
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
Points \( A(0,0) \) and \( B(1000,0) \), as well as points \( C_{1}(1,1) \), \( C_{2}(2,1) \), \(\ldots\), \( C_{999}(999,1) \), are marked on the coordinate plane. Then all possible lines \( A C_{i} \) and \( B C_{i} \) (for \( 1 \leq i \leq 999 \)) are drawn. How many integer-coordinate points of intersection do all these lines have? (An integer-coordinate point is one where both coordinates are integers.)

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