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>
#pragma GCC diagnostic ignored "-Wsign-compare"
#pragma GCC diagnostic ignored "-Wsign-conversion"

#define NDEBUG
using i32 = int32_t;
using i64 = int64_t;
using u32 = uint32_t;
using u64 = uint64_t;
using uint = unsigned int;
using usize = std::size_t;
using ll = long long;
using ull = unsigned long long;
using ld = long double;
template<typename T> constexpr T popcount(const T u) { return u ? static_cast<T>(__builtin_popcountll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T log2p1(const T u) { return u ? static_cast<T>(64 - __builtin_clzll(static_cast<u64>(u))) : static_cast<T>(0); }
template<typename T> constexpr T msbp1(const T u) { return log2p1(u); }
template<typename T> constexpr T lsbp1(const T u) { return __builtin_ffsll(u); }
template<typename T> constexpr T clog(const T u) { return u ? log2p1(u - 1) : static_cast<T>(u); }
template<typename T> constexpr bool ispow2(const T u) { return u and (static_cast<u64>(u) & static_cast<u64>(u - 1)) == 0; }
template<typename T> constexpr T ceil2(const T u) { return static_cast<T>(1) << clog(u); }
template<typename T> constexpr T floor2(const T u) { return u == 0 ? static_cast<T>(0) : static_cast<T>(1) << (log2p1(u) - 1); }
template<typename T> constexpr bool btest(const T mask, const usize ind) { return ((static_cast<u64>(mask) >> ind) & static_cast<u64>(1)); }
template<typename T> constexpr T bcut(const T mask, const usize ind) { return ind == 0 ? static_cast<T>(0) : static_cast<T>((static_cast<u64>(mask) << (64 - ind)) >> (64 - ind)); }
template<typename T> bool chmin(T& a, const T& b) { return (a > b ? a = b, true : false); }
template<typename T> bool chmax(T& a, const T& b) { return (a < b ? a = b, true : false); }
constexpr unsigned int mod                  = 1000000007;
template<typename T> constexpr T inf_v      = std::numeric_limits<T>::max() / 4;
template<typename Real> constexpr Real pi_v = Real{3.141592653589793238462643383279502884};


/**
 * http://xoshiro.di.unimi.it/xoshiro128starstar.c
 * http://xoshiro.di.unimi.it/xoshiro256starstar.c
 * http://xoshiro.di.unimi.it/splitmix64.c
 */
class xoshiro
{
public:
    using result_type = uint32_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro() : xoshiro(std::random_device{}()) {}
    xoshiro(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 33)) * 0x62A9D9ED799705F5, z = (z ^ (z >> 28)) * 0xCB24D0A5C88C35B3, s[i] = static_cast<result_type>(z >> 32); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 9;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 11), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (32 - k)); }
};
class xoshiro_64
{
public:
    using result_type = uint64_t;
    static constexpr result_type min() { return std::numeric_limits<result_type>::min(); }
    static constexpr result_type max() { return std::numeric_limits<result_type>::max(); }
    xoshiro_64() : xoshiro_64(std::random_device{}()) {}
    xoshiro_64(uint64_t seed)
    {
        uint64_t z = 0;
        for (int i = 0; i < 4; i++) { z = (seed += 0x9e3779b97f4a7c15), z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9, z = (z ^ (z >> 27)) * 0x94d049bb133111eb, s[i] = static_cast<result_type>(z ^ (z >> 31)); }
    }
    result_type operator()()
    {
        const result_type result = rotl(s[1] * 5, 7) * 9, t = s[1] << 17;
        return s[2] ^= s[0], s[3] ^= s[1], s[1] ^= s[2], s[0] ^= s[3], s[2] ^= t, s[3] = rotl(s[3], 45), result;
    }
    void discard(const usize rep)
    {
        for (usize i = 0; i < rep; i++) { (*this)(); }
    }

