
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 that \( a_1 + a_2 + \cdots + a_n = s \),

If \( a_i > \sqrt{2+\sqrt{5}} \) for \( i = 1, 2, \cdots, n \), then
\[
\left(a_1 + \frac{1}{a_1}\right) \left(a_2 + \frac{1}{a_2}\right) \cdots \left(a_n + \frac{1}{a_n}\right) \leqslant \left(\frac{s}{n} + \frac{n}{s}\right)^n
\]

If \( 0 < a_i < \sqrt{2+\sqrt{5}} \) for \( i = 1, 2, \cdots, n \), then the inequality is reversed.

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