CGMO 2017 Problem 2

Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$ Let $E$ be the intersection of $BD$ and the internal bisector of $\angle BAD$ . The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively. Prove that $A,C,X,Y$ are concyclic.