
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
Point \( A(3,0) \) is a fixed point on the circle \( x^2 + y^2 = 9 \). Two other points \( B \) and \( C \) are taken on the circle such that \( \angle BAC = \frac{\pi}{3} \). Find the equation of the trajectory of the centroid of \( \triangle ABC \).

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