BALKAN 2023 Problem 2

In triangle $A B C$, the incircle touches sides $B C, C A, A B$ at $D, E, F$ respectively. Assume there exists a point $X$ on the line $E F$ such that

$$\angle X B C=\angle X C B=45^{\circ} .$$

Let $M$ be the midpoint of the arc $B C$ on the circumcircle of $A B C$ not containing $A$. Prove that the line $M D$ passes through $E$ or $F$.