Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.