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

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