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<bits/stdc++.h>
using namespace std;
#define MAX 5005
int mod=1e9+7;

int add(int a,int b){
  return (a+b)%mod;
}

int N,L;
int x[MAX],a[MAX];
int dp[2][MAX];
int main(){
  cin>>N>>L;
  for(int i=0;i<N;i++)cin>>x[i],x[i]++;
  for(int i=0;i<N;i++)cin>>a[i];

  int ans=0;
  dp[1][0]=1;
  for(int i=0;i<N;i++){
    int ai=i&1,bi=1-ai;
    int sum=dp[bi][0];
    
    dp[ai][0]=0;
    for(int j=1;j<=L;j++){
      if(j<x[i]){
        dp[ai][j]=0;
      }else if(a[i]==0&&j!=x[i]){
        dp[ai][j]=0;
      }else if(a[i]>0&&(j-x[i])%a[i]!=0){
        dp[ai][j]=0;
      }else{
        dp[ai][j]=sum;
      }
      sum=add(sum,dp[bi][j]);

      if(i==N-1){
        ans=add(ans,dp[ai][j]);
      }
    }
  }
  cout<<ans<<endl;
  return 0;
}