
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
Point \( P \) lies inside equilateral triangle \( ABC \) so that \( \angle BPC = 120^\circ \) and \( AP \sqrt{2} = BP + CP \). The ratio \( \frac{AP}{AB} \) can be written as \( \frac{a \sqrt{b}}{c} \), where \( a, b, c \) are integers, \( c \) is positive, \( b \) is square-free, and \(\operatorname{gcd}(a, c) = 1 \). Find \( 100a + 10b + c \).

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