
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  $x_1,x_2,\ldots,x_n $  be real numbers, where  $n\ge 2$  is a given integer, and let  $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $  be a permutation of  $1,2,\ldots,n$ .
Find the maximum and  minimum of   $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$  (here  $\lfloor x\rfloor $  is the largest integer not greater than   $x$ ).

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