
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
For positive integers  $a$  and  $N$ , let  $r(a, N) \in \{0, 1, \dots, N - 1\}$  denote the remainder of  $a$  when divided by  $N$ . Determine the number of positive integers  $n \le 1000000$  for which
\[r(n, 1000) > r(n, 1001).\]

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