TSTST 2018 Problem 3

Let $A B C$ be an acute triangle with incenter $I$, circumcenter $O$, and circumcircle $\Gamma$. Let $M$ be the midpoint of $\overline{A B}$. Ray $A I$ meets $\overline{B C}$ at $D$. Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle B I C$ and $\triangle B A D$, respectively. Line $M O$ meets $\omega$ at $X$ and $Y$, while line $C O$ meets $\omega$ at $C$ and $Q$. Assume that $Q$ lies inside $\triangle A B C$ and $\angle A Q M=\angle A C B$. Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$. Given that $\angle B A C \neq 60^{\circ}$, prove that these four lines are concurrent on $\Gamma$.