TSTST 2024 Problem 4

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$ . Let $\omega_{1}$ be the circumcircle of $ADE$ and $\omega_{2}$ be the circumcircle of $BCE$ . The tangent to $\omega_{1}$ at $A$ and the tangent to $\omega_{2}$ at $C$ meet at $P$ . The tangent to $\omega_{1}$ at $D$ and the tangent to $\omega_{2}$ at $B$ meet at $Q$ . Show that $OP = OQ$ .