
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
Given the sides of \(\triangle ABC\) respectively as \(5\), \(12\), and \(13\), a circle with center \(P\) and radius \(2\) rolls inside along the edges of \(\triangle ABC\) while always being tangent to at least one of its sides. Compute the distance \(P\) travels when it returns to its original position for the first time.

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