HMMT November 2013 T5

In triangle $A B C, \angle B A C=60^{\circ}$. Let $\omega$ be a circle tangent to segment $A B$ at point $D$ and segment $A C$ at point $E$. Suppose $\omega$ intersects segment $B C$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$. Given that $A D=F G=4$ and $B F=\frac{1}{2}$, find the length of $C G$.