
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
For a given number \( n \geq 2 \), find the maximum and minimum values of the product

$$
x_{1} x_{2} \ldots x_{n}
$$

under the conditions that \( x_{i} \geq 1 / n \) for \( i = 1, 2, \ldots, n \) and \( x_{1}^{2} + x_{2}^{2} + \ldots + x_{n}^{2} = 1 \).

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