IR-TST 2023 Problem 2

$ABCD$ is cyclic quadrilateral and $O$ is the center of its circumcircle. Suppose that $AD \cap BC = E$ and $AC \cap BD = F$. Circle $\omega$ is tanget to line $AC$ and $BD$. $PQ$ is a diameter of $\omega$ that $F$ is orthocenter of $EPQ$. Prove that line $OE$ is passing through center of $\omega$

Proposed by Mahdi Etesami Fard