private:
    result_type s[4];
    static result_type rotl(const result_type x, const int k) { return (x << k) | (x >> (64 - k)); }
};
template<typename Rng>
class rng_base
{
public:
    using rng_type    = Rng;
    using result_type = typename rng_type::result_type;
    static constexpr result_type min() { return rng_type::min(); }
    static constexpr result_type max() { return rng_type::max(); }
    rng_base() : rng_base(std::random_device{}()) {}
    rng_base(const u64 seed) : rng(seed) {}
    ~rng_base() = default;
    result_type operator()() { return rng(); }
    template<typename Int>
    Int uniform_int(const Int max)
    {
        static_assert(std::is_integral<Int>::value, "Integer type is needed.");
        if (ispow2(max + 1)) { return static_cast<Int>(rng() & max); }
        const result_type mask = static_cast<result_type>(ceil2(static_cast<u64>(max + 1))) - 1;
        while (true) {
            const Int ans = static_cast<Int>(rng() & mask);
            if (ans <= max) { return ans; }
        }
    }
    template<typename Int> Int uniform_int(const Int min, const Int max) { return min + uniform_int(static_cast<Int>(max - min)); }
    bool uniform_bool() { return uniform_int<bool>(0, 1); }
    template<typename Real>
    Real uniform_real()
    {
        static_assert(std::is_floating_point<Real>::value, "FloatingPoint type is needed.");
        return static_cast<Real>(uniformInt(~static_cast<result_type>(0)) / static_cast<Real>(~static_cast<result_type>(0)));
    }
    template<typename Real> Real uniform_real(const Real min, const Real max) { return assert(min <= max), uniform_real<Real>() * (max - min) + min; }
    template<typename Int> std::pair<Int, Int> uniform_int_pair(const Int min, const Int max) { return std::minmax(uniform_int(min, max), uniform_int(min, max)); }
    template<typename Int>
    std::vector<Int> uniform_int_vec(const std::size_t size, const Int min, const Int max)
    {
        std::vector<Int> v(size);
        for (std::size_t i = 0; i < size; i++) { v[i] = uniformInt(min, max); }
        return v;
    }
    template<typename Real>
    std::vector<Real> uniform_real_vec(const std::size_t size, const Real min, const Real max)
    {
        std::vector<Real> v(size);
        for (std::size_t i = 0; i < size; i++) { v[i] = uniformReal(min, max); }
        return v;
    }
    void discard(const usize rep) { rng.discard(rep); }

private:
    Rng rng;
};
using rng_mt        = rng_base<std::mt19937>;
using rng_mt64      = rng_base<std::mt19937_64>;
using rng_xoshiro   = rng_base<xoshiro>;
using rng_xoshiro64 = rng_base<xoshiro_64>;
rng_mt g_rng_mt;
rng_mt64 g_rng_mt64;
rng_xoshiro g_rng_xo;
rng_xoshiro64 g_rng_xo64;
template<typename T>
T read()
{
    T v;
    return std::cin >> v, v;
}
template<typename T>
std::vector<T> read_vec(const std::size_t size)
{
    std::vector<T> v(size);
    for (auto& e : v) { std::cin >> e; }
    return v;
}
#define SHOW(...) static_cast<void>(0)
template<typename T>
std::vector<T> make_v(const std::size_t size, T v) { return std::vector<T>(size, v); }
template<class... Args>
auto make_v(const std::size_t size, Args... args) { return std::vector<decltype(make_v(args...))>(size, make_v(args...)); }


