#include using namespace std; using ll = long long; using ld = long double; using pll = pair; using tlll = tuple; constexpr ll INF = 1LL << 60; template bool chmin(T& a, T b) {if (a > b) {a = b; return true;} return false;} template bool chmax(T& a, T b) {if (a < b) {a = b; return true;} return false;} ll safemod(ll A, ll M) {ll res = A % M; if (res < 0) res += M; return res;} ll divfloor(ll A, ll B) {if (B < 0) A = -A, B = -B; return (A - safemod(A, B)) / B;} ll divceil(ll A, ll B) {if (B < 0) A = -A, B = -B; return divfloor(A + B - 1, B);} ll pow_ll(ll A, ll B) {if (A == 0 || A == 1) {return A;} if (A == -1) {return B & 1 ? -1 : 1;} ll res = 1; for (int i = 0; i < B; i++) {res *= A;} return res;} ll mul_limited(ll A, ll B, ll M = INF) { return A > M / B ? M : A * B; } ll pow_limited(ll A, ll B, ll M = INF) { if (A == 0 || A == 1) {return A;} ll res = 1; for (int i = 0; i < B; i++) {if (res > M / A) return M; res *= A;} return res;} ll logfloor(ll A, ll B) {assert(A >= 2); ll res = 0; for (ll tmp = 1; tmp <= B / A; tmp *= A) {res++;} return res;} ll logceil(ll A, ll B) {assert(A >= 2); ll res = 0; for (ll tmp = 1; tmp < B; tmp *= A) {res++;} return res;} ll arisum_ll(ll a, ll d, ll n) { return n * a + (n & 1 ? ((n - 1) >> 1) * n : (n >> 1) * (n - 1)) * d; } ll arisum2_ll(ll a, ll l, ll n) { return n & 1 ? ((a + l) >> 1) * n : (n >> 1) * (a + l); } ll arisum3_ll(ll a, ll l, ll d) { assert((l - a) % d == 0); return arisum2_ll(a, l, (l - a) / d + 1); } template void unique(vector &V) {V.erase(unique(V.begin(), V.end()), V.end());} template void sortunique(vector &V) {sort(V.begin(), V.end()); V.erase(unique(V.begin(), V.end()), V.end());} #define FINALANS(A) do {cout << (A) << '\n'; exit(0);} while (false) template void printvec(const vector &V) {int _n = V.size(); for (int i = 0; i < _n; i++) cout << V[i] << (i == _n - 1 ? "" : " ");cout << '\n';} template void printvect(const vector &V) {for (auto v : V) cout << v << '\n';} template void printvec2(const vector> &V) {for (auto &v : V) printvec(v);} //* #include using namespace atcoder; //using mint = modint998244353; //using mint = modint1000000007; //using mint = modint; //*/ // https://codeforces.com/blog/entry/61306 template vector BerlekampMassey(const vector &A) { int N = A.size(); vector B(0), C(0); int pos; T x; for (int i = 0; i < N; i++) { int d = C.size(); T y = A[i]; for (int j = 0; j < d; j++) y -= C[j] * A[i - 1 - j]; if (y == 0) continue; if (C.empty()) { C.assign(i + 1, 0); pos = i; x = y; continue; } T z = y / x; int d2 = i - pos + B.size(); vector tmp; if (d2 >= d) { tmp = C; C.resize(d2); } C[i - 1 - pos] += z; for (int j = 0; j < (int)B.size(); j++) C[i - pos + j] -= z * B[j]; if (d2 >= d) { pos = i; x = y; B = tmp; } } return C; } template vector convolution_anymod(const vector &A, const vector &B, const int MOD) { int N = A.size(), M = B.size(); if (min(N, M) <= 100) // 100 は適当, これから調整する { internal::barrett ba(MOD); vector C(N + M - 1, 0); for (int i = 0; i < N; i++) { for (int j = 0; j < M; j++) { C[i + j] += ba.mul(A[i], B[j]); if (C[i + j] >= MOD) C[i + j] -= MOD; } } return