
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 \( l \) passes through point \( A(2,0) \) and intersects the parabolic curve \( y = x^{2} + 2 \) at points \( B \) and \( C \). Points \( B_{1} \) and \( C_{1} \) are the feet of the perpendiculars from \( B \) and \( C \) to the \( x \)-axis, respectively. Point \( P \) is on the line segment \( BC \) such that \(\overrightarrow{B P} = \frac{\left|B B_{1}\right|}{\left|C C_{1}\right|} \cdot \overrightarrow{P C}\). Find the equation of the locus of the centroid \( G \) of triangle \( \triangle POA \) (where \( O \) is the origin).

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