#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } const std::vector> grid_dxs{{1, 0}, {-1, 0}, {0, 1}, {0, -1}}; int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template T1 floor_div(T1 num, T2 den) { return (num > 0 ? num / den : -((-num + den - 1) / den)); } template std::pair operator+(const std::pair &l, const std::pair &r) { return std::make_pair(l.first + r.first, l.second + r.second); } template std::pair operator-(const std::pair &l, const std::pair &r) { return std::make_pair(l.first - r.first, l.second - r.second); } template std::vector sort_unique(std::vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template IStream &operator>>(IStream &is, std::vector &vec) { for (auto &v : vec) is >> v; return is; } template OStream &operator<<(OStream &os, const std::vector &vec); template OStream &operator<<(OStream &os, const std::array &arr); template OStream &operator<<(OStream &os, const std::unordered_set &vec); template OStream &operator<<(OStream &os, const pair &pa); template OStream &operator<<(OStream &os, const std::deque &vec); template OStream &operator<<(OStream &os, const std::set &vec); template OStream &operator<<(OStream &os, const std::multiset &vec); template OStream &operator<<(OStream &os, const std::unordered_multiset &vec); template OStream &operator<<(OStream &os, const std::pair &pa); template OStream &operator<<(OStream &os, const std::map &mp); template OStream &operator<<(OStream &os, const std::unordered_map &mp); template OStream &operator<<(OStream &os, const std::tuple &tpl); template OStream &operator<<(OStream &os, const std::vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template std::istream &operator>>(std::istream &is, std::tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template OStream &operator<<(OStream &os, const std::tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } #endif template OStream &operator<<(OStream &os, const std::unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template OStream &operator<<(OStream &os, const std::set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::pair &pa) { return os << '(' << pa.first << ',' << pa.second << ')'; } template OStream &operator<<(OStream &os, const std::map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template OStream &operator<<(OStream &os, const std::unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl #define dbgif(cond, x) ((cond) ? std::cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << std::endl : std::cerr) #else #define dbg(x) ((void)0) #define dbgif(cond, x) ((void)0) #endif struct ModIntRuntime { private: static int md; public: using lint = long long; static int mod() { return md; } int val_; static std::vector &facs() { static std::vector facs_; return facs_; } static int &get_primitive_root() { static int primitive_root_ = 0; if (!primitive_root_) { primitive_root_ = [&]() { std::set fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModIntRuntime(g).power((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root_; } static void set_mod(const int &m) { if (md != m) facs().clear(); md = m; get_primitive_root() = 0; } ModIntRuntime &_setval(lint v) { val_ = (v >= md ? v - md : v); return *this; } int val() const noexcept { return val_; } ModIntRuntime() : val_(0) {} ModIntRuntime(lint v) { _setval(v % md + md); } explicit operator bool() const { return val_ != 0; } ModIntRuntime operator+(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ + x.val_); } ModIntRuntime operator-(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ - x.val_ + md); } ModIntRuntime operator*(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ * x.val_ % md); } ModIntRuntime operator/(const ModIntRuntime &x) const { return ModIntRuntime()._setval((lint)val_ * x.inv().val() % md); } ModIntRuntime operator-() const { return ModIntRuntime()._setval(md - val_); } ModIntRuntime &operator+=(const ModIntRuntime &x) { return *this = *this + x; } ModIntRuntime &operator-=(const ModIntRuntime &x) { return *this = *this - x; } ModIntRuntime &operator*=(const ModIntRuntime &x) { return *this = *this * x; } ModIntRuntime &operator/=(const ModIntRuntime &x) { return *this = *this / x; } friend ModIntRuntime operator+(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md + x.val_); } friend ModIntRuntime operator-(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md - x.val_ + md); } friend ModIntRuntime operator*(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md * x.val_ % md); } friend ModIntRuntime operator/(lint a, const ModIntRuntime &x) { return ModIntRuntime()._setval(a % md * x.inv().val() % md); } bool operator==(const ModIntRuntime &x) const { return val_ == x.val_; } bool operator!