
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
Let  $ S$  be a set of 2002 points in the coordinate plane, no two of which have the same  $ x\minus{}$  or  $ y\minus{}$ coordinate. For any two points  $ P,Q \in S$ , consider the rectangle with one diagonal  $ PQ$  and the sides parallel to the axes. Denote by  $ W_{PQ}$  the number of points of  $ S$  lying in the interior of this rectangle. Determine the maximum  $ N$  such that, no matter how the points of  $ S$  are distributed, there always exist points  $ P,Q$  in  $ S$  with  $ W_{PQ}\ge N$ .

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