
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 \( n \) such that \( n \geq 3 \). For \( n \) real numbers \( x_1, x_2, \cdots, x_n \), denote the minimum value of \( \left|x_i - x_j\right| \) (for \( 1 \leq i < j \leq n \)) as \( m \). If \( x_1^2 + x_2^2 + \cdots + x_n^2 = 1 \), determine the maximum value of \( m \).

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