PMO 2021 Problem 1

In convex quadrilateral $ABCD$ , $\angle CAB = \angle BCD$ . $P$ lies on line $BC$ such that $AP = PC$ , $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$ , and $S$ is the point of intersection of lines $AC$ and $QR$ . Line $AD$ meets the circumcircle of $AQS$ again at $T$ . Prove that $AB$ and $QT$ are parallel.