
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
In triangle \( \triangle ABC \), let the side lengths be denoted as \( \alpha, \beta, \gamma \), and the distances from the centroid to each vertex be denoted as \( \lambda, \mu, v \). Find the value of \( \frac{\alpha^{2} + \beta^{2} + \gamma^{2}}{\lambda^{2} + \mu^{2} + v^{2}} \).

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