
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
Let \( n \) be a given natural number where \( n \geq 3 \). For \( n \) given real numbers \( a_1, a_2, \cdots, a_n \), let the minimum value of \( |a_i - a_j| \) (for \( 1 \leq i < j \leq n \)) be \( m \). Find the maximum value of \( m \) under the condition \( \sum_{i=1}^n a_i^2 = 1 \).

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