
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
Circles \( \omega_1 \), \( \omega_2 \), and \( \omega_3 \) each have radius 6 and are placed in such a way that each circle is externally tangent to the other two. Points \( P_1 \), \( P_2 \), and \( P_3 \) are located on \( \omega_1 \), \( \omega_2 \), and \( \omega_3 \) respectively such that \( P_1P_2 = P_2P_3 = P_3P_1 \) and line \( P_iP_{i+1} \) is tangent to \( \omega_i \) for each \( i = 1, 2, 3 \), where \( P_4 = P_1 \). Determine the area of \( \triangle P_1P_2P_3 \) and express it in the form \( \sqrt{a} + \sqrt{b} \) for integers \( a \) and \( b \).

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