China Girls' Mathematical Olympiad

Let $ABCD$ be a convex quadrilateral. Let $O$ be the intersection of $AC$ and $BD$. Let $O$ and $M$ be the intersections of the circumcircle of $△OAD$ with the circumcircle of $△OBC$. Let $T$ and $S$ be the intersections of $OM$ with the circumcircle of $△OAB$ and $△OCD$ respectively. Prove that $M$ is the midpoint of $TS$.