ISL 2011 G1

Let $A B C$ be an acute triangle. Let $\omega$ be a circle whose center $L$ lies on the side $B C$. Suppose that $\omega$ is tangent to $A B$ at $B^{\prime}$ and to $A C$ at $C^{\prime}$. Suppose also that the circumcenter $O$ of the triangle $A B C$ lies on the shorter $\operatorname{arc} B^{\prime} C^{\prime}$ of $\omega$. Prove that the circumcircle of $A B C$ and $\omega$ meet at two points.