
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 natural number \( n \geq 2 \), find the smallest positive number \( \lambda \) such that for any positive numbers \( a_{1}, a_{2}, \ldots, a_{n} \) and any \( n \) numbers \( b_{1}, b_{2}, \ldots, b_{n} \) from the interval \(\left[0, \frac{1}{2}\right]\), provided \( a_{1} + a_{2} + \cdots + a_{n} = b_{1} + b_{2} + \cdots + b_{n} = 1 \), the inequality \( a_{1} a_{2} \cdots a_{n} \leq \lambda \left(a_{1} b_{1} + a_{2} b_{2} + \cdots + a_{n} b_{n} \right) \) holds.

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