verify/yukicoder/1559.test.cpp

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• Last update: 2024-11-09 02:16:49+09:00
• Problem: https://yukicoder.me/problems/no/1559

Depends on

• convolution (src/convolution/convolution.hpp)
• convolution_arbitrary (src/convolution/convolution_arbitrary.hpp)
• berlekamp_massey (src/fps/berlekamp_massey.hpp)
• bostan_mori (src/fps/bostan_mori.hpp)
• FormalPowerSeriesArbitrary (src/fps/formal_power_series_arbitrary.hpp)
• nth_term (src/fps/nth_term.hpp)
• pow_mod (src/math/pow_mod.hpp)
• primitive_root (src/math/primitive_root.hpp)
• StaticModint (src/template/static_modint.hpp)
• template (src/template/template.hpp)

Code

#define PROBLEM "https://yukicoder.me/problems/no/1559"
#include "../../src/template/template.hpp"
#include "../../src/template/static_modint.hpp"
#include "../../src/fps/formal_power_series_arbitrary.hpp"
#include "../../src/fps/nth_term.hpp"
using mint = modint1000000007;
using fps = FormalPowerSeriesArbitrary<mint>;
int main(void) {
    ll n, k;
    fps f(10);
    cin >> n >> f[0] >> f[1] >> k;
    rep(i, 2, 10) {
        f[i] = (f[i - 1] * f[i - 1] + k) / f[i - 2];
    }
    cout << nth_term(f, n - 1) << '\n';
}

#line 1 "verify/yukicoder/1559.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/1559"
#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/template/static_modint.hpp"
template <uint32_t m>
struct StaticModint {
    using mint = StaticModint;
    static constexpr uint32_t mod() {
        return m;
    }
    static constexpr mint raw(const uint32_t v) {
        mint a;
        a._v = v;
        return a;
    }
    constexpr StaticModint()
        : _v(0) {}
    template <class T>
    constexpr StaticModint(const T& v) {
        static_assert(is_integral_v<T>);
        if constexpr(is_signed_v<T>) {
            int64_t x = int64_t(v % int64_t(m));
            if(x < 0) x += m;
            _v = uint32_t(x);
        } else _v = uint32_t(v % m);
    }
    constexpr uint32_t val() const {
        return _v;
    }
    constexpr mint& operator++() {
        return *this += 1;
    }
    constexpr mint& operator--() {
        return *this -= 1;
    }
    constexpr mint operator++(int) {
        mint res = *this;
        ++*this;
        return res;
    }
    constexpr mint operator--(int) {
        mint res = *this;
        --*this;
        return res;
    }
    constexpr mint& operator+=(mint rhs) {
        if(_v >= m - rhs._v) _v -= m;
        _v += rhs._v;
        return *this;
    }
    constexpr mint& operator-=(mint rhs) {
        if(_v < rhs._v) _v += m;
        _v -= rhs._v;
        return *this;
    }
    constexpr mint& operator*=(mint rhs) {
        return *this = *this * rhs;
    }
    constexpr mint& operator/=(mint rhs) {
        return *this *= rhs.inv();
    }
    constexpr mint operator+() const {
        return *this;
    }
    constexpr mint operator-() const {
        return mint{} - *this;
    }
    constexpr mint pow(long long n) const {
        assert(0 <= n);
        if(n == 0) return 1;
        mint x = *this, r = 1;
        while(1) {
            if(n & 1) r *= x;
            n >>= 1;
            if(n == 0) return r;
            x *= x;
        }
    }
    constexpr mint inv() const {
        if constexpr(prime) {
            assert(_v);
            return pow(m - 2);
        } else {
            const auto eg = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(mint lhs, mint rhs) {
        return lhs += rhs;
    }
    friend constexpr mint operator-(mint lhs, mint rhs) {
        return lhs -= rhs;
    }
    friend constexpr mint operator*(mint lhs, mint rhs) {
        return uint64_t(lhs._v) * rhs._v;
    }
    friend constexpr mint operator/(mint lhs, mint rhs) {
        return lhs /= rhs;
    }
