TUYMAADA 2021 S7

An acute triangle $ABC$ is given, $AC \ne BC$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $AHB$ meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If $PX = QY$, prove that $AP + BQ \geqslant 2CH$.