
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 $\triangle ABC$, and $P$ be any point in the same plane. Then we have: 
\[ PA^{2} + PB^{2} + PC^{2} = GA^{2} + GB^{2} + GC^{2} + 3 PG^{2} = 3 PG^{2} + \frac{1}{3}(a^{2} + b^{2} + c^{2}) \]
(This latter equation is known as the Leibniz formula).

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