
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 real numbers \( x_{1}, x_{2}, \ldots, x_{n} \) (where \( n \geq 2 \)), with \( x_{1} \geq \frac{1}{n} \) for \( i = 1, 2, \ldots, n \) and \( x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} = 1 \), find whether the product \( P = x_{1} x_{2} x_{3} \cdots x_{n} \) has a maximum and/or minimum value, and if so, provide these values.

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