SHARYGIN 2021 Problem 13

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$ , point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$ . The excircle opposite to $A$ touches $\overline{BC}$ at point $E$ . Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$ . Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$ .