
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
Suppose $0<a_{1}<a_{2}<\cdots<a_{n}$. For any permutation $b_{1}, b_{2}, \cdots, b_{n}$ of $a_{1}, a_{2}, \cdots, a_{n}$, let
$$
M=\prod_{i=1}^{n}\left(a_{i}+\frac{1}{b_{i}}\right) .
$$

Find the permutation $b_{1}, b_{2}, \cdots, b_{n}$ that maximizes the value of $M$.

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