C; } const int MOD1 = 167772161, MOD2 = 469762049, MOD3 = 1224736769; using mint1 = dynamic_modint<100>; using mint2 = dynamic_modint<101>; using mint3 = dynamic_modint<102>; using mint4 = dynamic_modint<103>; mint1::set_mod(MOD1); mint2::set_mod(MOD2); mint3::set_mod(MOD3); mint4::set_mod(MOD); auto C1 = convolution(A, B); auto C2 = convolution(A, B); auto C3 = convolution(A, B); vector C(N + M - 1); for (ll i = 0; i < N + M - 1; i++) { int c1 = C1[i], c2 = C2[i], c3 = C3[i]; int t1 = ((mint2::raw(c2) - mint2::raw(c1)) / mint2::raw(MOD1)).val(); mint3 x2_m3 = mint3::raw(c1) + mint3::raw(t1) * mint3::raw(MOD1); mint4 x2_m = mint4::raw(c1) + mint4::raw(t1) * mint4::raw(MOD1); int t2 = ((mint3::raw(c3) - x2_m3) / (mint3::raw(MOD1) * mint3::raw(MOD2))).val(); C[i] = (x2_m + mint4::raw(t2) * mint4::raw(MOD1) * mint4::raw(MOD2)).val(); } return C; } template vector convolution_anymod(const vector &A, const vector &B) { int N = A.size(), M = B.size(); if (min(N, M) <= 100) // 100 は適当, これから調整する { internal::barrett ba(MOD); vector C(N + M - 1, 0); for (int i = 0; i < N; i++) { for (int j = 0; j < M; j++) { C[i + j] += ba.mul(A[i], B[j]); if (C[i + j] >= MOD) C[i + j] -= MOD; } } return C; } const int MOD1 = 167772161, MOD2 = 469762049, MOD3 = 1224736769; using mint1 = dynamic_modint<100>; using mint2 = dynamic_modint<101>; using mint3 = dynamic_modint<102>; using mint4 = dynamic_modint<103>; mint1::set_mod(MOD1); mint2::set_mod(MOD2); mint3::set_mod(MOD3); mint4::set_mod(MOD); auto C1 = convolution(A, B); auto C2 = convolution(A, B); auto C3 = convolution(A, B); vector C(N + M - 1); for (ll i = 0; i < N + M - 1; i++) { int c1 = C1[i], c2 = C2[i], c3 = C3[i]; int t1 = ((mint2::raw(c2) - mint2::raw(c1)) / mint2::raw(MOD1)).val(); mint3 x2_m3 = mint3::raw(c1) + mint3::raw(t1) * mint3::raw(MOD1); mint4 x2_m = mint4::raw(c1) + mint4::raw(t1) * mint4::raw(MOD1); int t2 = ((mint3::raw(c3) - x2_m3) / (mint3::raw(MOD1) * mint3::raw(MOD2))).val(); C[i] = (x2_m + mint4::raw(t2) * mint4::raw(MOD1) * mint4::raw(MOD2)).val(); } return C; } template vector> convolution_anymod(const vector> &A, const vector> &B) { int N = A.size(), M = B.size(); vector A2(N), B2(M); for (int i = 0; i < N; i++) A2[i] = A[i].val(); for (int i = 0; i < M; i++) B2[i] = B[i].val(); vector C2 = convolution_anymod(A2, B2); vector> C(N + M - 1); for (int i = 0; i < N + M - 1; i++) C[i] = static_modint::raw(C2[i]); return C; } template vector> convolution_anymod(const vector> &A, const vector> &B) { int N = A.size(), M = B.size(); vector A2(N), B2(M); for (int i = 0; i < N; i++) A2[i] = A[i].val(); for (int i = 0; i < M; i++) B2[i] = B[i].val(); vector C2 = convolution_anymod(A2, B2, dynamic_modint::mod()); vector> C(N + M - 1); for (int i = 0; i < N + M - 1; i++) C[i] = dynamic_modint::raw(C2[i]); return C; } // https://opt-cp.com/fps-implementation/ // https://qiita.com/hotman78/items/f0e6d2265badd84d429a // https://opt-cp.com/fps-fast-algorithms/ // https://maspypy.com/%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%83%BB%E5%BD%A2%E5%BC%8F%E7%9A%84%E3%81%B9%E3%81%8D%E7%B4%9A%E6%95%B0-%E9%AB%98%E9%80%9F%E3%81%AB%E8%A8%88%E7%AE%97%E3%81%A7%E3%81%8D%E3%82%8B%E3%82%82%E3%81%AE