TUYMAADA 2018 S3

A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$ . The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$ , respectively. The lines $PS$ and $QR$ meet at $L$ . Prove that the line $CL$ bisects the segment $PQ$ .

Proposed by A. Antropov