Problem Statement

In an N-dimensional space, the Manhattan distance $d(x,y) between two points x=(x_1, x_2, \dots, x_N)$ and $y = (y_1, y_2, \dots, y_N)$ is defined by:

$\displaystyle d(x,y)=\sum_{i=1}^n \vert x_i - y_i \vert$.

$A point x=(x_1, x_2, \dots, x_N)$ is said to be a lattice point if the components $x_1, x_2, \dots, x_N$ are all integers.
You are given lattice points $p=(p_1, p_2, \dots, p_N)$ and $q = (q_1, q_2, \dots, q_N)$ in an N-dimensional space.
How many lattice points $r satisfy d(p,r) \leq D$ and $d(q,r) \leq D$? Find the count modulo 998244353.

Input

Input is given from Standard Input in the following format:
N D
$p_1 p_2 \dots p_N$
$q_1 q_2 \dots q_N$

Output

Print the answer.

Constraints

• $1 \leq N \leq 100$
• $0 \leq D \leq 1000$
• -$1000 \leq p_i, q_i \leq 1000$
• All values in input are integers.

Sample Input 1

1 5
0
3

Sample Output 1

8

When N=1, we consider points in a one-dimensional space, that is, on a number line.
8 lattice points satisfy the conditions: -2,-1,0,1,2,3,4,5.

Sample Input 2

3 10
2 6 5
2 1 2

Sample Output 2

632

Sample Input 3

10 100
3 1 4 1 5 9 2 6 5 3
2 7 1 8 2 8 1 8 2 8

Sample Output 3

145428186

见题面 Input 部分

见题面 Output 部分