
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  $n$ , find the least  $\lambda>0$  such that for any  $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$ , the condition  $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$  implies  $\sum_{i=1}^{n}\cos x_i\le\lambda$ .

*Huang Yumin*

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