
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
Two sets of points in the coordinate plane are given: \(\{(-1,1),(-1,2), \ldots,(-1,2000)\}\) and \(\{(1,1),(1,2), \ldots,(1,2000)\}\). 2000 line segments are drawn connecting these points so that each point in the first set is connected to exactly one point in the second set, and vice versa. Find, with proof, the sum of the \(y\)-intercepts of the segments.

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