=(const ModIntRuntime &x) const { return val_ != x.val_; } bool operator<(const ModIntRuntime &x) const { return val_ < x.val_; } // To use std::map friend std::istream &operator>>(std::istream &is, ModIntRuntime &x) { lint t; return is >> t, x = ModIntRuntime(t), is; } friend std::ostream &operator<<(std::ostream &os, const ModIntRuntime &x) { return os << x.val_; } lint power(lint n) const { lint ans = 1, tmp = this->val_; while (n) { if (n & 1) ans = ans * tmp % md; tmp = tmp * tmp % md; n /= 2; } return ans; } ModIntRuntime pow(lint n) const { return power(n); } ModIntRuntime inv() const { return this->pow(md - 2); } ModIntRuntime fac() const { int l0 = facs().size(); if (l0 > this->val_) return facs()[this->val_]; facs().resize(this->val_ + 1); for (int i = l0; i <= this->val_; i++) facs()[i] = (i == 0 ? ModIntRuntime(1) : facs()[i - 1] * ModIntRuntime(i)); return facs()[this->val_]; } ModIntRuntime doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModIntRuntime(k * 2).fac() / (ModIntRuntime(2).pow(k) * ModIntRuntime(k).fac()) : ModIntRuntime(k).fac() * ModIntRuntime(2).pow(k); } ModIntRuntime nCr(const ModIntRuntime &r) const { return (this->val_ < r.val_) ? ModIntRuntime(0) : this->fac() / ((*this - r).fac() * r.fac()); } ModIntRuntime sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (power((md - 1) / 2) != 1) return 0; ModIntRuntime b = 1; while (b.power((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModIntRuntime x = power((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModIntRuntime z = b.power(m); while (y != 1) { int j = 0; ModIntRuntime t = y; while (t != 1) j++, t *= t; z = z.power(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModIntRuntime(std::min(x.val_, md - x.val_)); } }; int ModIntRuntime::md = 1; using mint = ModIntRuntime; namespace matrix_ { struct has_id_method_impl { template static auto check(T_ *) -> decltype(T_::id(), std::true_type()); template static auto check(...) -> std::false_type; }; template struct has_id : decltype(has_id_method_impl::check(nullptr)) {}; } // namespace matrix_ template struct matrix { int H, W; std::vector elem; typename std::vector::iterator operator[](int i) { return elem.begin() + i * W; } inline T &at(int i, int j) { return elem[i * W + j]; } inline T get(int i, int j) const { return elem[i * W + j]; } int height() const { return H; } int width() const { return W; } std::vector> vecvec() const { std::vector> ret(H); for (int i = 0; i < H; i++) { std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i])); } return ret; } operator std::vector>() const { return vecvec(); } matrix() = default; matrix(int H, int W) : H(H), W(W), elem(H * W) {} matrix(const std::vector> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) { for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem)); } template ::value>::type * = nullptr> static T2 _T_id() { return T2::id(); } template ::value>::type * = nullptr> static T2 _T_id() { return T2(1); } static matrix Identity(int N) { matrix ret(N, N); for (int i = 0; i < N; i++) ret.at(i, i) = _T_id(); return ret; } matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; } matrix operator*(const T &v) const { matrix ret = *this; for (auto &x : ret.elem) x *= v; return ret; } matrix operator/(const T &v) const { matrix ret = *this; const T vinv = _T_id() / v; for (auto &x : ret.elem) x *= vinv; return ret; } matrix operator+(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; } matrix operator-(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; } matrix operator*(const matrix &r) const { matrix ret(H, r.W); for (int i = 0; i < H; i++) { for (int k = 0; k < W; k++) { for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j); } } return ret; } matrix &operator*=(const T &v) { return *this = *this * v; } matrix &operator/=(const T &v) { return *this = *this / v; } matrix &operator+=(const matrix &r) { return *this = *this + r; } matrix &operator-=(const matrix &r) { return *this = *this - r; } matrix &operator*=(const matrix &r) { return *this = *this * r; } bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; } bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; } bool operator<(const matrix &r) const { return elem < r.elem; } matrix pow(int64_t