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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.util.Scanner;

class Main{

    static int N;
    static long A,B;
    static long[] S;

    static int findIndex(long s){
        int left=0,right=N+1;
        while(right-left>1){
            int center=(left+right)/2;
            if(s-S[center]>=B)left=center;
            else right=center;
        }
        return left;
    }

    public static void main(String[] args){
        Scanner scan = new Scanner(System.in);
        N = scan.nextInt();
        A = scan.nextLong();
        B = scan.nextLong();
        S = new long[N+1];
        long[] sum=new long[N+1];
        sum[0]=1;
        S[0] = -B;
        for(int i=1;i<=N;++i)S[i]=scan.nextLong();
        int lindex=-1;
        long mod = (long)1e9+7;
        if(A>B){
            long a = A;
            A=B;
            B=a;
        }
        for(int i=2;i<=N;++i)if(S[i]-S[i-2]<A){
            System.out.println(0);
            return;
        }

        for(int i=1;i<=N;++i){
            if(S[i]-S[i-1]<A){
                //怪しい findindexがlindexに対して小さかったら？
                int index = findIndex(S[i]);
                long dp = (index >= lindex ? (sum[findIndex(S[i])] - (lindex>=0 ? sum[lindex]:0)+mod)%mod : 0);
                sum[i]=(sum[i-1]+dp)%mod;
                lindex=i-2;
            }else if(S[i]-S[i-1]<B){
                int index = findIndex(S[i]);
                long dp = (index >= lindex ? (sum[findIndex(S[i])] - (lindex>=0 ? sum[lindex]:0)+mod)%mod : 0);
                sum[i]=(sum[i-1]+dp)%mod;
            }else{
                long dp = (sum[i-1] - (lindex>=0 ? sum[lindex]:0)+mod)%mod;
                sum[i]=(sum[i-1]+dp)%mod;
            }
        }
        System.out.println(((sum[N]-(lindex>=0 ? sum[lindex]:0)+mod)%mod));
    }
}