
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
For the circle with diameter $\overline{AB}$ equal to 36, point $C$ is another point on the circle, distinct from $A$ and $B$. $C$ traces the entire circle around $A$ and $B$. As $C$ moves, the centroid of $\triangle ABC$ also traces a path. Find the closest integer value for the area enclosed by the path that the centroid traces.

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