
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
Suppose 2012 reals are selected independently and at random from the unit interval  $[0,1]$ , and then written in nondecreasing order as  $x_1\le x_2\le\cdots\le x_{2012}$ . If the probability that  $x_{i+1} - x_i \le \frac{1}{2011}$  for  $i=1,2,\ldots,2011$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ , find the remainder when  $m+n$  is divided by 1000.

*Victor Wang.*

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