Problem: Problem statement

N-winged rabbit is on a balance beam of length L-1. The initial position of the i-th rabbit is the integer x_i, which satisfies 0 ≤ x_ {i} \ lt x_ {i + 1} ≤ L−1. The coordinates increase as you move to the right. Any i-th rabbit can jump to the right (ie, move from x_i to x_i + a_i) any number of times, just a distance a_i. However, you cannot jump over another rabbit or enter a position below -1 or above L. Also, at most one rabbit can jump at the same time, and at most one rabbit can exist at a certain coordinate.

How many possible states of x_ {0},…, x_ {N−1} after starting from the initial state and repeating the jump any number of times? Find by the remainder divided by 1 \, 000 \, 000 \, 007.

input

The input is given in the following format.


N L
x_ {0}… x_ {N−1}
a_ {0}… a_ {N−1}


Constraint

* All inputs are integers
* 1 \ ≤ N \ ≤ 5 \,000
* N \ ≤ L \ ≤ 5 \,000
* 0 \ ≤ x_ {i} \ lt x_ {i + 1} \ ≤ L−1
* 0 \ ≤ a_ {i} \ ≤ L−1



output

Print the answer in one line.

sample

Sample input 1


13
0
1


Sample output 1


3


If 1/0 is used to express the presence / absence of a rabbit, there are three ways: 100, 010, and 001.

Sample input 2


twenty four
0 1
1 2


Sample output 2


Four


There are four ways: 1100, 1001, 0101, 0011.

Sample input 3


10 50
0 1 2 3 4 5 6 7 8 9
1 1 1 1 1 1 1 1 1 1


Sample output 3


272278100


The binomial coefficient C (50,10) = 10 \, 272 \, 278 \, 170, and the remainder obtained by dividing it by 1 \, 000 \, 000 \, 007 is 272 \, 278 \, 100.





Example

Input

1 3
0
1


Output

3

Solution (in CPP):

Solution:
#include<iostream>
using namespace std;

#define MOD 1000000007

long long N, L;
long long x[6000], a[6000];
long long dp[6000][6000];

int main() {
	cin >> N >> L;
	for (int i = 0; i < N; i++) { cin >> x[i]; }
	for (int i = 0; i < N; i++) { cin >> a[i]; }
	int cnt = 0;
	for (int i = x[N - 1]; i < L; i += a[N - 1]) {
		cnt++;
		if (cnt >= 20000) { break; }
		dp[N - 1][i] = 1;
	}
	for (int i = L - 1; i >= 0; i--) {
		dp[N - 1][i] += dp[N - 1][i + 1];
	}
	for (int i = N - 2; i >= 0; i--) {
		cnt = 0;
		for (int j = x[i]; j < L; j += a[i]) {
			cnt++;
			if (cnt >= 20000) { break; }
			dp[i][j] = dp[i + 1][j + 1];
		}
		for (int j = L - 1; j >= 0; j--) {
			dp[i][j] += dp[i][j + 1];
			dp[i][j] %= MOD;
		}
	}
	cout << dp[0][0] << endl;
	return 0;
}