FKMO 2022 Problem 4

Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$ . The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$ , respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$ . Denote by $\Omega$ a circle centered at $O$ and passing through $L$ , and let $AL$ meet $\Omega$ again at $N$ .

Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$ .