
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 \( n \geq 2 \), find the maximum and minimum values of the product \( x_1 x_2 \cdots x_n \) under the conditions \( x_i \geq \frac{1}{n} \) for \( i=1, 2, \cdots, n \) and \( x_1^2 + x_2^2 + \cdots + x_n^2 = 1 \).

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