#pragma region competitive_programming #ifdef __LOCAL #define _GLIBCXX_DEBUG #endif #pragma GCC optimize("Ofast") #include //#include "modint_plus.hpp" //typedef atcoder::modint1000000007 mint; //typedef atcoder::modint998244353 mint; //#include //#include "Formal_Power_Series.hpp" //typedef tomo0608::Formal_Power_Series fps; #include namespace tomo0608 { #define is_prime(n) atcoder::internal::is_prime_constexpr(n) std::map prime_factorization(long long n) { std::map res; n = abs(n); assert(n > 1); for (long long i = 2; i * i <= n; i++) { if (n % i == 0) { int num = 0; while (n % i == 0) { num++; n /= i; } res[i] = num; } } if (n != 1)res[n] = 1; return res; } class Osa_k { public: const int MAX_N; std::vector min_prime; Osa_k(int size) : MAX_N(size), min_prime(size + 1) { if (MAX_N < 2)return; std::vector primes{ 2, 3 }; for (int i = 2; i <= MAX_N; i += 2)min_prime[i] = 2; for (int i = 3; i <= MAX_N; i += 6)min_prime[i] = 3; for (int i = 5, d = 4; i <= MAX_N; i += d = 6 - d) { if (!min_prime[i]) { min_prime[i] = i; primes.emplace_back(i); } for (int j = 2; j < (int)primes.size() && i * primes[j] <= MAX_N; j++) { min_prime[i * primes[j]] = primes[j]; if (primes[j] == min_prime[i])break; } } } std::vector> factorization(int n) { assert(0 < n && n <= MAX_N); std::vector> res; while (n > 1) { int p = min_prime[n]; int num = 0; while (min_prime[n] == p) { num++; n /= p; } res.emplace_back(p, num); } return res; } }; std::vector prime_enumerate(int N) { std::vector sieve(N / 3 + 1, 1); for (int p = 5, d = 4, i = 1, sqn = std::sqrt(N); p <= sqn; p += d = 6 - d, i++) { if (!sieve[i]) continue; for (int q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p, qe = sieve.size();q < qe; q += r = s - r)sieve[q] = 0; } std::vector ret{ 2, 3 }; for (int p = 5, d = 4, i = 1; p <= N; p += d = 6 - d, i++) if (sieve[i]) ret.push_back(p); while (!ret.empty() && ret.back() > N) ret.pop_back(); return ret; } template void div_zeta(std::vector& a) { std::vector sieve = prime_enumerate((int)a.size() - 1); for (auto& p : sieve)for (int i = 1; i * p < a.size(); i++)a[i * p] += a[i]; } template void div_mobius(std::vector& a) { std::vector sieve = prime_enumerate((int)a.size() - 1); for (auto& p : sieve)for (int i = ((int)a.size() - 1) / p; i > 0; i--)a[i * p] -= a[i]; } template void mul_zeta(std::vector& a) { std::vector sieve = prime_enumerate((int)a.size() - 1); for (auto& p : sieve)for (int i = ((int)a.size() - 1) / p; i > 0; i--)a[i] += a[i * p]; } template void mul_mobius(std::vector& a) { std::vector sieve = prime_enumerate((int)a.size() - 1); for (auto& p : sieve)for (int i = 1; i * p < a.size(); i++)a[i] -= a[i * p]; } template std::vector gcd_convolution(std::vector a, std::vector b){ mul_zeta(a); mul_zeta(b); for(int i = 0; i < a.size(); i++)a[i] *= b[i]; mul_mobius(a); return a; } template std::vector lcm_convolution(std::vector a, std::vector b){ div_zeta(a); div_zeta(b); for(int i = 0; i < a.size(); i++)a[i] *= b[i]; div_mobius(a); return a; } long long euler_phi(long long n) { long long res = n; for (long long i = 2; i * i <= n; i++) { if (n % i == 0) { res -= res / i; while (n % i == 0)n /= i; } } if (n > 1)res -= res / n; return res; } #if __cplusplus >= 201703L template long long discrete_logarithm(Monoid f, Set s, Set t, long long MAX_N) {// return the smallest integer n lower than MAX_N such that f^n s = t if exists, otherwise -1 static_assert(std::is_convertible_v>, "action must work as Set(Monoid, Set)"); static_assert(std::is_convertible_v>, "composition must work as Monoid(Monoid, Monoid)"); static_assert(std::is_convertible_v>, "id must work as Monoid()"); #else template long long discrete_logarithm(Monoid f, Set s, Set t, long long MAX_N) { #endif