HMMT November 2013 T7

In equilateral triangle $A B C$, a circle $\omega$ is drawn such that it is tangent to all three sides of the triangle. A line is drawn from $A$ to point $D$ on segment $B C$ such that $A D$ intersects $\omega$ at points $E$ and $F$. If $E F=4$ and $A B=8$, determine $|A E-F D|$.