HMMT 2013 G8

Let $A B C D$ be a convex quadrilateral. Extend line $C D$ past $D$ to meet line $A B$ at $P$ and extend line $C B$ past $B$ to meet line $A D$ at $Q$. Suppose that line $A C$ bisects $\angle B A D$. If $A D=\frac{7}{4}, A P=\frac{21}{2}$, and $A B=\frac{14}{11}$, compute $A Q$.