
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 \(\overline{AB}\) is a diameter of a circle with \(AB = 32\). Point \(C\), not equal to \(A\) or \(B\), lies on the circle. As point \(C\) moves around the circle, the centroid (center of mass) of \(\triangle ABC\) traces out a closed curve missing two points. Determine the exact area of the region bounded by this curve.

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