Problem:
Score : 1100 points

Problem Statement

There is a set consisting of N distinct integers. The i-th smallest element in this set is S_i. We want to divide this set into two sets, X and Y, such that:

The absolute difference of any two distinct elements in X is A or greater.

The absolute difference of any two distinct elements in Y is B or greater.

How many ways are there to perform such division, modulo 10^9 + 7? Note that one of X and Y may be empty.

Constraints

All input values are integers.

1 ≦ N ≦ 10^5

1 ≦ A , B ≦ 10^{18}

0 ≦ S_i ≦ 10^{18}(1 ≦ i ≦ N)

S_i < S_{i+1}(1 ≦ i ≦ N - 1)

Input

The input is given from Standard Input in the following format:

N A B
S_1
:
S_N

Output

Print the number of the different divisions under the conditions, modulo 10^9 + 7.

Sample Input 1

5 3 7
1
3
6
9
12

Sample Output 1

5

There are five ways to perform division:

X={1,6,9,12}, Y={3}

X={1,6,9}, Y={3,12}

X={3,6,9,12}, Y={1}

X={3,6,9}, Y={1,12}

X={3,6,12}, Y={1,9}

Sample Input 2

7 5 3
0
2
4
7
8
11
15

Sample Output 2

4

Sample Input 3

8 2 9
3
4
5
13
15
22
26
32

Sample Output 3

13

Sample Input 4

3 3 4
5
6
7

Sample Output 4

0

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include<bits/stdc++.h>
using namespace std;
using Int = long long;
//INSERT ABOVE HERE
signed main(){
  Int n,a,b;
  cin>>n>>a>>b;
  vector<Int> s(n+1);
  for(Int i=1;i<=n;i++) cin>>s[i];
  auto ok=[&](Int i,Int j,Int d){
    return !i || s[j]-s[i]>=d;
  };
  const Int MOD = 1e9+7;
  vector<Int> dp1(n,0),dp2(n,0);
  dp1[0]=dp2[0]=1;
  for(Int i=2;i<=n;i++){
    for(Int j=0;j<i-1;j++){
      dp1[i-1]+=dp2[j]*ok(j,i,a);
      dp2[i-1]+=dp1[j]*ok(j,i,b);
      dp1[i-1]%=MOD;dp2[i-1]%=MOD;
      
      dp1[j]*=ok(i-1,i,a);
      dp2[j]*=ok(i-1,i,b);
    }
  }
  
  Int ans=0;
  for(Int i=0;i<n;i++){
    ans+=dp1[i]+dp2[i];
    ans%=MOD;
  }
  cout<<ans<<endl;
  return 0;
}