
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
ABC is a triangle with centroid G and side lengths \( a = BC \), \( b = CA \), \( c = AB \). Find the point \( P \) in the plane which minimizes \( AP \cdot AG + BP \cdot BG + CP \cdot CG \) and find the minimum value in terms of \( a \), \( b \), and \( c \).

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