template<typename T> T gcd(const T& a, const T& b) { return (a > b ? gcd(b, a) : a == 0 ? b : gcd(b % a, a)); }
template<typename T> T lcm(const T& a, const T& b) { return a / gcd(a, b) * b; }
template<typename T>
constexpr std::pair<T, T> extgcd(const T a, const T b)
{
    if (b == 0) { return std::pair<T, T>{1, 0}; }
    const auto g = gcd(a, b), da = std::abs(b) / g;
    const auto p = extgcd(b, a % b);
    const auto x = (da + p.second % da) % da, y = (g - a * x) / b;
    return {x, y};
}
template<typename T>
constexpr T inverse(const T a, const T mod) { return extgcd(a, mod).first; }
template<uint mod_value, bool dynamic = false>
class modint_base
{
public:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, const UInt> mod() { return mod_ref(); }
    template<typename UInt = uint>
    static constexpr std::enable_if_t<not dynamic, const UInt> mod() { return mod_value; }
    template<typename UInt = uint>
    static void set_mod(const std::enable_if_t<dynamic, const UInt> mod) { mod_ref() = mod; }
    modint_base() : v{0} {}
    modint_base(const ll val) : v{norm(static_cast<uint>(val % static_cast<ll>(mod()) + static_cast<ll>(mod())))} {}
    modint_base(const modint_base& n) : v{n()} {}
    explicit operator bool() const { return v != 0; }
    bool operator!() const { return not static_cast<bool>(*this); }
    modint_base& operator=(const modint_base& m) { return v = m(), (*this); }
    modint_base& operator=(const ll val) { return v = norm(uint(val % static_cast<ll>(mod()) + static_cast<ll>(mod()))), (*this); }
    friend modint_base operator+(const modint_base& m) { return m; }
    friend modint_base operator-(const modint_base& m) { return make(norm(mod() - m.v)); }
    friend modint_base operator+(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + m2.v)); }
    friend modint_base operator-(const modint_base& m1, const modint_base& m2) { return make(norm(m1.v + mod() - m2.v)); }
    friend modint_base operator*(const modint_base& m1, const modint_base& m2) { return make(static_cast<uint>(static_cast<ll>(m1.v) * static_cast<ll>(m2.v) % static_cast<ll>(mod()))); }
    friend modint_base operator/(const modint_base& m1, const modint_base& m2) { return m1 * inv(m2.v); }
    friend modint_base operator+(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) - val}; }
    friend modint_base operator*(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const modint_base& m, const ll val) { return modint_base{static_cast<ll>(m.v) * inv(val)}; }
    friend modint_base operator+(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) + val}; }
    friend modint_base operator-(const ll val, const modint_base& m) { return modint_base{-static_cast<ll>(m.v) + val}; }
    friend modint_base operator*(const ll val, const modint_base& m) { return modint_base{static_cast<ll>(m.v) * (val % static_cast<ll>(mod()))}; }
    friend modint_base operator/(const ll val, const modint_base& m) { return modint_base{val * inv(static_cast<ll>(m.v))}; }
    friend modint_base& operator+=(modint_base& m1, const modint_base& m2) { return m1 = m1 + m2; }
    friend modint_base& operator-=(modint_base& m1, const modint_base& m2) { return m1 = m1 - m2; }
    friend modint_base& operator*=(modint_base& m1, const modint_base& m2) { return m1 = m1 * m2; }
    friend modint_base& operator/=(modint_base& m1, const modint_base& m2) { return m1 = m1 / m2; }
    friend modint_base& operator+=(modint_base& m, const ll val) { return m = m + val; }
    friend modint_base& operator-=(modint_base& m, const ll val) { return m = m - val; }
    friend modint_base& operator*=(modint_base& m, const ll val) { return m = m * val; }
    friend modint_base& operator/=(modint_base& m, const ll val) { return m = m / val; }
    friend modint_base operator^(const modint_base& m, const ll n) { return power(m.v, n); }
    friend modint_base& operator^=(modint_base& m, const ll n) { return m = m ^ n; }
    friend bool operator==(const modint_base& m1, const modint_base& m2) { return m1.v == m2.v; }
    friend bool operator!=(const modint_base& m1, const modint_base& m2) { return not(m1 == m2); }
    friend bool operator==(const modint_base& m, const ll val) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const modint_base& m, const ll val) { return not(m == val); }
    friend bool operator==(const ll val, const modint_base& m) { return m.v == norm(static_cast<uint>(static_cast<ll>(mod()) + val % static_cast<ll>(mod()))); }
    friend bool operator!=(const ll val, const modint_base& m) { return not(m == val); }
    friend std::istream& operator>>(std::istream& is, modint_base& m)
    {
        ll v;
        return is >> v, m = v, is;
    }
    friend std::ostream& operator<<(std::ostream& os, const modint_base& m) { return os << m(); }
    uint operator()() const { return v; }

