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 <bits/stdc++.h>
using namespace std;
 
#define int long long
#define dub long double
#define FOR(i, a, b) for (int (i)=(a); (i)<(b); ++(i))
#define ROF(i, a, b) for (int (i)=(a); (i)>(b); --(i))
#define CASET int _T; cin >> _T; FOR(caset, 1, _T+1)
#define pii pair<int,int>
#define t3i tuple<int,int,int>
#define mii map<int,int>
#define vi vector<int>
#define vb vector<bool>
#define vs vector<string>
#define vvi vector<vi>
#define vvb vector<vb>
#define vpii vector<pii>
#define vt3i vector<t3i>
#define vvpii vector<vpii>
#define sq(x) (x)*(x)
#define inv(x) powa(x, MOD-2)
#define ALL(x) (x).begin(), (x).end()
#define RALL(x) (x).rbegin(), (x).rend()
#define SIZE(x) (signed)(x).size()
#define SUM(x) accumulate(ALL(x), 0LL)
 
long long INF = 1LL<<60;
long long MOD = 1e9 + 7;
 
namespace output {
    void pr(short x) { cout << x; }
    void pr(signed x) { cout << x; }
    void pr(long x) { cout << x; }
    void pr(long long x) { cout << x; }
    void pr(unsigned short x) { cout << x; }
    void pr(unsigned x) { cout << x; }
    void pr(unsigned long x) { cout << x; }
    void pr(unsigned long long x) { cout << x; }
    void pr(float x) { cout << x; }
    void pr(double x) { cout << x; }
    void pr(long double x) { cout << x; }
    void pr(char x) { cout << x; }
    void pr(const char* x) { cout << x; }
    void pr(const string& x) { cout << x; }
    void pr(bool x) { cout << x; }
    template<size_t sz> void pr(const bitset<sz>& x) { cout << x; }
    template<class T> void pr(const complex<T>& x) { cout << x; }
    
    template<class T1, class T2> void pr(const pair<T1,T2>& x);
    template<class T1, class T2, class T3> void pr(const tuple<T1,T2,T3>& x);
    template<class T> void pr(const T& x);
    
    template<class T, class... Ts> void pr(const T& t, const Ts&... ts) { pr(t); pr(ts...); }
    template<class T1, class T2> void pr(const pair<T1,T2>& x) { pr("(",x.first,", ",x.second,")"); }
    template<class T1, class T2, class T3> void pr(const tuple<T1,T2,T3>& x) { pr("(",get<0>(x),", ",get<1>(x),", ",get<2>(x),")"); }
    template<class T> void pr(const T& x) { 
        pr("["); // const iterator needed for vector<bool>
        bool fst = 1; for (const auto& a: x) pr(!fst?", ":"",a), fst = 0; 
        pr("]");
    }
    
    void print() { pr("\n"); } // print w/ spaces
    template<class T, class... Ts> void print(const T& t, const Ts&... ts) { pr(t); if (sizeof...(ts)) pr(" "); print(ts...); }
}
using namespace output;
 
template<typename T = int> vector<T> create(size_t n){ return vector<T>(n); }
template<typename T, typename... Args> auto create(size_t n, Args... args){ return vector<decltype(create<T>(args...))>(n, create<T>(args...)); }
 
template <class T1, class T2> istream &operator>>(istream &is, pair<T1,T2> &x) { is >> x.first >> x.second; return is; }
template <class T1, class T2, class T3> istream &operator>>(istream &is, tuple<T1,T2,T3> &x) { is >> get<0>(x) >> get<1>(x) >> get<2>(x); return is; }
template <class T> istream &operator>>(istream &is, vector<T> &v) { for (auto &x : v) is >> x; return is; }
int powa(int base, int exp) {
    int res = 1;
    while (exp) {
        if (exp&1) res = res*base % MOD;
        base = base*base % MOD;
        exp >>= 1;
    }
    return res;
}
bool mini(int &a, int b) { return b < a ? a = b, 1 : 0; }
bool maxi(int &a, int b) { return b > a ? a = b, 1 : 0; }
 
signed main() {
    ios::sync_with_stdio(0), cin.tie(0);
    
    int n;
    vi x(2);
    cin >> n >> x;
    if (x[0] > x[1]) swap(x[0], x[1]);

    vi a(n);
    cin >> a;

    vi bad(n);
    FOR(i, 1, n) bad[i] = a[i] - a[i-1] < x[0];
    vi badpref = bad;
    FOR(i, 1, n) badpref[i] += badpref[i-1];

    auto qbad = [&](int l, int r) {
        if (l > r) return 0LL;
        return badpref[r] - (l ? badpref[l-1] : 0);
    };

    vi nxt(n);
    int j = 0;
    FOR(i, 0, n) {
        while (j < n && a[j] - a[i] < x[1]) ++j;
        nxt[i] = j;
    }

    vvi dp(n, vi(2));
    dp[n-1][0] = dp[n-1][1] = 1;

    ROF(i, n-2, -1) {
        if (a[i+1] - a[i] < x[0]) {
            if (i+2 >= n || a[i+2] - a[i] >= x[0]) {
                dp[i][0] += dp[i+1][1];
            }
        } else {
            dp[i][0] += dp[i+1][0] + dp[i+1][1];
        }
        dp[i][0] %= MOD;

        if (a[i+1] - a[i] < x[1]) {
            if (!qbad(i+2, nxt[i]-1)) {
                if (nxt[i] == n) {
                    dp[i][1] += 1;
                } else if (a[nxt[i]] - a[nxt[i]-1] < x[0]) {
                    if (nxt[i]+1 >= n || a[nxt[i]+1] - a[nxt[i]-1] >= x[0]) {
                        dp[i][1] += dp[nxt[i]][1];
                    }
                } else {
                    dp[i][1] += dp[nxt[i]][0] + dp[nxt[i]][1];
                }
            }
        } else {
            dp[i][1] += dp[i+1][0] + dp[i+1][1];
        }
        dp[i][1] %= MOD;
    }
    print((dp[0][0] + dp[0][1]) % MOD);
}