
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 an integer and  $n \geq 2$ ,  $x_1, x_2, \cdots , x_n$  are arbitrary real number, find the maximum value of   $$ 2\sum_{1\leq i<j \leq n}\left \lfloor x_ix_j \right \rfloor-\left (  n-1 \right )\sum_{i=1}^{n}\left \lfloor x_i^2 \right \rfloor   $$  

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