
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 that \( n \) is a fixed positive integer, for \( -1 \leq x_{i} \leq 1 \) (where \( i = 1, 2, \cdots, 2n \)), find the maximum value of
\[ 
\sum_{1 \leq r < s \leq 2n} (s - r - n) x_{r} x_{s}.
\]

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