CZ-SK-MO 2017 Problem 5

Given is the acute triangle $ABC$ with the intersection of altitudes $H$ . The angle bisector of angle $BHC$ intersects side $BC$ at point $D$ . Mark $E$ and $F$ the symmetrics of the point $D$ wrt lines $AB$ and $AC$ . Prove that the circle circumscribed around the triangle $AEF$ passes through the midpoint of the arc $BAC$