Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line passing through point $B$ intersects $\omega_1$ for the second time at point $C$ and $\omega_2$ at point $D$. The line $AC$ intersects circle $\omega_2$ for the second time at point $F$, and the line $AD$ intersects the circle $\omega_1$ for the second time at point $E$ . Let point $O$ be the center of the circle circumscribed around $\vartriangle AEF$. Prove that $OB \perp CD$.