Question 36
In triangle $ABC$, $\angle BAC = 60^\circ$. Let $\omega$ be a circle tangent to segment $AB$ at point $D$ and segment $AC$ at point $E$. Suppose $\omega$ intersects segment $BC$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$. Given that $AD = FG = 4$ and $BF = \frac{1}{2}$, find the length of $CG$.
