
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 \( n \geqslant 2 \), positive real numbers \( a_1, a_2, \ldots, a_n \), and non-negative real numbers \( b_1, b_2, \ldots, b_n \), which satisfy the following conditions:
(a) \( a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n = n \);
(b) \( a_1 a_2 \cdots a_n + b_1 b_2 \cdots b_n = \frac{1}{2} \).

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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