template struct FormalPowerSeries : vector { using vector::vector; using vector::operator=; using F = FormalPowerSeries; using S = vector>; FormalPowerSeries(const S &f, int n = -1) { if (n == -1) n = f.back().first + 1; (*this).assign(n, T(0)); for (auto [d, a] : f) (*this)[d] += a; } F operator-() const { F res(*this); for (auto &a : res) a = -a; return res; } F operator*=(const T &k) { for (auto &a : *this) a *= k; return *this; } F operator*(const T &k) const { return F(*this) *= k; } friend F operator*(const T k, const F &f) { return f * k; } F operator/=(const T &k) { *this *= k.inv(); return *this; } F operator/(const T &k) const { return F(*this) /= k; } F &operator+=(const F &g) { int n = (*this).size(), m = g.size(); (*this).resize(max(n, m), T(0)); for (int i = 0; i < m; i++) (*this)[i] += g[i]; return *this; } F operator+(const F &g) const { return F(*this) += g; } F &operator-=(const F &g) { int n = (*this).size(), m = g.size(); (*this).resize(max(n, m), T(0)); for (int i = 0; i < m; i++) (*this)[i] -= g[i]; return *this; } F operator-(const F &g) const { return F(*this) -= g; } F &operator<<=(const ll d) { int n = (*this).size(); (*this).insert((*this).begin(), min(ll(n), d), T(0)); (*this).resize(n); return *this; } F operator<<(const ll d) const { return F(*this) <<= d; } F &operator>>=(const ll d) { int n = (*this).size(); (*this).erase((*this).begin(), (*this).begin() + min(ll(n), d)); (*this).resize(n, T(0)); return *this; } F operator>>(const ll d) const { return F(*this) >>= d; } F &operator*=(const S &g) { int n = (*this).size(); auto [d, c] = g.front(); if (d != 0) c = 0; for (int i = n - 1; i >= 0; i--) { (*this)[i] *= c; for (auto &[j, b] : g) { if (j == 0) continue; if (j > i) break; (*this)[i] += (*this)[i - j] * b; } } return *this; } F operator*(const S &g) { return F(*this) *= g; } F &operator/=(const S &g) { int n = (*this).size(); auto [d, c] = g.front(); assert(d == 0 && c != T(0)); T inv_c = c.inv(); for (int i = 0; i < n; i++) { for (auto &[j, b] : g) { if (j == 0) continue; if (j > i) break; (*this)[i] -= (*this)[i - j] * b; } (*this)[i] *= inv_c; } return *this; } F operator/(const S &g) { return F(*this) /= g; } template F convolution2(const vector> &A, const vector> &B, const int d = -1) { F res; if (is_ntt_friendly) res = convolution(A, B); else res = convolution_anymod(A, B); if (d != -1 && (int)res.size() > d) res.resize(d); return res; } template F convolution2(const vector> &A, const vector> &B, const int d = -1) { F res; res = convolution_anymod(A, B); if (d != -1 && (int)res.size() > d) res.resize(d); return res; } F &operator*=(const F &g) { int n = (*this).size(); if (n == 0) return *this; *this = convolution2(*this, g, n); return *this; } F operator*(const F &g) const { return F(*this) *= g; } template void butterfly2(FormalPowerSeries, true> &A) const { internal::butterfly(A); } template void butterfly2(FormalPowerSeries, false> &A) const { assert(false); } template void butterfly2(FormalPowerSeries, false> &A) const { assert(false); } template void butterfly_inv2(FormalPowerSeries, true> &A) const { internal::butterfly_inv(A); } template void butterfly_inv2(FormalPowerSeries, false> &A) const { assert(false); } template void butterfly_inv2(FormalPowerSeries, false> &A) const { assert(false); } F inv(int d = -1) const { int n = (*this).size(); assert(n != 0 && (*this).front() != 0); if (d == -1) d = n; assert(d > 0); F g{(*this).front().inv()}; while (g.size() < d) { if (is_ntt_friendly) { int m = g.size(); F f = {(*this).begin(), (*this).begin() + min(n, 2 * m)}; F g2(g); f.resize(2 * m, T(0)), butterfly2(f); g2.resize(2 * m, T(0)), butterfly2(g2); for (int i = 0; i < 2 * m; i++) f[i] *= g2[i]; butterfly_inv2(f); f.erase(f.begin(), f.begin() + m); f.resize(2 * m, T(0)), butterfly2(f); for (int i = 0; i < 2 * m; i++) f[i] *= g2[i]; butterfly_inv2(f); T inv_z = T(2 * m).inv(); inv_z *= -inv_z; for (int i = 0; i < m; i++) f[i] *= inv_z; g.insert(g.end(), f.begin(), f.begin() + m); } else { g.resize(2 * g.size(), T(0)); g *= F{T(2)} - g * (*this); } } return {g.begin(), g.begin() + d}; } F &operator/=(const F &g) { *this *= g.inv(); return *this; } F operator/(const F &g) const { return F(*this) *= g.inv(); } }; // [x^N] P(x)/Q(x) を求める(P の次数は Q の次数より小さい) template T bostan_mori(const FormalPowerSeries &P, const FormalPowerSeries &Q, ll N) { using F = FormalPowerSeries; if (N == 0) return P[0] / Q[0]; int d = (int)Q.size() - 1; assert((int)P.size() <= d); F P2 = F(P); P2.resize(d, T(0)); F Q3 = F(Q); for (int i = 1; i <= d; i += 2) Q3[i] = -Q3[i]; F U, V; if (is_ntt_friendly) { int z = 1; while (z < (1 << (2 * d + 1))) z <<= 1; F Q2 = F(Q); P2.resize(z), Q2.resize(z), Q3.resize(z); P2.butterfly2(P2), Q2.butterfly2(Q2), Q3.butterfly2(Q3); for (int i = 0; i < z; i++) P2[i] *= Q3[i], Q2[i] *= Q3[i]; P2.butterfly_inv2(P2), Q2.butterfly_inv2(Q2); T iz = T(z).inv(); for (int i = 0; i <= 2 * d; i++) P2[i] *= iz, Q2[i] *= iz; U = F(P2), V = F(Q2); } else U = U.convolution2(P2, Q3), V = V.convolution2(Q, Q3); F U2(d), V2(d + 1); for (int i = 0; i <= d; i++) V2[i] = V[2 * i]; if (N & 1) { for (int i = 0; i < d; i++) U2[i] = U[2 * i + 1]; } else { for (int i = 0; i < d; i++) U2[i] = U[2 * i]; } return bostan_mori(U2, V2, N / 2); } // a_n = sum[i = 1..d] c_i a_{n-i}(n ≥ d)を満たすとき、a_N を求める(A は 0-indexed で C は 1-indexed) template T linear_recurrence(const vector &A, const vector &C, ll N) { using F = FormalPowerSeries; int d = C.size(); assert(A.size() >= d); F Ga(d), Q(d + 1); Q[0] = 1; for (int i = 0; i < d; i++) Ga[i] = A[i], Q[i + 1] = -C[i]; F P = Ga * Q; return bostan_mori(P, Q, N); } /* using mint = modint998244353; const bool ntt = true; //*/ //* using mint = modint1000000007; const bool ntt = false; //*/ /* using mint = modint; const bool ntt = false; //*/ using fps = FormalPowerSeries; int main() { ll N, X, Y, K; cin >> N >> X >> Y >> K; N--; vector A(100); A.at(0) = X; A.at(1) = Y; for (ll i = 2; i < 100; i++) { A.at(i) = (A.at(i - 1) * A.at(i - 1) + K) / A.at(i - 2); } vector C = BerlekampMassey(A); mint ans = linear_recurrence(A, C, N); cout << ans.val() << endl; }