private:
    template<typename UInt = uint>
    static std::enable_if_t<dynamic, UInt&> mod_ref()
    {
        static UInt mod = 0;
        return mod;
    }
    static uint norm(const uint x) { return x < mod() ? x : x - mod(); }
    static modint_base make(const uint x)
    {
        modint_base m;
        return m.v = x, m;
    }
    static modint_base power(modint_base x, ull n)
    {
        modint_base ans = 1;
        for (; n; n >>= 1, x *= x) {
            if (n & 1) { ans *= x; }
        }
        return ans;
    }
    static modint_base inv(const modint_base& x) { return modint_base{inverse(static_cast<ll>(x.v), static_cast<ll>(mod()))}; }

    uint v;
};
template<uint mod>
using modint = modint_base<mod, false>;
template<uint id>
using dynamic_modint = modint_base<id, true>;

template<typename MonoidAct>
class lazyseg
{
public:
    using monoid_act_type      = MonoidAct;
    using value_monoid_type    = typename monoid_act_type::value_monoid_type;
    using operator_monoid_type = typename monoid_act_type::operator_monoid_type;
    using value_type           = typename value_monoid_type::value_type;
    using operator_type        = typename operator_monoid_type::operator_type;
    lazyseg(const usize sz, const value_type initial = value_monoid_type::id()) : sz{sz}, depth{clog(sz)}, half{static_cast<usize>(1) << depth}, val(half << 1, value_monoid_type::id()), op(half << 1, operator_monoid_type::id())
    {
        if (initial != value_monoid_type::id()) {
            std::fill(val.begin() + half, val.end(), initial);
            for (usize i = half - 1; i >= 1; i--) { up(i); }
        }
    }
    template<typename InIt>
    lazyseg(const InIt first, const InIt last) : sz{static_cast<usize>(std::distance(first, last))}, depth{clog(sz)}, half{static_cast<usize>(1) << depth}, val(half << 1, value_monoid_type::id()), op(half << 1, operator_monoid_type::id())
    {
        std::copy(first, last, val.begin() + half);
        for (usize i = half - 1; i >= 1; i--) { up(i); }
    }
    value_type get(const usize a) { return assert(a < sz), fold(a, a + 1); }
    void set(usize a, const value_type& v)
    {
        assert(a < sz);
        top_down(a += half), top_down(a + 1), op[a] = operator_monoid_type::id(), val[a] = v;
        while (a >>= 1) { up(a); }
    }
    value_type fold(usize l, usize r)
    {
        assert(l < r), assert(r <= sz);
        top_down(l += half), top_down(r += half);
        value_type accl = value_monoid_type::id(), accr = value_monoid_type::id();
        for (; l < r; l >>= 1, r >>= 1) {
            if (l & 1) { accl = value_monoid_type::merge(accl, val[l++]); }
            if (r & 1) { accr = value_monoid_type::merge(val[--r], accr); }
        }
        return value_monoid_type::merge(accl, accr);
    }
    void act(usize l, usize r, const operator_type& f)
    {
        assert(l < r), assert(r <= sz);
        const usize lin = l + half, rin = r + half;
        top_down(l += half), top_down(r += half);
        for (usize ls = 1, rs = 1; l < r; l >>= 1, r >>= 1, ls <<= 1, rs <<= 1) {
            if (l & 1) { update(l++, f, ls); }
            if (r & 1) { update(--r, f, rs); }
        }
        bottom_up(lin), bottom_up(rin);
    }
    friend std::ostream& operator<<(std::ostream& os, lazyseg& lseg)
    {
        os << "[";
        for (usize i = 0; i < lseg.sz; i++) { os << lseg.get(i) << (i + 1 == lseg.sz ? "" : ","); }
        return (os << "]\n");
    }

