
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
Leibniz's theorem. Let $M$ be an arbitrary point in the plane, and $G$ be the centroid of triangle $ABC$. Then the following equality holds:
\[ 3|MG|^{2} = |MA|^{2} + |MB|^{2} + |MC|^{2} - \frac{1}{3}\left(|AB|^{2} + |BC|^{2} + |CA|^{2}\right). \]

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