
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 \(r(a, N) \in \{0,1, \ldots, N-1\}\) denote the remainder of \(a\) when divided by \(N\). Determine the number of positive integers \(n \leq 1000000\) for which
$$
r(n, 1000) > r(n, 1001).
$$

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