KöMaL problem A, 2024

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\triangle ABC$ touches side $B C$. The circumcircle of $\triangle B D E$ meets line $A B$ again at point $K$, the circumcircle of $\triangle C D F$ meets line $A C$ again at point $L$. The circumcircle of $\triangle B D E$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\triangle ABC$.