CZ-SK-MO 2023 Problem 3

In acute triangle $ABC$ let $H$ be its orthocenter and $I$ be its incenter. Let $D$ be the projection of point $I$ onto the line $BC$ and $E$ be the reflection of point $A$ in point $I$ . Further, let $F$ be the projection of point $H$ onto the line $ED$ . Prove that points $B, H, F$ and $C$ lie on circle.