
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 \(ABC\) be an equilateral triangle of side length 2. Let \(\omega\) be its circumcircle, and let \(\omega_{A}, \omega_{B}, \omega_{C}\) be circles congruent to \(\omega\) centered at each of its vertices. Let \(R\) be the set of all points in the plane contained in exactly two of these four circles. What is the area of \(R\)?

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