
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 circle with diameter $\overline{DE}$ of length 36. Point $F$, which is distinct from $D$ and $E$, is placed on this circle. As point $F$ moves all around the circle, the centroid (center of mass) of $\triangle DEF$ traces a path. Calculate the area of the region enclosed by this path traced by the centroid.

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