SHARYGIN 2015 Problem 21

A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$ . Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$ . Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$ with the circumcircles of triangles $CDM_1$ and $ABM_2$ . Prove that $X_1X_2 // Y_1Y_2$ .