    friend constexpr bool operator==(mint lhs, mint rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(mint lhs, mint rhs) {
        return lhs._v != rhs._v;
    }
    friend istream& operator>>(istream& in, mint& x) {
        long long a;
        in >> a;
        x = a;
        return in;
    }
    friend ostream& operator<<(ostream& out, const mint& x) {
        return out << x.val();
    }

   private:
    uint32_t _v = 0;
    static constexpr bool prime = []() -> bool {
        if(m == 1) return 0;
        if(m == 2 or m == 7 or m == 61) return 1;
        if(m % 2 == 0) return 0;
        uint32_t d = m - 1;
        while(d % 2 == 0) d /= 2;
        for(uint32_t a : {2, 7, 61}) {
            uint32_t t = d;
            mint y = mint(a).pow(t);
            while(t != m - 1 && y != 1 && y != m - 1) {
                y *= y;
                t <<= 1;
            }
            if(y != m - 1 && t % 2 == 0) return 0;
        }
        return 1;
    }();
    static constexpr pair<int32_t, int32_t> inv_gcd(const int32_t a, const int32_t b) {
        if(a == 0) return {b, 0};
        int32_t s = b, t = a, m0 = 0, m1 = 1;
        while(t) {
            const int32_t u = s / t;
            s -= t * u;
            m0 -= m1 * u;
            swap(s, t);
            swap(m0, m1);
        }
        if(m0 < 0) m0 += b / s;
        return {s, m0};
    }
};
using modint998244353 = StaticModint<998244353>;
using modint1000000007 = StaticModint<1000000007>;
#line 3 "src/math/pow_mod.hpp"
constexpr long long pow_mod(long long x, long long n, const long long mod) {
    assert(n >= 0 and mod >= 1);
    x %= mod;
    if(x < 0) x += mod;
    long long res = 1;
    while(n > 0) {
        if(n & 1) res = res * x % mod;
        x = x * x % mod;
        n >>= 1;
    }
    return res;
}
#line 4 "src/math/primitive_root.hpp"
constexpr int primitive_root(const int m) {
    if(m == 2) return 1;
    if(m == 167772161) return 3;
    if(m == 469762049) return 3;
    if(m == 754974721) return 11;
    if(m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while(x % 2 == 0) x /= 2;
    for(int i = 3; (long long)(i)*i <= x; i += 2) {
        if(x % i == 0) {
            divs[cnt++] = i;
            while(x % i == 0) {
                x /= i;
            }
        }
    }
    if(x > 1) {
        divs[cnt++] = x;
    }
    for(int g = 2;; ++g) {
        bool ok = true;
        for(int i = 0; i < cnt; ++i) {
            if(pow_mod(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if(ok) return g;
    }
}
#line 4 "src/convolution/convolution.hpp"
constexpr int countr_zero(const unsigned int n) {
    int res = 0;
    while(!(n & (1 << res))) ++res;
    return res;
}
template <typename mint, int g = primitive_root(mint::mod())>
struct FFT_Info {
    static constexpr int rank2 = countr_zero(mint::mod() - 1);
    array<mint, rank2 + 1> root;
    array<mint, rank2 + 1> iroot;
    array<mint, max(0, rank2 - 2 + 1)> rate2;
    array<mint, max(0, rank2 - 2 + 1)> irate2;
    array<mint, max(0, rank2 - 3 + 1)> rate3;
    array<mint, max(0, rank2 - 3 + 1)> irate3;
    FFT_Info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for(int i = rank2 - 1; i >= 0; --i) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }
        {
            mint prod = 1, iprod = 1;
            for(int i = 0; i <= rank2 - 2; ++i) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            mint prod = 1, iprod = 1;
            for(int i = 0; i <= rank2 - 3; ++i) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
};
template <typename mint>
void butterfly(vector<mint>& a) {
    const int n = (int)a.size();
    const int h = __builtin_ctz((unsigned int)n);
    static const FFT_Info<mint> info;
    int len = 0;
    while(len < h) {
        if(h - len == 1) {
            const int p = 1 << (h - len - 1);
