A15253. Coloring

You are given $n$ points on the plane, the coordinates of the $i$ -th point are $(x_i, y_i)$ . No two points have the same coordinates.

The distance between points $i$ and $j$ is defined as $d(i,j) = |x_i - x_j| + |y_i - y_j|$ .

For each point, you have to choose a color, represented by an integer from $1$ to $n$ . For every ordered triple of different points $(a,b,c)$ , the following constraints should be met:

- if $a$ , $b$ and $c$ have the same color, then $d(a,b) = d(a,c) = d(b,c)$ ;
- if $a$ and $b$ have the same color, and the color of $c$ is different from the color of $a$ , then $d(a,b) < d(a,c)$ and $d(a,b) < d(b,c)$ .

Calculate the number of different ways to choose the colors that meet these constraints.

The first line contains one integer $n$ ( $2 \le n \le 100$ ) — the number of points.

Then $n$ lines follow. The $i$ -th of them contains two integers $x_i$ and $y_i$ ( $0 \le x_i, y_i \le 10^8$ ).

No two points have the same coordinates (i. e. if $i \ne j$ , then either $x_i \ne x_j$ or $y_i \ne y_j$ ).

Print one integer — the number of ways to choose the colors for the points. Since it can be large, print it modulo $998244353$ .

In the first test, the following ways to choose the colors are suitable:

- $[1, 1, 1]$ ;
- $[2, 2, 2]$ ;
- $[3, 3, 3]$ ;
- $[1, 2, 3]$ ;
- $[1, 3, 2]$ ;
- $[2, 1, 3]$ ;
- $[2, 3, 1]$ ;
- $[3, 1, 2]$ ;
- $[3, 2, 1]$ .