BG-MO 2022 Problem 2

Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$ . A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$ . The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$ . The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$ . The segments $BP$ and $CQ$ intersect at point $T$ . Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$ .