
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 \( a_{1}, a_{2}, \cdots, a_{n} \), denote the smallest value of \( \left|a_{i} - a_{j}\right| \) (with \( 1 \leq i < j \leq n \)) 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