
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 real number $\lambda(n)$ that satisfies the following condition: For any $\theta_{i} \in \left(0, \frac{\pi}{2}\right)$ for $i = 1, 2, \cdots, n$, the following inequality holds:
$$
\begin{array}{l}
\left(\sum_{i=1}^{n} \tan \theta_{i}\right)\left(\sum_{i=1}^{n} \cot \theta_{i}\right) \\
\geqslant\left(\sum_{i=1}^{n} \sin \theta_{i}\right)^{2}+\left(\sum_{i=1}^{n} \cos \theta_{i}\right)^{2}+\lambda(n)\left(\theta_{1}-\theta_{n}\right)^{2}
\end{array}
$$

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