
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
Regular hexagon \( P_{1} P_{2} P_{3} P_{4} P_{5} P_{6} \) has side length 2. For \( 1 \leq i \leq 6 \), let \( C_{i} \) be a unit circle centered at \( P_{i} \) and \( \ell_{i} \) be one of the internal common tangents of \( C_{i} \) and \( C_{i+2} \), where \( C_{7} = C_{1} \) and \( C_{8} = C_{2} \). Assume that the lines \( \{ \ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6} \} \) bound a regular hexagon. The area of this hexagon can be expressed as \( \sqrt{\frac{a}{b}} \), where \( a \) and \( b \) are relatively prime positive integers. Compute \( 100a + b \).

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