            mint rot = 1;
            for(int s = 0; s < (1 << len); ++s) {
                const int offset = s << (h - len);
                for(int i = 0; i < p; ++i) {
                    const auto l = a[i + offset];
                    const auto r = a[i + offset + p] * rot;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                if(s + 1 != (1 << len)) rot *= info.rate2[__builtin_ctz(~(unsigned int)(s))];
            }
            ++len;
        } else {
            const int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for(int s = 0; s < (1 << len); ++s) {
                const mint rot2 = rot * rot;
                const mint rot3 = rot2 * rot;
                const int offset = s << (h - len);
                for(int i = 0; i < p; ++i) {
                    const auto mod2 = 1ULL * mint::mod() * mint::mod();
                    const auto a0 = 1ULL * a[i + offset].val();
                    const auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
                    const auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
                    const auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
                    const auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    const auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if(s + 1 != (1 << len)) rot *= info.rate3[__builtin_ctz(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}
template <typename mint>
void butterfly_inv(vector<mint>& a) {
    const int n = (int)a.size();
    const int h = __builtin_ctz((unsigned int)n);
    static const FFT_Info<mint> info;
    int len = h;
    while(len) {
        if(len == 1) {
            const int p = 1 << (h - len);
            mint irot = 1;
            for(int s = 0; s < (1 << (len - 1)); ++s) {
                const int offset = s << (h - len + 1);
                for(int i = 0; i < p; ++i) {
                    const auto l = a[i + offset];
                    const auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] = (unsigned long long)(mint::mod() + l.val() - r.val()) * irot.val();
                }
                if(s + 1 != (1 << (len - 1))) irot *= info.irate2[__builtin_ctz(~(unsigned int)(s))];
            }
            --len;
        } else {
            const int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for(int s = 0; s < (1 << (len - 2)); ++s) {
                const mint irot2 = irot * irot;
                const mint irot3 = irot2 * irot;
                const int offset = s << (h - len + 2);
                for(int i = 0; i < p; ++i) {
                    const auto a0 = 1ULL * a[i + offset + 0 * p].val();
                    const auto a1 = 1ULL * a[i + offset + 1 * p].val();
                    const auto a2 = 1ULL * a[i + offset + 2 * p].val();
                    const auto a3 = 1ULL * a[i + offset + 3 * p].val();
                    const auto a2na3iimag = 1ULL * mint((mint::mod() + a2 - a3) * iimag.val()).val();
                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] = (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] = (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val();
                    a[i + offset + 3 * p] = (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) * irot3.val();
                }
                if(s + 1 != (1 << (len - 2))) irot *= info.irate3[__builtin_ctz(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}
template <typename mint>
vector<mint> convolution_naive(const vector<mint>& a, const vector<mint>& b) {
    const int n = (int)a.size(), m = (int)b.size();
    vector<mint> res(n + m - 1);
    if(n < m) {
        for(int j = 0; j < m; ++j) {
            for(int i = 0; i < n; ++i) {
                res[i + j] += a[i] * b[j];
            }
        }
    } else {
