#include #ifdef _MSC_VER # include #else # include #endif #include #include namespace suisen { // ! utility template using constraints_t = std::enable_if_t, std::nullptr_t>; template constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) { if constexpr (cond_v) { return std::forward(then); } else { return std::forward(or_else); } } // ! function template using is_same_as_invoke_result = std::is_same, ReturnType>; template using is_uni_op = is_same_as_invoke_result; template using is_bin_op = is_same_as_invoke_result; template using is_comparator = std::is_same, bool>; // ! integral template >> constexpr int bit_num = std::numeric_limits>::digits; template struct is_nbit { static constexpr bool value = bit_num == n; }; template static constexpr bool is_nbit_v = is_nbit::value; // ? template struct safely_multipliable {}; template <> struct safely_multipliable { using type = long long; }; template <> struct safely_multipliable { using type = __int128_t; }; template <> struct safely_multipliable { using type = unsigned long long; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = __uint128_t; }; template <> struct safely_multipliable { using type = float; }; template <> struct safely_multipliable { using type = double; }; template <> struct safely_multipliable { using type = long double; }; template using safely_multipliable_t = typename safely_multipliable::type; template struct rec_value_type { using type = T; }; template struct rec_value_type> { using type = typename rec_value_type::type; }; template using rec_value_type_t = typename rec_value_type::type; } // namespace suisen // ! type aliases using i128 = __int128_t; using u128 = __uint128_t; template using pq_greater = std::priority_queue, std::greater>; // ! macros (internal) #define DETAIL_OVERLOAD2(_1,_2,name,...) name #define DETAIL_OVERLOAD3(_1,_2,_3,name,...) name #define DETAIL_OVERLOAD4(_1,_2,_3,_4,name,...) name #define DETAIL_REP4(i,l,r,s) for(std::remove_reference_t>i=(l);i<(r);i+=(s)) #define DETAIL_REP3(i,l,r) DETAIL_REP4(i,l,r,1) #define DETAIL_REP2(i,n) DETAIL_REP3(i,0,n) #define DETAIL_REPINF3(i,l,s) for(std::remove_reference_t>i=(l);;i+=(s)) #define DETAIL_REPINF2(i,l) DETAIL_REPINF3(i,l,1) #define DETAIL_REPINF1(i) DETAIL_REPINF2(i,0) #define DETAIL_RREP4(i,l,r,s) for(std::remove_reference_t>i=(l)+fld((r)-(l)-1,s)*(s);i>=(l);i-=(s)) #define DETAIL_RREP3(i,l,r) DETAIL_RREP4(i,l,r,1) #define DETAIL_RREP2(i,n) DETAIL_RREP3(i,0,n) #define DETAIL_CAT_I(a, b) a##b #define DETAIL_CAT(a, b) DETAIL_CAT_I(a, b) #define DETAIL_UNIQVAR(tag) DETAIL_CAT(tag, __LINE__) // ! macros #define REP(...) DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_REP4 , DETAIL_REP3 , DETAIL_REP2 )(__VA_ARGS__) #define RREP(...) DETAIL_OVERLOAD4(__VA_ARGS__, DETAIL_RREP4 , DETAIL_RREP3 , DETAIL_RREP2 )(__VA_ARGS__) #define REPINF(...) DETAIL_OVERLOAD3(__VA_ARGS__, DETAIL_REPINF3, DETAIL_REPINF2, DETAIL_REPINF1)(__VA_ARGS__) #define LOOP(n) for (std::remove_reference_t> DETAIL_UNIQVAR(loop_variable) = n; DETAIL_UNIQVAR(loop_variable) --> 0;) #define ALL(iterable) std::begin(iterable), std::end(iterable) #define INPUT(type, ...) type __VA_ARGS__; read(__VA_ARGS__) // ! debug #ifdef LOCAL # define debug(...) debug_internal(#__VA_ARGS__, __VA_ARGS__) template void debug_internal(const char* s, T&& first, Args&&... args) { constexpr const char* prefix = "[\033[32mDEBUG\033[m] "; constexpr const char* open_brakets = sizeof...(args) == 0 ? "" : "("; constexpr const char* close_brakets = sizeof...