In a triangle $ABC$ with circumcircle $\Gamma$, $M$ is the midpoint of $BC$ and point $D$ lies on segment $MC$. Point $G$ lies on ray $\overrightarrow{BC}$ past $C$ such that $\frac{BC}{DC} = \frac{BG}{GC}$, and let $N$ be the midpoint of $DG$. The points $P$, $S$, and $T$ are defined as follows: Line $CA$ meets the circumcircle $\Gamma_1$ of triangle $AGD$ again at point $P$. Line $PM$ meets $\Gamma_1$ again at $S$. Line $PN$ meets the line through $A$ that is parallel to $BC$ at $Q$. Line $CQ$ meets $\Gamma$ again at $T$. Prove that the points $P$, $S$, $T$, and $C$ are concyclic.