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 INF_LL (int64)1e18
#define INF (int32)1e9
#define REP(i, n) for(int64 i = 0;i < (n);i++)
#define FOR(i, a, b) for(int64 i = (a);i < (b);i++)
#define all(x) x.begin(),x.end()
#define fs first
#define sc second

using int32 = int_fast32_t;
using uint32 = uint_fast32_t;
using int64 = int_fast64_t;
using uint64 = uint_fast64_t;
using PII = pair<int32, int32>;
using PLL = pair<int64, int64>;

const double eps = 1e-10;

template<typename A, typename B>inline void chmin(A &a, B b){if(a > b) a = b;}
template<typename A, typename B>inline void chmax(A &a, B b){if(a < b) a = b;}

template<typename T>
vector<T> make_v(size_t a){return vector<T>(a);}

template<typename T,typename... Ts>
auto make_v(size_t a,Ts... ts){
  return vector<decltype(make_v<T>(ts...))>(a,make_v<T>(ts...));
}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value!=0>::type
fill_v(U &u,const V... v){u=U(v...);}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value==0>::type
fill_v(U &u,const V... v){
  for(auto &e:u) fill_v<T>(e,v...);
}


template<::std::uint_fast64_t mod>
class ModInt{
private:
  using value_type = ::std::uint_fast64_t;
  value_type n;
public:
  ModInt() : n(0) {}
  ModInt(value_type n_) : n(n_ % mod) {}
  ModInt(const ModInt& m) : n(m.n) {}

  template<typename T>
  explicit operator T() const { return static_cast<T>(n); }
  value_type get() const { return n; }

  friend ::std::ostream& operator<<(::std::ostream &os, const ModInt<mod> &a) {
    return os << a.n;
  }

  friend ::std::istream& operator>>(::std::istream &is, ModInt<mod> &a) {
    value_type x;
    is >> x;
    a = ModInt<mod>(x);
    return is;
  }

  bool operator==(const ModInt& m) const { return n == m.n; }
  bool operator!=(const ModInt& m) const { return n != m.n; }
  ModInt& operator*=(const ModInt& m){ n = n * m.n % mod; return *this; }

  ModInt pow(value_type b) const{
    ModInt ans = 1, m = ModInt(*this);
    while(b){
      if(b & 1) ans *= m;
      m *= m;
      b >>= 1;
    }
    return ans;
  }

  ModInt inv() const { return (*this).pow(mod-2); }
  ModInt& operator+=(const ModInt& m){ n += m.n; n = (n < mod ? n : n - mod); return *this; }
  ModInt& operator-=(const ModInt& m){ n += mod - m.n; n = (n < mod ? n : n - mod); return *this; }
  ModInt& operator/=(const ModInt& m){ *this *= m.inv(); return *this; }
  ModInt operator+(const ModInt& m) const { return ModInt(*this) += m; }
  ModInt operator-(const ModInt& m) const { return ModInt(*this) -= m; }
  ModInt operator*(const ModInt& m) const { return ModInt(*this) *= m; }
  ModInt operator/(const ModInt& m) const { return ModInt(*this) /= m; }
  ModInt& operator++(){ n += 1; return *this; }
  ModInt& operator--(){ n -= 1; return *this; }
  ModInt operator++(int){
    ModInt old(n);
    n += 1;
    return old;
  }
  ModInt operator--(int){
    ModInt old(n);
    n -= 1;
    return old;
  }
  ModInt operator-() const { return ModInt(mod-n); }
};
template<::std::size_t size, ::std::uint_fast64_t mod=1000000007>
class Factorial{
private:
  using value_type = ModInt<mod>;
  ::std::vector<value_type> fact, inv;
public:
  Factorial() : fact(size+1, 1), inv(size+1, 1){
    for(::std::size_t i = 1; i <= size; ++i){
      fact[i] = fact[i-1] * value_type(i);
      inv[i] = fact[i].inv();
    }
  }

  value_type comb(::std::int64_t a, ::std::int64_t b){
    assert(a >= b);
    assert(b >= 0);
    return fact[a]*inv[b]*inv[a-b];
  }

  value_type& operator[](::std::size_t k){ return fact[k]; }
};

class UnionFind{
private:
  ::std::vector<int32_t> par;
  size_t n;

public:
  UnionFind(){}
  UnionFind(size_t n):n(n){
    par.resize(n, -1);
  }

  uint32_t find(uint32_t x){
    return par[x] < 0 ? x : par[x] = find(par[x]);
  }

