
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 \( n \geq 2 \), find the largest positive real number \( \lambda(n) \) such that for any real numbers \( x_1, x_2, \ldots, x_n \), not all of which are positive, the following inequality holds:
\[
\prod_{i=1}^{n}\left(x_{i}^{2}-x_{i}+1\right) \geq \lambda(n) \cdot \left(\left(\prod_{i=1}^{n} x_{i}\right)^{2}-\prod_{i=1}^{n} x_{i}+1\right)
\]

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