
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
The incircle \(\omega\) of equilateral triangle \(ABC\) has radius 1. Three smaller circles are inscribed tangent to \(\omega\) and the sides of \(ABC\). Then, three smaller circles are inscribed tangent to the previous circles and to each of two sides of \(ABC\). This process is repeated infinitely. What is the total length of the circumferences of all the circles?

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