Infinite Tangent Circles

Let $A, B, C$ be points on the same plane with $\angle ACB=120^{\circ}$ . There is a sequence of circles $\omega_0, \omega_1, \omega_2, ...$ on the same plane (with corresponding radii $r_0, r_1, r_2, ...$ , where $r_0>r_1>r_2>...$ ) such that each circle is tangent to both segments $CA$ and $CB$ . Furthermore, $\omega_i$ is tangent to $\omega_{i-1}$ for all $i\ge1$ . If $r_0=3$ , find the value of $r_0+r_1+r_2+...$