
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, \( n \geq 3 \). For \( n \) given real numbers \( a_1, a_2, \cdots, a_n \), denote the minimum value of \( \left| a_i - a_j \right| \) for \( 1 \leq i < j \leq n \) as \( m \). Find the maximum value of \( m \) under the condition that \( a_1^2 + a_2^2 + \cdots + a_n^2 = 1 \).

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