        for(int i = 0; i < n; ++i) {
            for(int j = 0; j < m; ++j) {
                res[i + j] += a[i] * b[j];
            }
        }
    }
    return res;
}
template <typename mint>
vector<mint> convolution(vector<mint> a, vector<mint> b) {
    const int n = (int)a.size(), m = (int)b.size();
    if(n == 0 or m == 0) return {};
    int z = 1;
    while(z < n + m - 1) z *= 2;
    assert((mint::mod() - 1) % z == 0);
    if(min(n, m) <= 60) return convolution_naive(a, b);
    a.resize(z);
    b.resize(z);
    butterfly(a);
    butterfly(b);
    for(int i = 0; i < z; ++i) a[i] *= b[i];
    butterfly_inv(a);
    a.resize(n + m - 1);
    const mint iz = mint(z).inv();
    for(int i = 0; i < n + m - 1; ++i) a[i] *= iz;
    return a;
}
#line 5 "src/convolution/convolution_arbitrary.hpp"
template <typename mint>
vector<mint> convolution_arbitrary(const vector<mint>& a, const vector<mint>& b) {
    const int n = (int)a.size(), m = (int)b.size();
    if(!n or !m) return {};
    static constexpr long long MOD1 = 754974721;
    static constexpr long long MOD2 = 167772161;
    static constexpr long long MOD3 = 469762049;
    static constexpr long long M1_inv_M2 = 95869806;
    static constexpr long long M12_inv_M3 = 187290749;
    static const long long M12_mod = MOD1 * MOD2 % mint::mod();
    using mint1 = StaticModint<(uint32_t)MOD1>;
    using mint2 = StaticModint<(uint32_t)MOD2>;
    using mint3 = StaticModint<(uint32_t)MOD3>;
    vector<mint1> a1(n), b1(m);
    vector<mint2> a2(n), b2(m);
    vector<mint3> a3(n), b3(m);
    for(int i = 0; i < n; ++i) a1[i] = a[i].val();
    for(int i = 0; i < n; ++i) a2[i] = a[i].val();
    for(int i = 0; i < n; ++i) a3[i] = a[i].val();
    for(int i = 0; i < m; ++i) b1[i] = b[i].val();
    for(int i = 0; i < m; ++i) b2[i] = b[i].val();
    for(int i = 0; i < m; ++i) b3[i] = b[i].val();
    vector<mint1> x = convolution<mint1>(a1, b1);
    vector<mint2> y = convolution<mint2>(a2, b2);
    vector<mint3> z = convolution<mint3>(a3, b3);
    vector<mint> c(n + m - 1);
    for(int i = 0; i < n + m - 1; ++i) {
        long long v1 = ((long long)y[i].val() - (long long)x[i].val()) * M1_inv_M2 % MOD2;
        if(v1 < 0) v1 += MOD2;
        long long v2 = ((long long)z[i].val() - ((long long)x[i].val() + MOD1 * v1) % MOD3) * M12_inv_M3 % MOD3;
        if(v2 < 0) v2 += MOD3;
        c[i] = (long long)x[i].val() + MOD1 * v1 + M12_mod * v2;
    }
    return c;
}
#line 4 "src/fps/formal_power_series_arbitrary.hpp"
template <typename mint>
struct FormalPowerSeriesArbitrary : vector<mint> {
    using vector<mint>::vector;
    using F = FormalPowerSeriesArbitrary;
    F& operator=(const vector<mint>& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] = g[i];
        return (*this);
    }
    F& operator-() {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= -1;
        return (*this);
    }
    F& operator+=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] += g[i];
        return (*this);
    }
    F& operator+=(const mint& r) {
        if((*this).empty()) (*this).resize(1);
        (*this)[0] += r;
        return (*this);
    }
    F& operator-=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] -= g[i];
        return (*this);
    }
    F& operator-=(const mint& r) {
        if((*this).empty()) (*this).resize(1);
        (*this)[0] -= r;
        return (*this);
    }
    F& operator*=(const F& g) {
        (*this) = convolution_arbitrary((*this), g);
        return (*this);
    }
    F& operator*=(const mint& r) {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= r;
        return (*this);
    }
    F& operator/=(const F& g) {
        if((*this).size() < g.size()) {
            (*this).clear();
            return (*this);
        }
        const int n = (*this).size() - g.size() + 1;
