
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 \(n \geqslant 2\), find the maximum and minimum values of the product \(x_{1} x_{2} \cdots x_{n}\) under the following conditions: (1) \(x_{i} \geqslant \frac{1}{n}\) for \(i=1, 2, \cdots, n\); (2) \(x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} = 1\).

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