if (s == t)return 0; long long SQRT_N = sqrt(MAX_N) + 1; std::set T; Set f_t = t; for (int i = 1; i <= SQRT_N; i++) { f_t = action(f, f_t); T.emplace(f_t); } Monoid f_m = id(); { long long n = SQRT_N; Monoid _f = f; while(n > 0){ if(n&1)f_m = composition(f_m, _f); _f = composition(_f, _f); n >>= 1; } } int num_of_failures = 0; Set pre_s = s; for (int k = 1; k <= SQRT_N; k++) { Set now_s = action(f_m, pre_s); if (T.count(now_s)) { Set lhs = pre_s; for (int i = 0; i < SQRT_N; i++, lhs = action(f, lhs)) { if (lhs == t)return (k - 1) * SQRT_N + i; } num_of_failures++; if (num_of_failures == 2)return -1; } std::swap(pre_s, now_s); } return -1; }; } namespace tomo0608 { typedef long long ll; typedef long double ld; template using V = std::vector; template using VV = V>; template using VVV = V>; typedef std::pair pii; typedef std::pair pll; templatevoid input(T&... a) { (std::cin >> ... >> a); }; #define INT(...) int __VA_ARGS__; IN(__VA_ARGS__) #define LL(...) long long __VA_ARGS__; IN(__VA_ARGS__) #define CHAR(...) char __VA_ARGS__; IN(__VA_ARGS__) #define STR(...) string __VA_ARGS__; IN(__VA_ARGS__) #define DBL(...) double __VA_ARGS__; IN(__VA_ARGS__) #define VEC(type, name, size) std::vector name(size);IN(name) #define VECVEC(type, name, h, w) std::vector> name(h, std::vector(w));IN(name) template std::istream& operator>>(std::istream& is, std::pair& p) { is >> p.first >> p.second; return is; } template std::ostream& operator<<(std::ostream& os, const std::pair& p) { os << '(' << p.first << ", " << p.second << ')'; return os; } template std::istream& operator>>(std::istream& is, std::vector& v) { for (auto& e : v) is >> e; return is; } template std::ostream& operator<<(std::ostream& os, const std::vector& v) { for (auto& e : v) os << e << ' '; return os; } template std::ostream& operator << (std::ostream& os, std::set& set_var) { os << "{"; for (auto itr = set_var.begin(); itr != set_var.end(); itr++) { os << *itr;++itr;if (itr != set_var.end()) os << ", ";itr--; }os << "}";return os; } template std::ostream& operator<<(std::ostream& os, std::map& map_var) { os << "{";for (auto itr = map_var.begin(); itr != map_var.end(); itr++) { os << *itr;itr++;if (itr != map_var.end()) os << ", ";itr--; }os << "}";return os; } template void print_tuple(std::ostream&, const T&) {} template void print_tuple(std::ostream& os, const T& t) { if (i){os << ", ";} os << std::get(t); print_tuple(os, t); } template std::ostream& operator<<(std::ostream& os, const std::tuple& t) { os << "{"; print_tuple<0, std::tuple, Args...>(os, t); return os << "}"; } void IN() {} template void IN(Head& head, Tail &...tail) { std::cin >> head; IN(tail...); } void print() { std::cout << '\n'; } templatevoid print(const T& a, const Ts&... b) { std::cout << a; (std::cout << ... << (std::cout << ' ', b)); std::cout << '\n'; } void drop() { std::cout << '\n';exit(0); } templatevoid drop(const T& a, const Ts&... b) { std::cout << a; (std::cout << ... << (std::cout << ' ', b)); std::cout << '\n';exit(0); } #ifdef __LOCAL void debug_out() { std::cerr << std::endl; } template < class Head, class... Tail> void debug_out(Head H, Tail... T) { std::cerr << ' ' << H; debug_out(T...); } #define debug(...) std::cerr << 'L' << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__) #define dump(x) std::cerr << 'L' << __LINE__ << " " << #x << " = " << (x) << std::endl #else #define debug(...) (void(0)) #define dump(x) (void(0)) #endif #define rep1(a) for(long long i = 0; i < (a); i++) #define rep2(i, a) for(long long i = 0; i < (a); i++) #define rep3(i, a, b) for(long long i = (a); i < (b); i++) #define rep4(i, a, b, c) for(long long i = (a); i < (b); i += (c)) #define drep1(a) for(long long i = (a)-1; i >= 0; i--) #define drep2(i, a) for(long long i = (a)-1; i >= 0; i--) #define drep3(i, a, b) for(long long i = (a)-1; i >= (b); i--) #define drep4(i, a, b, c) for(long long i = (a)-1; i >= (b); i -= (c)) #define overload4(a, b, c, d, e, ...) e #define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__) #define drep(...) overload4(__VA_ARGS__, drep4, drep3, drep2, drep1)(__VA_ARGS__) #define endl '\n' struct io_setup { io_setup() { std::cin.tie(0); std::ios_base::sync_with_stdio(false); std::cout << std::setprecision(20); } }io_setup; } // namespace tomo0608 namespace tomo0608 { #define ALL(x) x.begin(),x.end() template T SUM(const S& v) { return accumulate(ALL(v), T(0)); } #define MIN(v) *min_element(ALL(v)) #define MAX(v) *max_element(ALL(v)) #define SORT(v) sort(ALL(v)) #define REVERSE(v) reverse(ALL(v)) #define RSORT(v) sort(ALL(v)); reverse(ALL(v)) #define UNIQUE(x) SORT(x), x.erase(unique(ALL(x)), x.end()) #define lb(c, x) distance((c).begin(), lower_bound(ALL(c), (x))) #define ub(c, x) distance((c).begin(), upper_bound(ALL(c), (x))) template void zip(std::vector& x) { std::vector y = x;UNIQUE(y);for (int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); } } template using priority_queue_rev = std::priority_queue, std::greater>; template inline bool chmax(T& a, const U& b) { if (a < b) { a = b; return 1; } return 0; } template inline bool chmin(T& a, const U& b) { if (a > b) { a = b; return 1; } return 0; } template inline int count_between(std::vector& a, T l, T r) { return lower_bound(ALL(a), r) - lower_bound(ALL(a), l); } // [l, r) template int bittest(T n, int k) { return (n >> k) & 1; } int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); } int topbit(long long t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); } int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); } int lowbit(long long a) { return a == 0 ? 64 : __builtin_ctzll(a); } #define perm(v) for(bool permrepflag = true; (permrepflag ? exchange(permrepflag, false) : next_permutation(ALL(v)));) template T ceil(T x, S y) { assert(y); return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y)); } template T floor(T x, S y) { assert(y); return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1))); } template std::vector id_sort(const std::vector& v, bool greater = false) { int n = v.size(); std::vector ret(n); std::iota(ALL(ret), 0); std::sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; }); return ret; } } //using namespace atcoder; using namespace std; using namespace tomo0608; int dx[8] = { 1, 0, -1, 0, 1, 1, -1, -1 }; int dy[8] = { 0, 1, 0, -1, 1, -1, -1, 1 }; // インタラクティブ問題のときは出力するたびにcout.flush();を忘れない!!!!! void solve(); int main() { int codeforces = 1; //cin >> codeforces; while (codeforces--) { solve(); } return 0; } #pragma endregion typedef array Set; typedef array Monoid; const ll MOD = 998244353; Set action(Monoid M, Set s){ Set res; res[0] = M[0] * s[0] + M[1] * s[1]; res[1] = M[2] * s[0] + M[3] * s[1]; res[0] = (res[0] + MOD)%MOD; res[1] = (res[1] + MOD)%MOD; return res; }; Monoid composition(Monoid M_1, Monoid M_2){ Monoid res; res[0] = M_1[0] * M_2[0] + M_1[1] * M_2[2]; res[1] = M_1[0] * M_2[1] + M_1[1] * M_2[3]; res[2] = M_1[2] * M_2[0] + M_1[3] * M_2[2]; res[3] = M_1[2] * M_2[1] + M_1[3] * M_2[3]; res[0] = (res[0] + MOD)%MOD; res[1] = (res[1] + MOD)%MOD; res[2] = (res[2] + MOD)%MOD; res[3] = (res[3] + MOD)%MOD; return res; }; Monoid id(){ Monoid e; e[0] = 1; e[1] = 0; e[2] = 0; e[3] = 1; return e; }; void solve() { LL(a, b, p, q); Monoid f; f[0] = a; f[1] = MOD-b; f[2] = 1; f[3] = 0; Set s, t; s[0] = (a*a-2*b+MOD)%MOD; s[1] = a; t[0] = p; t[1] = q; ll MAX_N = 10000000001; print(discrete_logarithm(f, s, t, MAX_N)+2); }