HMMT November 2011 U33

Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C$. $\Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$.