SHARYGIN 2021 Problem 4

Let $ABCD$ be a square with center $O$ , and $P$ be a point on the minor arc $CD$ of its circumcircle. The tangents from $P$ to the incircle of the square meet $CD$ at points $M$ and $N$ . The lines $PM$ and $PN$ meet segments $BC$ and $AD$ respectively at points $Q$ and $R$ . Prove that the median of triangle $OMN$ from $O$ is perpendicular to the segment $QR$ and equals to its half.