Score : $500$ points

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.

1 5
0
3

8

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

3 10
2 6 5
2 1 2

632

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

145428186