
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
Given a set \( S \) consisting of 2002 points on the \( xOy \) plane where no two points are collinear with the coordinate axes, consider any two distinct points \( P \) and \( Q \) in \( S \). Define \( W_{PQ} \) as the number of points from \( S \) that lie inside the rectangle with \( PQ \) as the diagonal (excluding \( P \) and \( Q \)). Determine the maximum value of \( N \) such that there exists at least one pair of points \( P \) and \( Q \) in \( S \) for which \( W_{PQ} \geq N \).

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