Example 23 (2002 Japan Mathematical Olympiad (Second Round) Problem) Given a set of 2002 points on the $x O y$ plane denoted as $S$, and any two points in $S$ are connected by a line that is not parallel to the coordinate axes. For two different points $P, Q$ in $S$, consider the rectangle with $P Q$ as its diagonal, whose sides are respectively parallel to the coordinate axes, $W_{P Q}$ represents the number of points in $S$ within this rectangle (excluding $P, Q$).

When the statement "regardless of how the points in $S$ are distributed on the coordinate plane, there exists at least one pair of points $P, Q$ in $S$ such that $W_{\mathrm{PQ}} \geqslant N$" is true, find the maximum value of $N$.
Here we only discuss rectangles parallel to the coordinate axes. For any two points $P, Q$ in $S$, $R_{P Q}$ denotes the rectangle with $P Q$ as its diagonal, and $W_{P Q}$ denotes the number of points in $S$ within the rectangle $R_{P Q}$. Here, $P$ and $Q$ do not have to be distinct; if $P$ and $Q$ are the same, then there are no points in the rectangle $R_{P Q}$, and we denote $W_{P Q}=0$.
We will prove that the maximum value of $N$ is 400.
(1) Regardless of how the 2002 points in $S$ are placed, there always exist $P, Q \in S$ such that $W_{P Q} \geqslant 400$;
(2) If the 2002 points in $S$ are placed in a certain way, then for all $P, Q \in S$, we have $W_{P Q} \leqslant 400$.
First, we prove (1). Given a point set $S$ with 2002 points, let the 4 points in $S$ with the smallest x-coordinate, smallest y-coordinate, largest x-coordinate, and largest y-coordinate be denoted as $A, B, C, D$, respectively. These 4 points may include duplicates.

Thus, we can assert that any point in $S$ other than $A, B, C, D$ (at least 1998 points) is contained in at least one of the rectangles $R_{A B}, R_{B C}, R_{D D}, R_{D A}, R_{A C}$. This conclusion is derived from the following three cases:
(i) For a point $M$ with a y-coordinate less than those of $A$ and $C$. If the x-coordinate of $M$ is less than the x-coordinate of $B$, then $M$ is inside $R_{A B}$; otherwise, $M$ is inside $R_{B C}$.
(ii) For a point $M$ with a y-coordinate greater than those of $A$ and $C$. If the x-coordinate of $M$ is less than the x-coordinate of $D$, then $M$ is inside $R_{D A}$; otherwise, $M$ is inside $R_{C D}$.
(iii) For a point with a y-coordinate between those of $A$ and $C$, this point is inside $R_{A C}$.
This indicates that at least one of the 5 rectangles $R_{A B}, R_{B C}, R_{C D}, R_{D A}, R_{A C}$ contains more than 399 points in $S$. Otherwise, $S$ would have at most $399 \times 5 = 1995$ points other than $A, B, C, D$, which contradicts the fact that $S$ has 2002 points. Therefore, (1) is established.
Next, we prove (2).
Place the 2002 points in $S$ as shown in Figure 5-25, divided into 5 point groups $E, F, G, H, I$. Then we have:
(i) For any rectangle $R_{P Q} (P, Q \in S)$, it contains at most one point group;
(ii) This point group is either $I$ or a point group that includes at least one of $P, Q$.
This means that a rectangle with a diagonal formed by two points in $S$ contains at most 400 points in $S$.
(1) and (2) are thus proven. Therefore, the maximum value of $N$ is 400.