
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 \( x_{i} \in \mathbf{R}^{+} \) (for \( i=1,2, \cdots, n \)) and \( x_{1} x_{2} \cdots x_{n}=1 \), where \( n \) is a given positive integer, find the minimum positive value of \( \lambda \) such that the inequality \(\frac{1}{\sqrt{1+2 x_{1}}}+\frac{1}{\sqrt{1+2 x_{2}}}+\cdots+\frac{1}{\sqrt{1+2 x_{n}}} \leqslant \lambda\) always holds.

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