
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 real number \( k \) ( \( k > 2 \) ) and a positive integer \( n \) ( \( n \geq 3 \) ), find the largest positive real number \( \lambda \) such that for any positive real numbers \( a_i \) ( \( 1 \leq i \leq n \) ) satisfying \( \left(\sum_{i=1}^{n} a_i\right)\left(\sum_{i=1}^{n} \frac{1}{a_i}\right) < \lambda \), it always holds that \( a_1 + a_2 < k a_3 \).

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