        (*this) = ((*this).rev().pre(n) * g.rev().inv(n)).pre(n).rev();
        return (*this);
    }
    F& operator/=(const mint& r) {
        const int n = (*this).size();
        const mint inv_r = r.inv();
        for(int i = 0; i < n; ++i) (*this)[i] *= inv_r;
        return (*this);
    }
    F& operator%=(const F& g) {
        (*this) -= (*this) / g * g;
        shrink();
        return (*this);
    }
    F operator*(const mint& g) const {
        return F(*this) *= g;
    }
    F operator-(const mint& g) const {
        return F(*this) -= g;
    }
    F operator+(const mint& g) const {
        return F(*this) += g;
    }
    F operator/(const mint& g) const {
        return F(*this) /= g;
    }
    F operator*(const F& g) const {
        return F(*this) *= g;
    }
    F operator-(const F& g) const {
        return F(*this) -= g;
    }
    F operator+(const F& g) const {
        return F(*this) += g;
    }
    F operator/(const F& g) const {
        return F(*this) /= g;
    }
    F operator%(const F& g) const {
        return F(*this) %= g;
    }
    F operator<<(const int d) const {
        F ret(*this);
        ret.insert(ret.begin(), d, mint(0));
        return ret;
    }
    F operator>>(const int d) const {
        const int n = (*this).size();
        if(n <= d) return {};
        F ret(*this);
        ret.erase(ret.begin(), ret.begin() + d);
        return ret;
    }
    void shrink() {
        while(!(*this).empty() and (*this).back() == mint(0)) (*this).pop_back();
    }
    F rev() const {
        F ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }
    F pre(const int deg) const {
        assert(deg >= 0);
        F ret(begin(*this), begin(*this) + min((int)(*this).size(), deg));
        if((int)ret.size() < deg) ret.resize(deg);
        return ret;
    }
    mint eval(const mint& a) const {
        const int n = (*this).size();
        mint x = 1, ret = 0;
        for(int i = 0; i < n; ++i) {
            ret += (*this)[i] * x;
            x *= a;
        }
        return ret;
    }
    void onemul(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n + d;
        if(deg > n) (*this).resize(deg);
        for(int i = deg - d - 1; i >= 0; --i) {
            (*this)[i + d] += (*this)[i] * c;
        }
    }
    void onediv(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(deg > n) (*this).resize(deg + 1);
        for(int i = 0; i < deg - d; ++i) {
            (*this)[i + d] -= (*this)[i] * c;
        }
    }
    F diff() const {
        const int n = (*this).size();
        F ret(max(0, n - 1));
        for(int i = 1; i < n; ++i) ret[i - 1] = (*this)[i] * i;
        return ret;
    }
    F integral() const {
        const int n = (*this).size(), mod = mint::mod();
        F ret(n + 1);
        ret[0] = mint(0);
        if(n > 0) ret[1] = mint(1);
        for(int i = 2; i <= n; ++i) ret[i] = (-ret[mod % i]) * (mod / i);
        for(int i = 0; i < n; ++i) ret[i + 1] *= (*this)[i];
        return ret;
    }
    F inv(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] != mint(0));
        if(deg == -1) deg = n;
        F ret({mint(1) / (*this)[0]});
        for(int i = 1; i < deg; i <<= 1) {
            ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
        }
        return ret.pre(deg);
    }
    F log(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] == mint(1));
        if(deg == -1) deg = n;
        return ((*this).diff() * (*this).inv(deg)).pre(deg - 1).integral();
    }
    F exp(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n == 0 or (*this)[0] == mint(0));
        if(deg == -1) deg = n;
        F ret({mint(1)});
        for(int i = 1; i < deg; i <<= 1) {
            ret = (ret * (pre(i << 1) + mint(1) - ret.log(i << 1))).pre(i << 1);
        }
        return ret.pre(deg);
