USAMO 2016 Problem 5

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$ , $Q\in\overline{AC}$ , and $N,P\in\overline{BC}$ . Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$ . Denote by $\ell$ the angle bisector of $\angle MSQ$ .

Prove that $\overline{OI}$ is parallel to $\ell$ , where $O$ is the circumcenter of triangle $ABC$ , and $I$ is the incenter of triangle $ABC$ .