ELMO 2021 Problem 1

In $\triangle ABC$ , points $P$ and $Q$ lie on sides $AB$ and $AC$ , respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$ . Let $E$ lie on side $BC$ such that $BD = EC$ . Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$ , and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$ . Prove that $D$ , $E$ , $X$ , and $Y$ are concyclic.