  size_t size(uint32_t x){
    return -par[find(x)];
  }

  bool unite(uint32_t x, uint32_t y){
    x = find(x);
    y = find(y);
    if(x == y) return false;
    if(size(x) < size(y)) std::swap(x, y);
    par[x] += par[y];
    par[y] = x;
    return true;
  }

  bool same(uint32_t x, uint32_t y){
    return find(x) == find(y);
  }
};


template<class ValueMonoid, class OperatorMonoid, class Modifier,
  template<class...> class Container=::std::vector>
class LazySegTree{
public:
  using value_structure = ValueMonoid;
  using value_type = typename value_structure::value_type;
  using operator_structure = OperatorMonoid;
  using operator_type = typename operator_structure::value_type;
  using modifier = Modifier;
  using const_reference = const value_type &;
  using container_value_type = Container<value_type>;
  using container_operator_type = Container<operator_type>;
  using size_type = typename container_value_type::size_type;

private:
  container_value_type tree;
  container_operator_type lazy;
  size_type size_, height;

  static size_type getsize(const size_type x){
    size_type ret = 1;
    while(ret < x)
      ret <<= 1;
    return ret;
  }

  static size_type getheight(const size_type x){
    size_type ret = 0;
    while((static_cast<size_type>(1) << ret) < x){
      ret++;
    }
    return ret;
  }

  inline static value_type calc(const value_type a, const value_type b){
    return value_structure::operation(a, b);
  }

  inline static void apply(operator_type &data, const operator_type a){
    data = operator_structure::operation(data, a);
  }

  inline static value_type reflect(const value_type v, const operator_type o){
    return modifier::operation(v, o);
  }

  void push(const size_type index){
    tree[index] = reflect(tree[index], lazy[index]);
    apply(lazy[index << 1], lazy[index]);
    apply(lazy[index << 1 | 1], lazy[index]);
    lazy[index] = operator_structure::identity();
  }

  void calc_node(const size_type index){
    if(tree.size() <= (index << 1 | 1)) return;
    assert(0 < index);
    tree[index] = calc(reflect(tree[index << 1],  lazy[index << 1]),
                       reflect(tree[index << 1 | 1], lazy[index << 1 | 1]));
  }

  void build(size_type index){
    while(index >>= 1){
      calc_node(index);
    }
  }

  void propagate(const size_type index){
    for(size_type shift = height; shift ; --shift){
      push(index >> shift);
    }
  }

  void rebuild(){
    for(size_type i = size_-1;i > 0;--i){
      calc_node(i);
    }
  }
public:
  LazySegTree() : size_(0), height(0), tree(), lazy(){}
  LazySegTree(const size_type size)
    : size_(size), height(getheight(size)),
      tree(size << 1, value_structure::initializer()),
      lazy(size << 1, operator_structure::identity()){
    rebuild();
  }
  template<class InputIterator>
  LazySegTree(InputIterator first, InputIterator last)
    : size_(::std::distance(first, last)){
    height = getheight(size_);
    tree = container_value_type(size_, value_structure::identity());
    lazy = container_operator_type(size_ << 1, operator_structure::identity());
    tree.insert(tree.end(), first, last);
    rebuild();
  }

  size_type size() const { return size_; }
  const_reference operator[](const size_type k){
    assert(k < size_);
    propagate(k+size_);
    tree[k+size_] = reflect(tree[k+size_], lazy[k+size_]);
    lazy[k+size_] = operator_structure::identity();
    return tree[k+size_];
  }

  value_type query(size_type l, size_type r){
    assert(l <= r);
    assert(0 <= l && l < size_);
    assert(0 <= r && r <= size_);
    value_type retl = value_structure::identity(),
      retr = value_structure::identity();
    l += size_;
    r += size_;
    propagate(l);
    propagate(r-1);
    build(l);
    build(r-1);
    for(; l < r ; l >>= 1, r >>= 1){
      if(l&1){
        retl = calc(retl, reflect(tree[l], lazy[l]));
        l++;
      }
      if(r&1){
        r--;
        retr = calc(reflect(tree[r], lazy[r]), retr);
      }
    }
    return calc(retl, retr);
  }

  void update(size_type l, size_type r, const operator_type& data){
    assert(l <= r);
    assert(0 <= l && l < size_);
    assert(0 <= r && r <= size_);
    l += size_;
    r += size_;
    propagate(l);
    propagate(r - 1);
    for(size_type l_ = l, r_ = r; l_ < r_ ; l_ >>= 1, r_ >>= 1){
      if(l_ & 1) apply(lazy[l_++], data);
      if(r_ & 1) apply(lazy[--r_], data);
    }
    build(l);
    build(r - 1);
  }

  template<class F>
  void update(size_type index, const F& f){
    assert(0 <= index && index < size());
    index += size_;
    propagate(index);
    tree[index] = f(::std::move(tree[index]));
    lazy[index] = operator_structure::identity();
    build(index);
  }

