
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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

Present the answer in LaTex format: \boxed{Your answer}
user