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.

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.

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.

Sample Input 1

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

Sample Output 1

5

The following 5 paths are possible:

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

Sample Input 2

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

Sample Output 2

0

Sample Input 3

300 0
0 0 1 0 0 1

Sample Output 3

292172978

见题面 Input 部分

见题面 Output 部分