Middle European Mathematical Olympiad 2023 Problem I-3

Let $ABC$ be a triangle with incenter $I$. The incircle $\omega$ of $ABC$ is tangent to the line $BC$ at point $D$. Denote by $E$ and $F$ the points satisfying $AI \parallel BE \parallel CF$ and $\angle BEI = \angle CFI = 90°$. Lines $DE$ and $DF$ intersect $\omega$ again at points $E'$ and $F'$, respectively. Prove that $E'F' \perp AI$.