private:
    void up(const usize i) { val[i] = value_monoid_type::merge(val[i << 1], val[i << 1 | 1]); }
    void update(const usize a, const operator_type& f, const usize l) { op[a] = operator_monoid_type::compose(f, op[a]), val[a] = monoid_act_type::apply(f, val[a], l); }
    void down(const usize a, const usize l) { update(a << 1, op[a], l >> 1), update(a << 1 | 1, op[a], l >> 1), op[a] = operator_monoid_type::id(); }
    void top_down(const usize a)
    {
        const usize b = (a / (a & -a)) >> 1;
        for (usize i = 0, l = half; i < depth; i++, l >>= 1) {
            const usize v = a >> (depth - i);
            if (v > b) { break; }
            down(v, l);
        }
    }
    void bottom_up(usize a)
    {
        a = (a / (a & -a)) >> 1;
        for (; a >= 1; a >>= 1) { up(a); }
    }
    const usize sz, depth, half;
    std::vector<value_type> val;
    std::vector<operator_type> op;
};
template<typename Element>
struct sum
{
    using element_type = Element;
    using value_type   = element_type;
    sum()              = delete;
    static value_type merge(const value_type& a, const value_type& b) { return a + b; }
    static constexpr value_type id() { return value_type{}; }
};
template<typename ElemType>
struct assign
{
    using element_type = ElemType;
    using value_type   = element_type;
    assign()           = delete;
    static value_type merge(const value_type& a, const value_type&) { return a; }
};
template<typename BaseSemigroup>
struct monoid
{
    using base_semigroup_type = BaseSemigroup;
    using value_type          = std::pair<bool, typename base_semigroup_type::value_type>;
    using operator_type       = value_type;
    monoid()                  = delete;
    static value_type merge(const value_type& a, const value_type& b) { return a.first ? (b.first ? value_type{true, base_semigroup_type::merge(a.second, b.second)} : a) : (b.first ? b : id()); }
    static operator_type compose(const operator_type& a, const operator_type& b) { return merge(a, b); }
    static constexpr value_type id() { return value_type{}; }
};
using mint = modint<mod>;
struct monoid_act
{
    using value_monoid_type    = sum<mint>;
    using operator_monoid_type = monoid<assign<mint>>;
    using value_type           = typename value_monoid_type::value_type;
    using operator_type        = typename operator_monoid_type::operator_type;
    monoid_act()               = delete;
    template<typename Ind>
    static value_type apply(const operator_type& f, const value_type& x, const Ind l) { return f.first ? f.second * l : x; }
};
int main()
{
    const auto n = read<usize>();
    const auto a = read<ll>(), b = read<ll>();
    std::vector<ll> l{a, b};
    std::vector<ll> s{-inf_v<ll>, -inf_v<ll> / 2};
    for (usize i = 0; i < n; i++) { s.push_back(read<ll>()); }
    SHOW(s);
    auto r = make_v(n + 2, 2, 0UL);
    for (usize i = 2; i < n + 2; i++) {
        for (usize j = 0; j < 2; j++) {
            const ll sup = s[i] - l[j];
            r[i][j]      = std::upper_bound(s.begin(), s.end(), sup) - s.begin();
        }
    }
    lazyseg<monoid_act> dp0(n + 2, 0), dp1(n + 2, 0);
    dp0.set(0, 1);
    for (usize i = 2; i < n + 2; i++) {
        std::vector<mint> sums(2);
        for (usize j = 0; j < 2; j++) {
            const usize imax = r[i][j];
            sums[j]          = (j == 0 ? dp1 : dp0).fold(0, imax);
        }
        for (usize j = 0; j < 2; j++) {
            if (s[i] - s[i - 1] < l[j]) { (j == 0 ? dp0 : dp1).act(0, n + 2, {true, 0}); }
            (j == 0 ? dp0 : dp1).set(i - 1, (j == 0 ? dp0 : dp1).get(i - 1) + sums[j]);
        }
    }
    std::cout << dp0.fold(0, n + 2) + dp1.fold(0, n + 2) << std::endl;
    return 0;
}