Coloring
Description
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 $998244353$.
Input
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 $y_i \ne y_j$).
Output
Print one integer — the number of ways to choose the colors for the points. Since it can be large, print it modulo $998244353$.
Samples
3
1 0
3 0
2 1
9
5
1 2
2 4
3 4
4 4
1 3
240
4
1 0
3 0
2 1
2 0
24
Note
In the first test, the following ways to choose the colors are suitable:
- $[1, 1, 1]$;
- $[2, 2, 2]$;
- $[3, 3, 3]$;
- $[1, 2, 3]$;
- $[1, 3, 2]$;
- $[2, 1, 3]$;
- $[2, 3, 1]$;
- $[3, 1, 2]$;
- $[3, 2, 1]$.