OIM 2019 Problem 4

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$ . Let $P$ and $Q$ be two points on segment $AB$ ( $A$ , $P$ , $Q$ , $B$ appear in that order and are distinct) such that $AP=QB$ . Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$ , respectively. Lines $AB$ and $EF$ intersect at $G$ . Prove that line $DG$ is tangent to $\Gamma$ .