    }
    F pow(const long long k, int deg = -1) const {
        assert(deg >= -1);
        assert(k >= 0);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(k == 0) {
            F ret(deg);
            if(deg) ret[0] = 1;
            return ret;
        }
        for(int i = 0; i < n; ++i) {
            if((*this)[i] != mint(0)) {
                const mint rev = mint(1) / (*this)[i];
                F ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(deg);
                if((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
            if(__int128_t(i + 1) * k >= deg) return F(deg, mint(0));
        }
        return F(deg, mint(0));
    }
    F shift(const mint& c) const {
        const int n = (*this).size();
        vector<mint> fact(n), ifact(n);
        fact[0] = ifact[0] = mint(1);
        for(int i = 1; i < n; ++i) fact[i] = fact[i - 1] * i;
        ifact[n - 1] = mint(1) / fact[n - 1];
        for(int i = n - 1; i > 1; --i) ifact[i - 1] = ifact[i] * i;
        F ret(*this);
        for(int i = 0; i < n; ++i) ret[i] *= fact[i];
        ret = ret.rev();
        F bs(n, mint(1));
        for(int i = 1; i < n; ++i) bs[i] = bs[i - 1] * c * ifact[i] * fact[i - 1];
        ret = (ret * bs).pre(n);
        ret = ret.rev();
        for(int i = 0; i < n; ++i) ret[i] *= ifact[i];
        return ret;
    }
};
#line 3 "src/fps/berlekamp_massey.hpp"
template <template <typename> typename FPS, typename mint>
FPS<mint> berlekamp_massey(const FPS<mint>& s) {
    const int n = (int)s.size();
    FPS<mint> b = {mint(-1)}, c = {mint(-1)};
    mint y = mint(1);
    for(int ed = 1; ed <= n; ++ed) {
        int l = (int)c.size(), m = (int)b.size();
        mint x = 0;
        for(int i = 0; i < l; ++i) x += c[i] * s[ed - l + i];
        b.emplace_back(0);
        ++m;
        if(x == mint(0)) continue;
        const mint freq = x / y;
        if(l < m) {
            const auto tmp = c;
            c.insert(begin(c), m - l, mint(0));
            for(int i = 0; i < m; ++i) c[m - 1 - i] -= freq * b[m - 1 - i];
            b = tmp;
            y = x;
        } else {
            for(int i = 0; i < m; ++i) c[l - 1 - i] -= freq * b[m - 1 - i];
        }
    }
    c.pop_back();
    c = c.rev();
    return c;
}
#line 3 "src/fps/bostan_mori.hpp"
template <template <typename> typename FPS, typename mint>
mint bostan_mori(const FPS<mint>& a, const FPS<mint>& c, long long k) {
    assert(k >= 0);
    if(k < (int)a.size()) return a[k];
    assert(a.size() >= c.size());
    FPS<mint> q = FPS<mint>{1} - (c << 1);
    FPS<mint> p = (a * q).pre((int)c.size());
    if(p.empty()) return 0;
    while(k > 0) {
        auto q2 = q;
        for(int i = 1; i < (int)q2.size(); i += 2) q2[i] = -q2[i];
        const auto s = p * q2;
        const auto t = q * q2;
        if(k & 1) {
            for(int i = 1; i < (int)s.size(); i += 2) p[i >> 1] = s[i];
            for(int i = 0; i < (int)t.size(); i += 2) q[i >> 1] = t[i];
        } else {
            for(int i = 0; i < (int)s.size(); i += 2) p[i >> 1] = s[i];
            for(int i = 0; i < (int)t.size(); i += 2) q[i >> 1] = t[i];
        }
        k >>= 1;
    }
    return p[0];
}
#line 5 "src/fps/nth_term.hpp"
template <template <typename> typename FPS, typename mint>
mint nth_term(const FPS<mint>& s, const long long n) {
    assert(n >= 0);
    const FPS<mint> c = berlekamp_massey(s);
    return bostan_mori(s, c, n);
}
#line 6 "verify/yukicoder/1559.test.cpp"
using mint = modint1000000007;
using fps = FormalPowerSeriesArbitrary<mint>;
int main(void) {
    ll n, k;
    fps f(10);
    cin >> n >> f[0] >> f[1] >> k;
    rep(i, 2, 10) {
        f[i] = (f[i - 1] * f[i - 1] + k) / f[i - 2];
    }
    cout << nth_term(f, n - 1) << '\n';
}

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