
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 a $\triangle DEF$ with sides of lengths $9$, $12$, and $15$. A circle with center $Q$ and radius $2$ rolls around the inside of $\triangle DEF$, always remaining tangent to at least one side of the triangle. When $Q$ first returns to its original position, through what distance has $Q$ traveled?

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