
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 \( n (n \geq 2) \) be a given positive integer, and let the real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) satisfy \( x_{1}+2 x_{2}+\cdots+n x_{n}=0 \). Find the minimum real number \( \lambda(n) \) such that
$$
\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \leq \lambda(n) \left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right).
$$

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