
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  $N$  denote the number of ordered 2011-tuples of positive integers  $(a_1,a_2,\ldots,a_{2011})$  with  $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$  such that there exists a polynomial  $f$  of degree  $4019$  satisfying the following three properties:
 
- $f(n)$  is an integer for every integer  $n$ ;
- $2011^2 \mid f(i) - a_i$  for  $i=1,2,\ldots,2011$ ;
- $2011^2 \mid f(n+2011) - f(n)$  for every integer  $n$ .

Find the remainder when  $N$  is divided by  $1000$ .

*Victor Wang*

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