BALKAN-SL 2017 G7

Let $A B C$ be an acute triangle with $A B \neq A C$ and circumcirle $\Gamma$. The angle bisector of $B A C$ intersects $B C$ and $\Gamma$ at $D$ and $E$ respectively. Circle with diameter $D E$ intersects $\Gamma$ again at $F \neq E$. Point $P$ is on $A F$ such that $P B=P C$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $A B$ and $A C$ respectively. Let $H$ and $H^{\prime}$ be the ortocenters of $A B C$ and $A X Y$ respectively. $A H$ meets $\Gamma$ again at $Q$. If $A H^{\prime}$ and $H H^{\prime}$ intersect the circle with diameter $A H$ again at points $S$ and $T$, respectively, prove that the lines $A T, H S$ and $F Q$ are concurrent.