
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 \( n \) be a given positive integer, and consider the sum
\[ S = \sum_{1 \leq i < j \leq n} \left| x_i - x_j \right| \quad \text{where} \quad 0 < x_i < 1, \quad i = 1, 2, \ldots, n. \]

Find the maximum possible value of the sum.

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