HMMT 2023 T7

Let $A B C$ be a triangle. Point $D$ lies on segment $B C$ such that $\angle B A D=\angle D A C$. Point $X$ lies on the opposite side of line $B C$ as $A$ and satisfies $X B=X D$ and $\angle B X D=\angle A C B$. Analogously, point $Y$ lies on the opposite side of line $B C$ as $A$ and satisfies $Y C=Y D$ and $\angle C Y D=\angle A B C$. Prove that lines $X Y$ and $A D$ are perpendicular.