TSTST 2024 Problem 8

Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD = \angle DAC$ . Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$ . Prove that lines $DS$ and $AT$ are parallel.