
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 $n \geq 2$, find the smallest real number $c(n)$ such that for any positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$, the following inequality holds:
\[
\frac{\sum_{i=1}^{n} a_{i}}{n}-\sqrt[n]{\prod_{i=1}^{n} a_{i}} \leq c(n) \cdot \max_{1 \leq i < j \leq n}\left\{\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^{2}\right\}.
\]

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