
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 $\triangle DEF$ with sides having lengths $9$, $12$, and $15$. A circle with a radius of $2$ units rolls around the inside of $\triangle DEF$, always remaining tangent to at least one side of the triangle. Determine the total distance that the center of the circle, point $Q$, travels by the time it returns to its starting position.

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