IGO 2021 I4

Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$ . Line $AI$ intersects $\Gamma$ for the second time at $M$ . Let $N$ be the midpoint of $BC$ and $T$ be the point on $\Gamma$ such that $IN \perp MT$ . Finally, let $P$ and $Q$ be the intersection points of $TB$ and $TC$ , respectively, with the line perpendicular to $AI$ at $I$ . Show that $PB = CQ$ . Proposed by Patrik Bak - Slovakia