
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 an integer $\mathrm{n} \geq 2$, find the largest real number $\lambda$ such that for any complex number $\mathrm{z}_{\mathrm{k}}$ with modulus not greater than 1 $(1 \leq \mathrm{k} \leq \mathrm{n})$, the following inequality holds:
\[1+\sum_{\mathrm{k}=1}^{\mathrm{n}}\left|\sigma_{\mathrm{k}}\right| \geq \lambda\left(\sum_{\mathrm{k}=1}^{\mathrm{n}}\left|\mathrm{z}_{\mathrm{k}}\right|\right).\]

Here, $\sigma_{\mathrm{k}}=\sum_{1 \leq j_{1} \leqslant \mathrm{i}_{2}<\ldots<\mathrm{ik}_{\mathrm{k}} \leq \mathrm{n}} \Pi_{\mathrm{i}=1}^{\mathrm{k}} \mathrm{z}_{11} \circ$.

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