ELMO SL 2013 G5

Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$ . Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$ . The two circles tangent to $\omega_2$ , passing through $O$ and $A$ , touch $\omega_2$ at $F$ and $G$ . Prove that $FGOB$ is cyclic.

Proposed by Eric Chen