Neighbours

The Great Bytean Mountains occupy a rectangle in the Cartesian plane whose opposite corners are $(0, 0)$ and $(w, h)$, where $w$ and $h$ are positive integers. Inside the rectangle there are $n$ mountain peaks, each located at a grid point (a point with integer coordinates).

Tourists want to build houses on grid points. The park rules are strict: at most one house may stand on each grid point, and no house may be built on a peak. There are therefore $(w + 1)\cdot(h + 1) - n$ possible house locations.

Some locations are better than others. A grid point $(x, y)$ that is not a peak has a northern neighbour if there is a peak at $(x, y + d)$ for some positive integer $d$. Southern, eastern, and western neighbours are defined analogously — a peak in the same column below it, or in the same row to its east or west. Every non-peak grid point thus has between $0$ and $4$ neighbours, and the more neighbours it has, the better the view.

Count how many non-peak grid points have exactly $0$, $1$, $2$, $3$, and $4$ neighbours.

The first line contains three integers $w$, $h$, and $n$ ($1 \le w, h \le 10^9$, $1 \le n \le 500\,000$), separated by single spaces. Each of the following $n$ lines contains two integers $x$ and $y$ ($0 \le x \le w$, $0 \le y \le h$), separated by a single space, giving the location of one peak. All peaks are at distinct grid points.

Print five integers separated by single spaces: the numbers of non-peak grid points that have exactly $0$, $1$, $2$, $3$, and $4$ neighbours, in that order.

In the example, the non-peak points with exactly two neighbours are $(3, 1)$ and $(3, 3)$; those with three neighbours are $(1, 1)$, $(0, 2)$, and $(1, 3)$; the point $(2, 2)$ has four neighbours; the point $(4, 0)$ has none; and every remaining non-peak point has exactly one neighbour.