Length in a Perpendicular Bisector Configuration

Point $D$ is the foot of the altitude from $A$ of an acute triangle $ABC$ to side $BC$. The perpendicular bisector of $BC$ meets lines $AC$ and $AB$ at $E$ and $P$, respectively. The line through $E$ parallel to $BC$ meets line $DP$ at $X$, and lines $AX$ and $BE$ meet at $Q$. Given that $AX=14$ and $XQ=6$, find $AP$.