Hong Kong Preliminary Selection Contest

Let $A$, $B$, $C$ be points on the same plane with $\angle ACB=120^{\circ }$. There is a sequence of circles ${\omega }_0,{\omega }_1,{\omega }_2,\dots$ on the same plane (with corresponding radii $r_0,r_1,r_2,\dots$, where $r_0>r_1>r_2>\dots$) such that each circle is tangent to both segments $CA$ and $CB$. Furthermore, ${\omega }_i$ is tangent to ${\omega }_{i-1}$ for all $i\ge 1$. If $r_0=3$, find the value of $r_0+r_1+r_2+\dots$.