the circumcircle of $\triangle ABE$ intersects $AC$ and $BC$ at $N$ and $F$ . Moreover, let the circumcircle of $\triangle DNE$ intersects $CD$ at $G$ . Suppose $FG$ intersects $AB$ at $P$ . Prove that $PM = PN$ .