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E - Warp

Score : $500$ points

Problem Statement

Takahashi is at the origin of a two-dimensional plane.
Takahashi will repeat teleporting $N$ times. In each teleportation, he makes one of the following moves:

• Move from the current coordinates $(x,y)$ to $(x+A,y+B)$
• Move from the current coordinates $(x,y)$ to $(x+C,y+D)$
• Move from the current coordinates $(x,y)$ to $(x+E,y+F)$

There are obstacles on $M$ points $(X_1,Y_1),\ldots,(X_M,Y_M)$ on the plane; he cannot teleport to these coordinates.

How many paths are there resulting from the $N$ teleportations? Find the count modulo $998244353$.

Constraints

• $1 \leq N \leq 300$
• $0 \leq M \leq 10^5$
• $-10^9 \leq A,B,C,D,E,F \leq 10^9$
• $(A,B)$, $(C,D)$, and $(E,F)$ are distinct.
• $-10^9 \leq X_i,Y_i \leq 10^9$
• $(X_i,Y_i)\neq(0,0)$
• $(X_i,Y_i)$ are distinct.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A$ $B$ $C$ $D$ $E$ $F$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_M$ $Y_M$

Output

Print the answer.

2 2
1 1 1 2 1 3
1 2
2 2

5

The following $5$ paths are possible:

• $(0,0)\to(1,1)\to(2,3)$
• $(0,0)\to(1,1)\to(2,4)$
• $(0,0)\to(1,3)\to(2,4)$
• $(0,0)\to(1,3)\to(2,5)$
• $(0,0)\to(1,3)\to(2,6)$

10 3
-1000000000 -1000000000 1000000000 1000000000 -1000000000 1000000000
-1000000000 -1000000000
1000000000 1000000000
-1000000000 1000000000

0

300 0
0 0 1 0 0 1

292172978