
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 \( a_{1}, a_{2}, \cdots, a_{n} \) and any \( n \) numbers \( b_{1}, b_{2}, \cdots, b_{n} \) from the interval \(\left[0, \frac{1}{2}\right]\), if \( a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+\cdots+b_{n}=1 \), then \( a_{1} a_{2} \cdots a_{n} \leq \lambda\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right) \).

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