DE-MO 2022 Problem 3

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$ , respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$ .

Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$ . Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$ .