HMMT 2013 G10

Triangle $A B C$ is inscribed in a circle $\omega$. Let the bisector of angle $A$ meet $\omega$ at $D$ and $B C$ at $E$. Let the reflections of $A$ across $D$ and $C$ be $D^{\prime}$ and $C^{\prime}$, respectively. Suppose that $\angle A=60^{\circ}, A B=3$, and $A E=4$. If the tangent to $\omega$ at $A$ meets line $B C$ at $P$, and the circumcircle of $A P D^{\prime}$ meets line $B C$ at $F$ (other than $P$ ), compute $F C^{\prime}$.