
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
Let \(ABC\) be a triangle with centroid \(G\). Determine, with proof, the position of the point \(P\) in the plane of \(ABC\) such that
\[ AP \cdot AG + BP \cdot BG + CP \cdot CG \]
is a minimum, and express this minimum value in terms of the side lengths of \(ABC\).

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