
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 \( n (n \geq 2) \). There are \( 2n \) positive real numbers \( a_{1}, a_{2}, \cdots, a_{2n} \) that satisfy:
\[ \sum_{k=1}^{n} a_{2k-1} \cdot \sum_{k=1}^{n} a_{2k} = \prod_{k=1}^{n} a_{2k-1} + \prod_{k=1}^{n} a_{2k}. \]
Find the minimum value of \( S = \sum_{k=1}^{2n} \frac{a_{k}^{n-1}}{a_{k+1}} \), where \( a_{2n+1} = a_{1} \).

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