
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 fixed positive integer. To any choice of real numbers satisfying \[0\le x_{i}\le 1,\quad i=1,2,\ldots, n,\] there corresponds the sum \[\sum_{1\le i<j\le n}|x_{i}-x_{j}|.\] Let  $S(n)$  denote the largest possible value of this sum. Find  $S(n)$ .

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