Mongolian Mathematical Olympiad

Let $\omega$ be the circumcircle of $ABC$ where $AB\ne AC$, and let $M$ be the midpoint of side $BC$. Tangent lines drawn at points $B$ and $C$ of circle $\omega$ intersect at point $T$. The circumcircle of triangle $AMT$ intersects line $BC$ again at point $N$. Let $S$ be the midpoint of $NT$. Prove that $SA$ is tangent to the circle $\omega$. (Gerelkhuu Erdenetugs)