OMCC 2021 Problem 2

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$ . Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$ , and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$ . Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$ . Show that $E,\ G,\ F$ and $C$ lie on the same circle.