CF1290F.Making Shapes

You are given $n$ pairwise non-collinear two-dimensional vectors. You can make shapes in the two-dimensional plane with these vectors in the following fashion:

1. Start at the origin $(0, 0)$.
2. Choose a vector and add the segment of the vector to the current point. For example, if your current point is at $(x, y)$ and you choose the vector $(u, v)$, draw a segment from your current point to the point at $(x + u, y + v)$ and set your current point to $(x + u, y + v)$.
3. Repeat step 2 until you reach the origin again.

You can reuse a vector as many times as you want.

Count the number of different, non-degenerate (with an area greater than $0$) and convex shapes made from applying the steps, such that the shape can be contained within a $m \times m$ square, and the vectors building the shape are in counter-clockwise fashion. Since this number can be too large, you should calculate it by modulo $998244353$.

Two shapes are considered the same if there exists some parallel translation of the first shape to another.

A shape can be contained within a $m \times m$ square if there exists some parallel translation of this shape so that every point $(u, v)$ inside or on the border of the shape satisfies $0 \leq u, v \leq m$.

给你 $n$ 个两两不共线的二维向量。你可以用这些向量在二维平面上构造图形，方式如下：

1. 从原点 $(0, 0)$ 出发；
2. 选择一个向量，并将该向量对应的线段添加到当前点。例如，若当前点为 $(x, y)$，你选择了向量 $(u, v)$，则从当前点向点 $(x + u, y + v)$ 画一条线段，并将当前点更新为 $(x + u, y + v)$；
3. 重复步骤 2，直到再次回到原点。

每个向量可被重复使用任意多次。

请你统计满足以下所有条件的不同非退化（面积严格大于 $0$）且凸的图形的个数：

• 图形可通过上述步骤构造；
• 图形可被包含于一个 $m \times m$ 的正方形内；
• 构成图形的向量按逆时针顺序排列。

由于答案可能非常大，请对 $998244353$ 取模输出。

若存在某个平移变换能将第一个图形变为第二个图形，则认为这两个图形相同。

一个图形可被包含于一个 $m \times m$ 的正方形内，当且仅当存在某个平移变换，使得该图形内部或边界上的任意点 $(u, v)$ 均满足 $0 \leq u, v \leq m$。

The first line contains two integers $n$ and $m$ — the number of vectors and the size of the square ($1 \leq n \leq 5$, $1 \leq m \leq 10^9$).

Each of the next $n$ lines contains two integers $x_i$ and $y_i$ — the $x$-coordinate and $y$-coordinate of the $i$-th vector ($|x_i|, |y_i| \leq 4$, $(x_i, y_i) \neq (0, 0)$).

It is guaranteed, that no two vectors are parallel, so for any two indices $i$ and $j$ such that $1 \leq i \lt j \leq n$, there is no real value $k$ such that $x_i \cdot k = x_j$ and $y_i \cdot k = y_j$.

第一行包含两个整数 $n$ 和 $m$ —— 向量的个数以及正方形的边长（$1 \leq n \leq 5$，$1 \leq m \leq 10^9$）。

接下来的 $n$ 行中，每行包含两个整数 $x_i$ 和 $y_i$ —— 第 $i$ 个向量的 $x$ 坐标和 $y$ 坐标（$|x_i|, |y_i| \leq 4$，且 $(x_i, y_i) \neq (0, 0)$）。

保证任意两个向量均不平行，即对任意满足 $1 \leq i < j \leq n$ 的下标 $i$ 和 $j$，都不存在实数 $k$，使得 $x_i \cdot k = x_j$ 且 $y_i \cdot k = y_j$。

Output a single integer — the number of satisfiable shapes by modulo $998244353$.

输出一个整数——满足条件的形状数量对 $998244353$ 取模的结果。

The shapes for the first sample are:

The only shape for the second sample is:

The only shape for the fourth sample is:

第一个样例的图形如下：

第二个样例的唯一图形是：

第四个样例的唯一图形是：