Question 49
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 $r_0 \gt r_1 \gt r_2 \gt \dots$) 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 $r_0 + r_1 + r_2 + \dots$.
