
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
Given a positive integer \( n \geq 2 \), find the largest \(\lambda = \lambda(n)\) such that \(\lambda \cdot \max_{1 \leq k \leq n}\left|z_{k}\right|^{2} \leq \sum_{k=1}^{n}\left|z_{k+1}-z_{k}\right|^{2}\) holds for any complex numbers \( z_1, z_2, \ldots, z_n \) satisfying \(\sum_{k=1}^{n} z_{k} = 0\) and \( z_{n+1} = z_1 \).

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