  /*
  template<class F>
  size_type search(const F& f) const { // [0, result) is True and [0, result-1) is not.
    if(f(value_structure::identity()))
      return 0;
    if(!f(tree[1]))
      return size_+1;
    value_type acc = value_structure::identity();
    size_type i = 1;
    while(i <
  }
  */
};

constexpr int64 mod = 1e9+7;
using Mint = ModInt<mod>;

class v_monoid {
public:
  using value_type = Mint;
  static value_type identity() { return 0; }
  static value_type initializer() { return 0; }
  static value_type operation(const value_type& a, const value_type& b) {
    return a + b;
  }
};

class o_monoid {
public:
  using value_type = Mint;
  static value_type identity() { return 0; }
  static value_type operation(const value_type& a, const value_type& b) {
    return a + b;
  }
};

class modifier {
public:
  static Mint operation(const Mint& a, const Mint& b) {
    return a + b;
  }
};

template<typename T>
T sum(const vector<T>& v, int64 l, int64 r) { // sum[l, r)
  if (r <= 0) return 0;
  if (l <= 0) return v[r-1];
  return v[r-1]-v[l-1];
}

int main(void) {
  int64 N, A[2];
  cin >> N >> A[0] >> A[1];
  vector<int64> S(N+1), DS[2];
  S[0] = -INF_LL;
  DS[0].resize(N+2);
  DS[1].resize(N+2);
  DS[0][0] = 0;
  DS[1][0] = 0;
  REP(i, N) {
    cin >> S[i+1];
    DS[0][i+1] += (S[i+1] - S[i]) < A[0];
    DS[1][i+1] += (S[i+1] - S[i]) < A[1];
    DS[0][i+2] += DS[0][i+1];
    DS[1][i+2] += DS[1][i+1];
  }

  S.push_back(INF_LL);
  LazySegTree<v_monoid, o_monoid, modifier> dp[2], dp2[2];
  REP(i, 2) {
    dp[i] = LazySegTree<v_monoid, o_monoid, modifier>(N+2);
    dp2[i] = LazySegTree<v_monoid, o_monoid, modifier>(N+2);
  }
//  REP(i, 2) {
//    REP(j, N+1) {
//      cout << DS[i][j] << " ";
//    }
//    cout << endl;
//  }

  auto dpp = make_v<Mint>(2, N+2);
  dp[0].update(0, 1, 1);
  dp[1].update(0, 1, 1);
  dp2[0].update(0, 1, 1);
  dp2[1].update(0, 1, 1);

  FOR(i, 1, N+1) {
    REP(j, 2) {
      if (S[i] - S[i-1] >= A[j]) dpp[j][i] += dpp[j][i-1] + dp[j].query(i-1, i);

      int64 l = lower_bound(all(DS[1-j]), DS[1-j][i-1])-DS[1-j].begin()-1, r = upper_bound(all(S), S[i]-A[j])-S.begin();
      chmin(r, i-1);
      if (l < 0) l = 0;
//      cout << i << " " << j << ": ";
      if (l <= r) {
//        cout << "(" << l << " " << r << ") ";
        dp[j].update(i, i+1, dp2[j].query(l, r));
      }

      l = lower_bound(all(DS[j]), DS[j][i])-DS[j].begin(); r = upper_bound(all(S), S[i+1]-A[1-j])-S.begin()+1;
      if (l < 0) l = 0;
      chmin(r, N+1);
//      cout << "(" << l << " " << r << ") " << endl;
      if (l <= r) {
        dp2[j].update(i, i+1, dp[j].query(l, r));
      }
    }
  }
//  REP(i, 2) {
//    REP(j, N+1) {
//      cout << "(" << dp[i].query(j, j+1) << " " << dpp[i][j] << ") ";
//    }
//    cout << endl;
//  }
//  REP(i, 2) {
//    REP(j, N+1) {
//      cout << dp2[i].query(j, j+1) << " ";
//    }
//    cout << endl;
//  }
  cout << dp[0].query(N, N+1) + dp[1].query(N, N+1) + dpp[0][N] + dpp[1][N] << endl;
}