Iranian Mathematical Olympiad

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be midpoints of the sides $AD$ and $BC$, respectively. Suppose that ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ are circumcircles of triangles $ADP$, $BCP$ and $OMN$, respectively. Let $E$ and $F$ be intersection points of ${\omega }_1$ and ${\omega }_3$, which is not on the arc $APD$ of ${\omega }_1$, and the intersection point of ${\omega }_2$ and ${\omega }_3$, which is not on the arc $BPC$ of ${\omega }_2$, respectively. Prove that $OE=OF$.