PL-OM 2022 Problem 1

Let $ABC$ be an acute triangle with $AB<AC$ . The angle bisector of $BAC$ intersects the side $BC$ and the circumcircle of $ABC$ at $D$ and $M\neq A$ , respectively. Points $X$ and $Y$ are chosen so that $MX \perp AB$ , $BX \perp MB$ , $MY \perp AC$ , and $CY \perp MC$ . Prove that the points $X,D,Y$ are collinear.