CF1697E.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 $d(a,b) \lt d(a,c)$ and $d(a,b) \lt d(b,c)$.

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

给定平面上的 $n$ 个点，其中第 $i$ 个点的坐标为 $(x_i, y_i)$。任意两个点的坐标均不相同。

点 $i$ 与点 $j$ 之间的距离定义为 $d(i,j) = |x_i - x_j| + |y_i - y_j|$。

对每个点，你需要为其分配一种颜色，颜色用 $1$ 到 $n$ 之间的整数表示。对于每组互不相同的有序三元组 $(a,b,c)$，需满足以下约束条件：

• 若 $a$、$b$、$c$ 颜色相同，则 $d(a,b) = d(a,c) = d(b,c)$；
• 若 $a$ 与 $b$ 颜色相同，而 $c$ 的颜色与 $a$ 不同，则 $d(a,b) \lt d(a,c)$ 且 $d(a,b) \lt d(b,c)$。

请计算满足上述约束条件的颜色分配方案总数。

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 $y_i \ne y_j$).

第一行包含一个整数 $n$（$2 \le n \le 100$）—— 点的数量。

接下来有 $n$ 行。其中第 $i$ 行包含两个整数 $x_i$ 和 $y_i$（$0 \le x_i, y_i \le 10^8$）。

任意两个点的坐标均不相同（即若 $i \ne j$，则 $x_i \ne x_j$ 或 $y_i \ne y_j$）。

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

输出一个整数——为这些点选择颜色的方案数。由于该数可能很大，请对 $998244353$ 取模后输出。

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

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

在第一个测试中，以下颜色选择方式是合适的：

• $[1, 1, 1]$;
• $[2, 2, 2]$;
• $[3, 3, 3]$;
• $[1, 2, 3]$;
• $[1, 3, 2]$;
• $[2, 1, 3]$;
• $[2, 3, 1]$;
• $[3, 1, 2]$;
• $[3, 2, 1]$。