
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 $n(\geq 3)$ be a given positive integer. For $n$ given real numbers $a_{1}, a_{2}, \cdots, a_{n}$, denote the minimum value of $\left|a_{i} - a_{j}\right| (1 \leq i < j \leq n)$ by $m$. Under the condition that $a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} = 1$, find the maximum value of the aforementioned $m$.

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