
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 real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) and non-negative real numbers \(b_{1}, b_{2}, \cdots, b_{n}\) satisfying:
1. \(a_{1} + a_{2} + \cdots + a_{n} + b_{1} + b_{2} + \cdots + b_{n} = n\)
2. \(a_{1} a_{2} \cdots a_{n} + b_{1} b_{2} \cdots b_{n} = \frac{1}{2}\)

Find the maximum value of \(a_{1} a_{2} \cdots a_{n} \left( \frac{b_{1}}{a_{1}} + \frac{b_{2}}{a_{2}} + \cdots + \frac{b_{n}}{a_{n}} \right)\).

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