
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
Line segment $\overline{DE}$ is a diameter of a circle with $DE = 30$. Point $F$, not equal to $D$ or $E$, lies on the circle. As point $F$ moves around the circle, the centroid (center of mass) of $\triangle DEF$ traces out a closed curve missing two points. What is the area of the region bounded by this curve?

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