HMMT 2005 G10

Let $A B$ be the diameter of a semicircle $\Gamma$. Two circles, $\omega_{1}$ and $\omega_{2}$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $A B$ at $P$ and $Q$, respectively, and to semicircular arc $A B$ at $C$ and $D$, respectively, with $A P<A Q$. Suppose $F$ lies on $\Gamma$ such that $\angle F Q B=\angle C Q A$ and that $\angle A B F=80^{\circ}$. Find $\angle P D Q$ in degrees.