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# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Text.RegularExpressions;
using System.Text;
 
public class Program
{
    public void Proc() {
        Reader.IsDebug = false;
        long[] inpt = Reader.ReadLine().Split(' ').Select(a=>long.Parse(a)).ToArray();

        this.NumList = new long[(int)inpt[0]];
        for(int i=0; i<this.NumList.Length; i++) {
            this.NumList[i] = long.Parse(Reader.ReadLine());
        }
        this.DiffA = inpt[1];
        this.DiffB = inpt[2];

        long ans = GetAns(-1, -1, 0);
        Console.WriteLine(ans);
    }

    private long[] NumList;
    private long DiffA;
    private long DiffB;
    private Dictionary<long, Dictionary<long, long>> dic = new Dictionary<long, Dictionary<long, long>>();

    private long GetAns(long prevA, long prevB, int idx) {
        if(!dic.ContainsKey(prevA)) {
            dic.Add(prevA, new Dictionary<long, long>());
        }
        if(dic[prevA].ContainsKey(prevB)) {
            return dic[prevA][prevB];
        }

        if(idx == this.NumList.Length) {
            dic[prevA][prevB] = 1;
            return 1;
        }

        long ans = 0;
        if(prevA < 0 || prevA + DiffA <= NumList[idx]) {
            ans+=GetAns(NumList[idx], prevB, idx+1);
        }
        if(prevB < 0 || prevB + DiffB <= NumList[idx]) {
            ans+=GetAns(prevA, NumList[idx], idx+1);
        }
        ans = ans % mod;
        dic[prevA][prevB] = ans;
        return ans;
    }

    private const long mod = 1000000000 + 7;

    public class Reader {
        public static bool IsDebug = true;
        private static System.IO.StringReader SReader;
        private static string InitText = @"


8 2 9
3
4
5
13
15
22
26
32


";
        public static string ReadLine() {
            if(IsDebug) {
                if(SReader == null) {
                    SReader = new System.IO.StringReader(InitText.Trim());
                }
                return SReader.ReadLine();
            } else {
                return Console.ReadLine();
            }
        }
    }
    public static void Main(string[] args)
    {
        Program prg = new Program();
        prg.Proc();
    }
}