Circles on diameters CX and BY
Problem
Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$, whose incircle touches $AC$ and $AB$ at $E$ and $F$. The circumcircle of triangle $AEF$ meets $\Gamma$ again at $S \neq A$. The reflection of the line $SI$ in the line $EF$ meets the lines $AC$ and $AB$ at $P$ and $Q$. Let $X$ and $Y$ be the points on the line $EF$ with $PX \perp AC$ and $QY \perp AB$. Prove that the circles with diameters $CX$ and $BY$ have a common point on the line $PQ$ and a common point on $\Gamma$.
The diagram
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