MMO2025 Round 4

Let $I$ be the incenter of triangle $ABC$, which is inscribed in the circle $\omega$ centered at $O$. Suppose the line $BI$ intersects $\omega$ again at point $M$. Let $I_B$ be the reflection of $I$ over the line $AC$. Suppose that the line $M{I}_B$ intersects $\omega$ again at point $D\ne M$, and the line $DO$ intersects $\omega$ again at point $E$. Prove that the lines $OI$ and $BE$ are parallel. (Batzorig Undrakh)