
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 number \( k \) (where \( k > 2 \)) and a positive integer \( n \) (where \( n \geqslant 3 \)), find the largest positive number \( \lambda \) such that if the positive numbers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfy
$$
\left(a_{1}+a_{2}+\cdots+a_{n}\right)\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)< \lambda \text {, }
$$
then it must hold that \( a_{1}+a_{2}<k a_{3} \).

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