Middle European Mathematical Olympiad 2023 Problem T-6

Let $ABC$ be an acute triangle with $AB < AC$. Let $J$ be the center of the $A$-excircle of $ABC$. Let $D$ be the projection of $J$ on line $BC$. The internal bisectors of angles $BDJ$ and $JDC$ intersect lines $BJ$ and $JC$ at $X$ and $Y$, respectively. Segments $XY$ and $JD$ intersect at $P$. Let $Q$ be the projection of $A$ on line $BC$. Prove that the internal angle bisector of $\measuredangle QAP$ is perpendicular to line $XY$.

Remark. The $A$-excircle of the triangle $ABC$ is the circle outside the triangle which is tangent to the lines $AB$, $AC$, and the line segment $BC$.