
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
Consider a right triangle $\triangle ABC$ with sides of lengths $9, 12,$ and $15$. A circle of radius $2$ rolls around the inside of this triangle, always touching at least one side of the triangle. Calculate the total distance traveled by the center $P$ of the circle during a complete return to its original starting point.

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