n) const { matrix ret = Identity(H); bool ret_is_id = true; if (n == 0) return ret; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!ret_is_id) ret *= ret; if ((n >> i) & 1) ret *= (*this), ret_is_id = false; } return ret; } std::vector pow_vec(int64_t n, std::vector vec) const { matrix x = *this; while (n) { if (n & 1) vec = x * vec; x *= x; n >>= 1; } return vec; }; matrix transpose() const { matrix ret(W, H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j); } return ret; } // Gauss-Jordan elimination // - Require inverse for every non-zero element // - Complexity: O(H^2 W) template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) != T2()) return j; } return -1; } matrix gauss_jordan() const { int c = 0; matrix mtr(*this); std::vector ws; ws.reserve(W); for (int h = 0; h < H; h++) { if (c == W) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < W; w++) { std::swap(mtr[piv][w], mtr[h][w]); mtr.at(piv, w) *= -_T_id(); // To preserve sign of determinant } } ws.clear(); for (int w = c; w < W; w++) { if (mtr.at(h, w) != T()) ws.emplace_back(w); } const T hcinv = _T_id() / mtr.at(h, c); for (int hh = 0; hh < H; hh++) if (hh != h) { const T coeff = mtr.at(hh, c) * hcinv; for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff; mtr.at(hh, c) = T(); } c++; } return mtr; } int rank_of_gauss_jordan() const { for (int i = H * W - 1; i >= 0; i--) { if (elem[i] != 0) return i / W + 1; } return 0; } int rank() const { return gauss_jordan().rank_of_gauss_jordan(); } T determinant_of_upper_triangle() const { T ret = _T_id(); for (int i = 0; i < H; i++) ret *= get(i, i); return ret; } int inverse() { assert(H == W); std::vector> ret = Identity(H), tmp = *this; int rank = 0; for (int i = 0; i < H; i++) { int ti = i; while (ti < H and tmp[ti][i] == 0) ti++; if (ti == H) { continue; } else { rank++; } ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]); T inv = _T_id() / tmp[i][i]; for (int j = 0; j < W; j++) ret[i][j] *= inv; for (int j = i + 1; j < W; j++) tmp[i][j] *= inv; for (int h = 0; h < H; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } friend std::vector operator*(const matrix &m, const std::vector &v) { assert(m.W == int(v.size())); std::vector ret(m.H); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j]; } return ret; } friend std::vector operator*(const std::vector &v, const matrix &m) { assert(int(v.size()) == m.H); std::vector ret(m.W); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j); } return ret; } std::vector prod(const std::vector &v) const { return (*this) * v; } std::vector prod_left(const std::vector &v) const { return v * (*this); } template friend OStream &operator<<(OStream &os, const matrix &x) { os << "[(" << x.H << " * " << x.W << " matrix)"; os << "\n[column sums: "; for (int j = 0; j < x.W; j++) { T s = 0; for (int i = 0; i < x.H; i++) s += x.get(i, j); os << s << ","; } os << "]"; for (int i = 0; i < x.H; i++) { os << "\n["; for (int j = 0; j < x.W; j++) os << x.get(i, j) << ","; os << "]"; } os << "]\n"; return os; } template friend IStream &operator>>(IStream &is, matrix &x) { for (auto &v : x.elem) is >> v; return is; } }; int run(int n, lint m) { // alpha^m / sqrtd の小数部分は beta^m / sqrtd の小数部分と等しい // double beta = quadratic_solver(1, -n, -1).first; // auto ret = pow(beta, m) / sqrt(D); // if (ret < 0) ret += 1; // return floorl(ret * 10000); static vector init{0, 1}; static matrix mat(2, 2); mat[0][1] = 1; mat[1][0] = 1; mat[1][1] = mint(n); auto ret = mat.pow_vec(m, init).front(); if (m % 2 == 1) ret -= 1; return ret.val(); } void jikken(int n) { double d = n * n + 4; double sqrtd = sqrt(d); double p = (n + sqrtd) / 2; double q = (n - sqrtd) / 2; dbg(d); dbg(make_pair(p, q)); dbg((p - q) / sqrtd); dbg((p * p - q * q) / sqrtd); dbg((p * p * p - q * q * q) / sqrtd); dbg((p * p * p * p - q * q * q * q) / sqrtd); dbg((p * p * p * p * p - q * q * q * q * q) / sqrtd); } int main() { mint::set_mod(10000); // jikken(2); // jikken(3); // jikken(4); // jikken(5); // jikken(6); int N; lint M, L; cin >> N >> M >> L; vector memo(10000, -1); vector f(10000, -1); memo.at(N) = 0; bool skipped = false; for (lint t = 1; t <= L; ++t) { if (f.at(N) >= 0) { N = f[N]; } else { f.at(N) = run(N, M); N = f.at(N); } if (!skipped and memo.at(N) >= 0) { const lint rec = memo.at(N) - t; const lint sk = floor_div(L - t, rec); if (sk > 0) t += sk * rec; skipped = true; } memo.at(N) = t; } string ret = to_string(N); while (ret.size() < 4) ret = "0" + ret; cout << ret << endl; }