
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
Points  $A$ ,  $B$ , and  $O$  lie in the plane such that  $\measuredangle AOB = 120^\circ$ . Circle  $\omega_0$  with radius  $6$  is constructed tangent to both  $\overrightarrow{OA}$  and  $\overrightarrow{OB}$ . For all  $i \ge 1$ , circle  $\omega_i$  with radius  $r_i$  is constructed such that  $r_i < r_{i - 1}$  and  $\omega_i$  is tangent to  $\overrightarrow{OA}$ ,  $\overrightarrow{OB}$ , and  $\omega_{i - 1}$ . If 
		\[
		S = \sum_{i = 1}^\infty r_i,
		\]
		then  $S$  can be expressed as  $a\sqrt{b} + c$ , where  $a, b, c$  are integers and  $b$  is not divisible by the square of any prime. Compute  $100a + 10b + c$ .

*Proposed by Aaron Lin*

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