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: Accepted)

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<algorithm>
using namespace std;

typedef long long LL;

const int N=100005;
const int MOD=1000000007;

int n,f[N],g[N];
LL a,b,s[N];

int main()
{
    scanf("%d%lld%lld",&n,&a,&b);
    if (a>b) swap(a,b);
    for (int i=1;i<=n;i++) scanf("%lld",&s[i]);
    sort(s+1,s+n+1);
    for (int i=3;i<=n;i++) if (s[i]-s[i-2]<a) {puts("0");return 0;}
    f[0]=g[0]=1;int p=0,lim=0;
    for (int i=1;i<=n;i++)
    {
        while (p<i&&s[i]-s[p+1]>=b) p++;
        if (lim<=p) f[i]=(g[p]+MOD-(lim?g[lim-1]:0))%MOD;
        g[i]=(g[i-1]+f[i])%MOD;
        if (i>1&&s[i]-s[i-1]<a) lim=i-1;
    }
    int ans=0;
    for (int i=n;i>=0;i--)
    {
        (ans+=f[i])%=MOD;
        if (i<n&&s[i+1]-s[i]<a) break;
    }
    printf("%d",ans);
    return 0;
}