
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
Let \( x_{i} > 0 \) for \( i = 1, 2, \ldots, n \) and \( n > 2 \). What is the minimum value of \( S = \sum_{i, j, k \text{ mutually distinct}} \frac{x_{i}}{x_{j} + x_{k}} \), where \( 1 \leq i, j, k \leq n \)?

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