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;
const int mod = 1000000007;

struct BinaryIndexedTree
{
  vector< int > data;
  BinaryIndexedTree(int sz)
  {
    data.assign(++sz, 0);
  }
  int Query(int a)
  {
    int ret = 0;
    for(++a; a > 0; a -= a & -a) (ret += data[a]) %= mod;
    return(ret);
  }
  void Add(int k, int x)
  {
    for(++k; k < data.size(); k += k & -k) (data[k] += x) %= mod;
  }
};

int main()
{
  int N, L, x[5000], a[5000];
  vector< BinaryIndexedTree > dp(5001, BinaryIndexedTree(5005));

  cin >> N >> L; ++L;
  for(int i = 0; i < N; i++) {
    cin >> x[i]; ++x[i];
  }
  for(int i = 0; i < N; i++) {
    cin >> a[i];
  }
  dp[0].Add(0, 1);
  for(int i = 0; i < N; i++) {
    if(a[i] == 0) {
      dp[i + 1].Add(x[i], dp[i].Query(x[i] - 1));
    } else for(int j = L - 1; j >= x[i]; j--) {
      if((j - x[i]) % a[i] == 0) {
        dp[i + 1].Add(j, dp[i].Query(j - 1));
      }
    }
  }
  cout << dp[N].Query(L) << endl;
}