SHARYGIN 2022 Problem 95

Chords $AB$ and $CD$ of a circle $\omega$ meet at point $E$ in such a way that $AD = AE = EB$ . Let $F$ be a point of segment $CE$ such that $ED = CF$ . The bisector of angle $AFC$ meets an arc $DAC$ at point $P$ . Prove that $A$ , $E$ , $F$ , and $P$ are concyclic.