OMCC 2021 Problem 6

Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$ . A point $P$ (other than $B$ ) is chosen on the segment $BC$ in such a way that $AB=AP$ . Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$ , and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$ . Let $K$ be the intersection of lines $AP$ and $DE$ . Let $F$ be a point on $BC$ (other than $P$ ) such that $KP=KF$ . Show that $C,\ D,\ E$ and $F$ lie on the same circle.