
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 $n(n > 1)$ be a given integer. Find the greatest constant $\lambda(n)$ such that for any $n$ nonzero complex numbers $z_1, z_2, \cdots, z_n$, the following inequality holds:
\[
\sum_{k=1}^{n} \left| z_k \right|^2 \geq \lambda(n) \cdot \min_{1 \leq k \leq n} \left\{ \left| z_{k+1} - z_k \right|^2 \right\} \quad \text{where } z_{n+1} = z_1.
\]


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