
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 6 inscribed in a circle \(\omega\). Let \(A_1, A_2\) be the points (distinct from \(A\)) where the lines through \(A\) passing through the two trisection points of \(BC\) meet \(\omega\). Define \(B_1, B_2, C_1, C_2\) similarly. Given that \(A_1, A_2, B_1, B_2, C_1, C_2\) appear on \(\omega\) in that order, find the area of hexagon \(A_1 A_2 B_1 B_2 C_1 C_2\).

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