HMMT 2013 T3

Let $A B C$ be a triangle with circumcenter $O$ such that $A C=7$. Suppose that the circumcircle of $A O C$ is tangent to $B C$ at $C$ and intersects the line $A B$ at $A$ and $F$. Let $F O$ intersect $B C$ at $E$. Compute BE.