
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. Consider the sum
\[ S = \sum_{1 \leqslant i < j \leqslant n} \left|x_i - x_j \right| \quad (0 < x_i < 1, \, i = 1, 2, \ldots, n). \]
Find the maximum possible value of this sum.
(6th Canadian Mathematical Olympiad)

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