
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
In a circle, segment $\overline{DE}$ is a diameter with $DE = 30$. Points $F$ and $G$ lie on the circle such that $\overline{DE}$, $\overline{DF}$, and $\overline{EF}$ form a triangle. What is the area of the circle traced by the centroid of $\triangle DEF$ as $F$ moves on the circumference of the circle excluding the points $D$ and $E$?

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