Triangle $PQR$ has side lengths $PQ=10$, $QR=16$, and $RP=2\sqrt{129}$. The bisector of $\angle PQR$ intersects $\overline{PR}$ in point $S$, and intersects the circumcircle of $△PQR$ in point $T\ne Q$. The circumcircle of $△QST$ intersects the line $QR$ in points $Q$ and $U\ne Q$. What is $PU$?