Figure: Voronoi Diagram of size 4.

In the 2-dimensional Cartesian coordinate system, we define the Voronoi Diagram of a non-empty set of points $$$S$$$, as a diagram that divides the plane by the criteria "which point in the set $$$S$$$ is closest in this location?". More precisely, the Voronoi diagram of a given non-empty point set $$$\{P_1, P_2, \cdots, P_n\}$$$ is a collection of regions: A point $$$K$$$ is included in region $$$i$$$ if and only if $$$d(P_i, K) \le d(P_j, K)$$$ holds for all $$$1 \le j \le n$$$.

While the usual Voronoi Diagram uses Euclidean distance, we use Manhattan distance in this problem. $$$d(X, Y)$$$ denotes the Manhattan distance between point $$$X$$$ and $$$Y$$$. Manhattan distance between two points is the sum of the absolute differences of their $$$X$$$, $$$Y$$$ coordinates. Thus, the Manhattan distance between two points $$$(X_1, Y_1)$$$, $$$(X_2, Y_2)$$$ can be written as $$$|X_2 - X_1| + |Y_2 - Y_1|$$$.

For example, in the picture above, every location over the plane is colored by the closest point with such location. The points which belongs to a single region is colored by a light color indicating a region, and the points which belongs to more than one region forms lines and points colored black.

The region is unbounded if for any real number $$$R$$$, there exists point $$$P$$$ in the region such that $$$d(O, P) \gt R$$$ where $$$O$$$ is the origin. You have to find the number of unbounded regions in the Voronoi Diagram.