
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 largest constant \( \lambda \) such that for all positive real numbers \( x_1, x_2, \cdots, x_{2n} \) satisfying
\[ 
\frac{1}{2n} \sum_{i=1}^{2n} (x_i + 2)^n \geq \prod_{i=1}^{2n} x_i 
\]
we have 
\[ 
\frac{1}{2n} \sum_{i=1}^{2n} (x_i + 1)^n \geq \lambda \prod_{i=1}^{2n} x_i.
\]

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