Two smaller circles $\omega_1$ and $\omega_2$ are drawn internally tangent to a circle $\omega$ of radius $2012$, with distinct points of tangency $A$ and $B$. If $\omega_1$ and $\omega_2$ meet in two distinct points, and the line $AB$ passes through one of these points, find the sum of the radii of $\omega_1$ and $\omega_2$.
Now, prove the implication that the sum of the radii is invariant for all possible $\omega_1$ and $\omega_2$.