
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 positive numbers \(a_{1} < a_{2} < \ldots < a_{n}\). For which permutation \((b_{1}, b_{2}, \ldots, b_{n})\) of these numbers is the product

\[
\prod_{i=1}^{n}\left(a_{i} + \frac{1}{b_{i}}\right)
\]

maximized?

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