Olympiad Geometry · Medium problem

Lines $\ell$ and $m$ are parallel. Two transversals through point $P$ (between the lines) form $\triangle PAB$ (with $A$ on $\ell$) and $\triangle PCD$ (with $C$, $D$ on $m$), where $A,P,C$ are collinear and $B,P,D$ are collinear. $PA = 2x + 1$, $PC = 12$, $PB = 9$, and $PD = 12$. Find $x$.