
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 $n$ with $n \geq 2$, find the smallest real number $\lambda(n)$ such that for any real numbers $x_{1}, x_{2}, \ldots, x_{n}$ satisfying \( x_{1}+2 x_{2}+\cdots+n x_{n}=0 \), the following inequality holds:
$$
\left(\sum_{i=1}^{n} x_{i}\right)^{2} \leq \lambda(n)\left(\sum_{i=1}^{n} x_{i}^{2}\right).
$$

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