
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 \( \omega_1 \) be a circle of radius 5, and let \( \omega_2 \) be a circle of radius 2 whose center lies on \( \omega_1 \). Let the two circles intersect at \( A \) and \( B \), and let the tangents to \( \omega_2 \) at \( A \) and \( B \) intersect at \( P \). If the area of \( \triangle ABP \) can be expressed as \( \frac{a \sqrt{b}}{c} \), where \( b \) is square-free and \( a, c \) are relatively prime positive integers, compute \( 100a + 10b + c \).

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