
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 \(n \geq 2\), find the maximum constant \(\lambda(n)\) such that for a sequence of real numbers \(a_0, a_1, \cdots, a_n\) satisfying:
$$
\begin{array}{l}
0=a_0 \leq a_1 \leq \cdots \leq a_n, \\
2a_i \geq a_{i+1} + a_{i-1},
\end{array}
$$
the following inequality holds:
$$
\left(\sum_{i=1}^{n} i a_i \right)^2 \geq \lambda(n) \sum_{i=1}^{n} a_i^2.
$$

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