CGMO 2009 Problem 6

Circle $\Gamma_{1},$ with radius $r,$ is internally tangent to circle $\Gamma_{2}$ at $S.$ Chord $AB$ of $\Gamma_{2}$ is tangent to $\Gamma_{1}$ at $C.$ Let $M$ be the midpoint of arc $AB$ (not containing $S$ ), and let $N$ be the foot of the perpendicular from $M$ to line $AB.$ Prove that $AC\cdot CB=2r\cdot MN.$