
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 $AB = 30$. Point $C$, located on the circle but different from points $A$ and $B$, moves around the circle. As $C$ traces the circumference, the centroid of $\triangle ABC$ also traces a closed curve missing two points. What is the area, rounded to the nearest positive integer, of the region enclosed by this curve?

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