
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 \( n \geq 3 \). Find the minimum real number \( k \) such that for any positive real numbers \( a_{1}, a_{2}, \cdots, a_{n} \), the following inequality holds:
$$
\sum_{i=1}^{n-1} \frac{a_{i}}{s-a_{i}} + \frac{k a_{n}}{s-a_{n}} \geq \frac{n-1}{n-2},
$$

where \( s = a_{1} + a_{2} + \cdots + a_{n} \).

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