BALTIC-WAY 2019 Problem 14

Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$ , and let $H$ be the foot of the altitude from $B$ . The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$ , respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$ , respectively. The segments $AQ$ and $CP$ intersect at the point $R$ . Prove that the line $BR$ passes through the midpoint of the segment $MN$ .