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 $998244353$.

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.

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.

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