(args) == 0 ? "" : ")"; std::cerr << prefix << open_brakets << s << close_brakets << ": " << open_brakets << std::forward(first); ((std::cerr << ", " << std::forward(args)), ...); std::cerr << close_brakets << "\n"; } #else # define debug(...) void(0) #endif // ! I/O utilities // __int128_t std::ostream& operator<<(std::ostream& dest, __int128_t value) { std::ostream::sentry s(dest); if (s) { __uint128_t tmp = value < 0 ? -value : value; char buffer[128]; char* d = std::end(buffer); do { --d; *d = "0123456789"[tmp % 10]; tmp /= 10; } while (tmp != 0); if (value < 0) { --d; *d = '-'; } int len = std::end(buffer) - d; if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); } } return dest; } // __uint128_t std::ostream& operator<<(std::ostream& dest, __uint128_t value) { std::ostream::sentry s(dest); if (s) { char buffer[128]; char* d = std::end(buffer); do { --d; *d = "0123456789"[value % 10]; value /= 10; } while (value != 0); int len = std::end(buffer) - d; if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); } } return dest; } // pair template std::ostream& operator<<(std::ostream& out, const std::pair& a) { return out << a.first << ' ' << a.second; } // tuple template std::ostream& operator<<(std::ostream& out, const std::tuple& a) { if constexpr (N >= std::tuple_size_v>) return out; else { out << std::get(a); if constexpr (N + 1 < std::tuple_size_v>) out << ' '; return operator<<(out, a); } } // vector template std::ostream& operator<<(std::ostream& out, const std::vector& a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } // array template std::ostream& operator<<(std::ostream& out, const std::array& a) { for (auto it = a.begin(); it != a.end();) { out << *it; if (++it != a.end()) out << ' '; } return out; } inline void print() { std::cout << '\n'; } template inline void print(const Head& head, const Tail &...tails) { std::cout << head; if (sizeof...(tails)) std::cout << ' '; print(tails...); } template auto print_all(const Iterable& v, std::string sep = " ", std::string end = "\n") -> decltype(std::cout << *v.begin(), void()) { for (auto it = v.begin(); it != v.end();) { std::cout << *it; if (++it != v.end()) std::cout << sep; } std::cout << end; } __int128_t stoi128(const std::string& s) { __int128_t ret = 0; for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0'; if (s[0] == '-') ret = -ret; return ret; } __uint128_t stou128(const std::string& s) { __uint128_t ret = 0; for (int i = 0; i < int(s.size()); i++) if ('0' <= s[i] and s[i] <= '9') ret = 10 * ret + s[i] - '0'; return ret; } // __int128_t std::istream& operator>>(std::istream& in, __int128_t& v) { std::string s; in >> s; v = stoi128(s); return in; } // __uint128_t std::istream& operator>>(std::istream& in, __uint128_t& v) { std::string s; in >> s; v = stou128(s); return in; } // pair template std::istream& operator>>(std::istream& in, std::pair& a) { return in >> a.first >> a.second; } // tuple template std::istream& operator>>(std::istream& in, std::tuple& a) { if constexpr (N >= std::tuple_size_v>) return in; else return operator>>(in >> std::get(a), a); } // vector template std::istream& operator>>(std::istream& in, std::vector& a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } // array template std::istream& operator>>(std::istream& in, std::array& a) { for (auto it = a.begin(); it != a.end(); ++it) in >> *it; return in; } template void read(Args &...args) { (std::cin >> ... >> args); } // ! integral utilities // Returns pow(-1, n) template constexpr inline int pow_m1(T n) { return -(n & 1) | 1; } // Returns pow(-1, n) template <> constexpr inline int pow_m1(bool n) { return -int(n) | 1; } // Returns floor(x / y) template constexpr inline T fld(const T x, const T y) { return (x ^ y) >= 0 ? x / y : (x - (y + pow_m1(y >= 0))) / y; } template constexpr inline T cld(const T x, const T y) { return (x ^ y) <= 0 ? x / y : (x + (y + pow_m1(y >= 0))) / y; } template >, std::nullptr_t> = nullptr> __attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u32(x); } template , std::nullptr_t> = nullptr> __attribute__((target("popcnt"))) constexpr inline int popcount(const T x) { return _mm_popcnt_u64(x); } template >, std::nullptr_t> = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clz(x) : suisen::bit_num; } template , std::nullptr_t> = nullptr> constexpr inline int count_lz(const T x) { return x ? __builtin_clzll(x) : suisen::bit_num; } template >, std::nullptr_t> = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctz(x) : suisen::bit_num; } template , std::nullptr_t> = nullptr> constexpr inline int count_tz(const T x) { return x ? __builtin_ctzll(x) : suisen::bit_num; } template constexpr inline int floor_log2(const T x) { return suisen::bit_num - 1 - count_lz(x); } template constexpr inline int ceil_log2(const T x) { return floor_log2(x) + ((x & -x) != x); } template constexpr inline int kth_bit(const T x, const unsigned int k) { return (x >> k) & 1; } template constexpr inline int parity(const T x) { return popcount(x) & 1; } // ! container template auto priqueue_comp(const Comparator comparator) { return std::priority_queue, Comparator>(comparator); } template void sort_unique_erase(Container& a) { std::sort(a.begin(), a.end()); a.erase(std::unique(a.begin(), a.end()), a.end()); } template auto foreach_adjacent_values(InputIterator first, InputIterator last, BiConsumer f) -> decltype(f(*first++, *last), void()) { if (first != last) for (auto itr = first, itl = itr++; itr != last; itl = itr++) f(*itl, *itr); } template auto foreach_adjacent_values(Container &&c, BiConsumer f) -> decltype(c.begin(), c.end(), void()) { foreach_adjacent_values(c.begin(), c.end(), f); } // ! other utilities // x <- min(x, y). returns true iff `x` has chenged. template inline bool chmin(T& x, const T& y) { return y >= x ? false : (x = y, true); } // x <- max(x, y). returns true iff `x` has chenged. template inline bool chmax(T& x, const T& y) { return y <= x ? false : (x = y, true); } template , std::nullptr_t> = nullptr> std::string bin(T val, int bit_num = -1) { std::string res; if (bit_num != -1) { for (int bit = bit_num; bit-- > 0;) res += '0' + ((val >> bit) & 1); } else { for (; val; val >>= 1) res += '0' + (val & 1); std::reverse(res.begin(), res.end()); } return res; } template , std::nullptr_t> = nullptr> std::vector digits_low_to_high(T val, T base = 10) { std::vector res; for (; val; val /= base) res.push_back(val % base); if (res.empty()) res.push_back(T{ 0 }); return res; } template , std::nullptr_t> = nullptr> std::vector digits_high_to_low(T val, T base = 10) { auto res = digits_low_to_high(val, base); std::reverse(res.begin(), res.end()); return res; } template std::string join(const std::vector& v, const std::string& sep, const std::string& end) { std::ostringstream ss; for (auto it = v.begin(); it != v.end();) { ss << *it; if (++it != v.end()) ss << sep; } ss << end; return ss.str(); } template auto transform_to_vector(const Func &f, const Seq &s) { std::vector> v; v.reserve(std::size(s)), std::transform(std::begin(s), std::end(s), std::back_inserter(v), f); return v; } template auto copy_to_vector(const Seq &s) { std::vector v; v.reserve(std::size(s)), std::copy(std::begin(s), std::end(s), std::back_inserter(v)); return v; } template Seq concat(Seq s, const Seq &t) { s.reserve(std::size(s) + std::size(t)); std::copy(std::begin(t), std::end(t), std::back_inserter(s)); return s; } template std::vector split(const Seq s, typename Seq::value_type delim) { std::vector res; for (auto itl = std::begin(s), itr = itl;; itl = ++itr) { while (itr != std::end(s) and *itr != delim) ++itr; res.emplace_back(itl, itr); if (itr == std::end(s)) return res; } } int digit_to_int(char c) { return c - '0'; } int lowercase_to_int(char c) { return c - 'a'; } int uppercase_to_int(char c) { return c - 'A'; } std::vector digit_str_to_ints(const std::string &s) { return transform_to_vector(digit_to_int, s); } std::vector lowercase_str_to_ints(const std::string &s) { return transform_to_vector(lowercase_to_int, s); } std::vector uppercase_str_to_ints(const std::string &s) { return transform_to_vector(uppercase_to_int, s); } const std::string Yes = "Yes", No = "No", YES = "YES", NO = "NO"; namespace suisen {} using namespace suisen; using namespace std; struct io_setup { io_setup(int precision = 20) { std::ios::sync_with_stdio(false); std::cin.tie(nullptr); std::cout << std::fixed << std::setprecision(precision); } } io_setup_ {}; // ! code from here #include using mint = atcoder::modint998244353; namespace atcoder { std::istream& operator>>(std::istream& in, mint &a) { long long e; in >> e; a = e; return in; } std::ostream& operator<<(std::ostream& out, const mint &a) { out << a.val(); return out; } } // namespace atcoder #include #include #include #include #include #include #include /** * refernce: https://37zigen.com/tonelli-shanks-algorithm/ * calculates x s.t. x^2 = a mod p in O((log p)^2). */ template std::optional safe_sqrt(mint a) { static int p = mint::mod(); if (a == 0) return std::make_optional(0); if (p == 2) return std::make_optional(a); if (a.pow((p - 1) / 2) != 1) return std::nullopt; mint b = 1; while (b.pow((p - 1) / 2) == 1) ++b; static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz; mint x = a.pow((q + 1) / 2); b = b.pow(q); for (int shift = 2; x * x != a; ++shift) { mint e = a.inv() * x * x; if (e.pow(1 << (tlz - shift)) != 1) x *= b; b *= b; } return std::make_optional(x); } /** * calculates x s.t. x^2 = a mod p in O((log p)^2). * if not exists, raises runtime error. */ template auto sqrt(mint a) -> decltype(mint::mod(), mint()) { return *safe_sqrt(a); } template auto log(mint a) -> decltype(mint::mod(), mint()) { assert(a == 1); return 0; } template auto exp(mint a) -> decltype(mint::mod(), mint()) { assert(a == 0); return 1; } template auto pow(mint a, T b) -> decltype(mint::mod(), mint()) { return a.pow(b); } template auto inv(mint a) -> decltype(mint::mod(), mint()) { return a.inv(); } namespace suisen { template class inv_mods { public: inv_mods() {} inv_mods(int n) { ensure(n); } const mint& operator[](int i) const { ensure(i); return invs[i]; } static void ensure(int n) { int sz = invs.size(); if (sz < 2) invs = { 0, 1 }, sz = 2; if (sz < n + 1) { invs.resize(n + 1); for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i]; } } private: static std::vector invs; static constexpr int mod = mint::mod(); }; template std::vector inv_mods::invs{}; template std::vector get_invs(const std::vector& vs) { const int n = vs.size(); mint p = 1; for (auto& e : vs) { p *= e; assert(e != 0); } mint ip = p.inv(); std::vector rp(n + 1); rp[n] = 1; for (int i = n - 1; i >= 0; --i) { rp[i] = rp[i + 1] * vs[i]; } std::vector res(n); for (int i = 0; i < n; ++i) { res[i] = ip * rp[i + 1]; ip *= vs[i]; } return res; } } namespace suisen { template struct FPSNaive : std::vector { static inline int MAX_SIZE = std::numeric_limits::max() / 2; using value_type = T; using element_type = rec_value_type_t; using std::vector::vector; FPSNaive(const std::initializer_list l) : std::vector::vector(l) {} FPSNaive(const std::vector& v) : std::vector::vector(v) {} static void set_max_size(int n) { FPSNaive::MAX_SIZE = n; } const value_type operator[](int n) const { return n <= deg() ? unsafe_get(n) : value_type{ 0 }; } value_type& operator[](int n) { return ensure_deg(n), unsafe_get(n); } int size() const { return std::vector::size(); } int deg() const { return size() - 1; } int normalize() { while (size() and this->back() == value_type{ 0 }) this->pop_back(); return deg(); } FPSNaive& cut_inplace(int n) { if (size() > n) this->resize(std::max(0, n)); return *this; } FPSNaive cut(int n) const { FPSNaive f = FPSNaive(*this).cut_inplace(n); return f; } FPSNaive operator+() const { return FPSNaive(*this); } FPSNaive operator-() const { FPSNaive f(*this); for (auto& e : f) e = -e; return f; } FPSNaive& operator++() { return ++(*this)[0], * this; } FPSNaive& operator--() { return --(*this)[0], * this; } FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; } FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; } FPSNaive& operator+=(const FPSNaive& g) { ensure_deg(g.deg()); for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i); return *this; } FPSNaive& operator-=(const FPSNaive& g) { ensure_deg(g.deg()); for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i); return *this; } FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; } FPSNaive& operator*=(const value_type x) { for (auto& e : *this) e *= x; return *this; } FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; } FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; } FPSNaive& operator<<=(const int shamt) { this->insert(this->begin(), shamt, value_type{ 0 }); return *this; } FPSNaive& operator>>=(const int shamt) { if (shamt > size()) this->clear(); else this->erase(this->begin(), this->begin() + shamt); return *this; } friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; } friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; } friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; } friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; } friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) { if (f.empty() or g.empty()) return FPSNaive{}; const int n = f.size(), m = g.size(); FPSNaive h(std::min(MAX_SIZE, n + m - 1)); for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) { if (i + j >= MAX_SIZE) break; h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j); } return h; } friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; } friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); } friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); } friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; } friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; } friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; } std::pair div_mod(FPSNaive g) const { FPSNaive f = *this; const int fd = f.normalize(), gd = g.normalize(); assert(gd >= 0); if (fd < gd) return { FPSNaive{}, f }; if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} }; const int k = f.deg() - gd; value_type head_inv = g.unsafe_get(gd).inv(); FPSNaive q(k + 1); for (int i = k; i >= 0; --i) { value_type div = f.unsafe_get(i + gd) * head_inv; q.unsafe_get(i) = div; for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j); } return { q, f.cut_inplace(gd) }; } friend bool operator==(const FPSNaive& f, const FPSNaive& g) { const int n = f.size(), m = g.size(); if (n < m) return g == f; for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false; for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false; return true; } friend bool operator!=(const FPSNaive& f, const FPSNaive& g) { return not (f == g); } FPSNaive mul(const FPSNaive& g, int n = -1) const { if (n < 0) n = size(); if (this->empty() or g.empty()) return FPSNaive{}; const int m = size(), k = g.size(); FPSNaive h(std::min(n, m + k - 1)); for (int i = 0; i < m; ++i) { for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) { h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j); } } return h; } FPSNaive diff() const { if (this->empty()) return {}; FPSNaive g(size() - 1); for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i; return g; } FPSNaive intg() const { const int n = size(); FPSNaive g(n + 1); for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1]; if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE); return g; } FPSNaive inv(int n = -1) const { if (n < 0) n = size(); FPSNaive g(n); const value_type inv_f0 = ::inv(unsafe_get(0)); g.unsafe_get(0) = inv_f0; for (int i = 1; i < n; ++i) { for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j]; g.unsafe_get(i) *= inv_f0; } return g; } FPSNaive exp(int n = -1) const { if (n < 0) n = size(); assert(unsafe_get(0) == value_type{ 0 }); FPSNaive g(n); g.unsafe_get(0) = value_type{ 1 }; for (int i = 1; i < n; ++i) { for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j]; g.unsafe_get(i) *= invs[i]; } return g; } FPSNaive log(int n = -1) const { if (n < 0) n = size(); assert(unsafe_get(0) == value_type{ 1 }); FPSNaive g(n); g.unsafe_get(0) = value_type{ 0 }; for (int i = 1; i < n; ++i) { g.unsafe_get(i) = i * (*this)[i]; for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j]; g.unsafe_get(i) *= invs[i]; } return g; } FPSNaive pow(const long long k, int n = -1) const { if (n < 0) n = size(); if (k == 0) { FPSNaive res(n); res[0] = 1; return res; } int z = 0; while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z; if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0); const int m = n - z * k; FPSNaive g(m); const value_type inv_f0 = ::inv(unsafe_get(z)); g.unsafe_get(0) = unsafe_get(z).pow(k); for (int i = 1; i < m; ++i) { for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j]; g.unsafe_get(i) *= inv_f0 * invs[i]; } g <<= z * k; return g; } std::optional safe_sqrt(int n = -1) const { if (n < 0) n = size(); int dl = 0; while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl; if (dl == size()) return FPSNaive(n, 0); if (dl & 1) return std::nullopt; const int m = n - dl / 2; FPSNaive g(m); auto opt_g0 = ::safe_sqrt((*this)[dl]); if (not opt_g0.has_value()) return std::nullopt; g.unsafe_get(0) = *opt_g0; value_type inv_2g0 = ::inv(2 * g.unsafe_get(0)); for (int i = 1; i < m; ++i) { g.unsafe_get(i) = (*this)[dl + i]; for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j); g.unsafe_get(i) *= inv_2g0; } g <<= dl / 2; return g; } FPSNaive sqrt(int n = -1) const { if (n < 0) n = size(); return *safe_sqrt(n); } value_type eval(value_type x) const { value_type y = 0; for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i); return y; } private: static inline inv_mods invs; void ensure_deg(int d) { if (deg() < d) this->resize(d + 1, value_type{ 0 }); } const value_type& unsafe_get(int i) const { return std::vector::operator[](i); } value_type& unsafe_get(int i) { return std::vector::operator[](i); } }; } // namespace suisen template suisen::FPSNaive sqrt(suisen::FPSNaive a) { return a.sqrt(); } template suisen::FPSNaive log(suisen::FPSNaive a) { return a.log(); } template suisen::FPSNaive exp(suisen::FPSNaive a) { return a.exp(); } template suisen::FPSNaive pow(suisen::FPSNaive a, T b) { return a.pow(b); } template suisen::FPSNaive inv(suisen::FPSNaive a) { return a.inv(); } namespace suisen { template * = nullptr> struct FormalPowerSeries : std::vector { using base_type = std::vector; using value_type = typename base_type::value_type; using base_type::vector; FormalPowerSeries(const std::initializer_list l) : std::vector::vector(l) {} FormalPowerSeries(const std::vector& v) : std::vector::vector(v) {} int size() const noexcept { return base_type::size(); } int deg() const noexcept { return size() - 1; } void ensure(int n) { if (size() < n) this->resize(n); } value_type safe_get(int d) const { return d <= deg() ? (*this)[d] : 0; } value_type& safe_get(int d) { ensure(d + 1); return (*this)[d]; } FormalPowerSeries& cut_trailing_zeros() { while (size() and this->back() == 0) this->pop_back(); return *this; } FormalPowerSeries& cut(int n) { if (size() > n) this->resize(std::max(0, n)); return *this; } FormalPowerSeries cut_copy(int n) const { FormalPowerSeries res(this->begin(), this->begin() + std::min(size(), n)); res.ensure(n); return res; } FormalPowerSeries cut_copy(int l, int r) const { if (l >= size()) return FormalPowerSeries(r - l, 0); FormalPowerSeries res(this->begin() + l, this->begin() + std::min(size(), r)); res.ensure(r - l); return res; } /* Unary Operations */ FormalPowerSeries operator+() const { return *this; } FormalPowerSeries operator-() const { FormalPowerSeries res = *this; for (auto& e : res) e = -e; return res; } FormalPowerSeries& operator++() { return ++safe_get(0), * this; } FormalPowerSeries& operator--() { return --safe_get(0), * this; } FormalPowerSeries operator++(int) { FormalPowerSeries res = *this; ++(*this); return res; } FormalPowerSeries operator--(int) { FormalPowerSeries res = *this; --(*this); return res; } /* Binary Operations With Constant */ FormalPowerSeries& operator+=(const value_type& x) { return safe_get(0) += x, *this; } FormalPowerSeries& operator-=(const value_type& x) { return safe_get(0) -= x, *this; } FormalPowerSeries& operator*=(const value_type& x) { for (auto& e : *this) e *= x; return *this; } FormalPowerSeries& operator/=(const value_type& x) { return *this *= x.inv(); } friend FormalPowerSeries operator+(FormalPowerSeries f, const value_type& x) { f += x; return f; } friend FormalPowerSeries operator+(const value_type& x, FormalPowerSeries f) { f += x; return f; } friend FormalPowerSeries operator-(FormalPowerSeries f, const value_type& x) { f -= x; return f; } friend FormalPowerSeries operator-(const value_type& x, FormalPowerSeries f) { f -= x; return -f; } friend FormalPowerSeries operator*(FormalPowerSeries f, const value_type& x) { f *= x; return f; } friend FormalPowerSeries operator*(const value_type& x, FormalPowerSeries f) { f *= x; return f; } friend FormalPowerSeries operator/(FormalPowerSeries f, const value_type& x) { f /= x; return f; } /* Binary Operations With Formal Power Series */ FormalPowerSeries& operator+=(const FormalPowerSeries& g) { const int n = g.size(); ensure(n); for (int i = 0; i < n; ++i) (*this)[i] += g[i]; return *this; } FormalPowerSeries& operator-=(const FormalPowerSeries& g) { const int n = g.size(); ensure(n); for (int i = 0; i < n; ++i) (*this)[i] -= g[i]; return *this; } FormalPowerSeries& operator*=(const FormalPowerSeries& g) { return *this = *this * g; } FormalPowerSeries& operator/=(const FormalPowerSeries& g) { return *this = *this / g; } FormalPowerSeries& operator%=(const FormalPowerSeries& g) { return *this = *this % g; } friend FormalPowerSeries operator+(FormalPowerSeries f, const FormalPowerSeries& g) { f += g; return f; } friend FormalPowerSeries operator-(FormalPowerSeries f, const FormalPowerSeries& g) { f -= g; return f; } friend FormalPowerSeries operator*(const FormalPowerSeries& f, const FormalPowerSeries& g) { const int siz_f = f.size(), siz_g = g.size(); if (siz_f < siz_g) return g * f; if (std::min(siz_f, siz_g) <= 60) return atcoder::convolution(f, g); const int deg = siz_f + siz_g - 2; int fpow2 = 1; while ((fpow2 << 1) <= deg) fpow2 <<= 1; if (const int dif = deg - fpow2 + 1; dif <= 10) { FormalPowerSeries h = atcoder::convolution(std::vector(f.begin(), f.end() - dif), g); h.resize(h.size() + dif); for (int i = siz_f - dif; i < siz_f; ++i) for (int j = 0; j < siz_g; ++j) { h[i + j] += f[i] * g[j]; } return h; } return atcoder::convolution(f, g); } friend FormalPowerSeries operator/(FormalPowerSeries f, FormalPowerSeries g) { if (f.size() < 60) return FPSNaive(f).div_mod(g).first; f.cut_trailing_zeros(), g.cut_trailing_zeros(); const int fd = f.deg(), gd = g.deg(); assert(gd >= 0); if (fd < gd) return {}; if (gd == 0) { f /= g[0]; return f; } std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end()); const int qd = fd - gd; FormalPowerSeries q = f * g.inv(qd + 1); q.cut(qd + 1); std::reverse(q.begin(), q.end()); return q; } friend FormalPowerSeries operator%(const FormalPowerSeries& f, const FormalPowerSeries& g) { return f.div_mod(g).second; } std::pair div_mod(const FormalPowerSeries& g) const { if (size() < 60) { auto [q, r] = FPSNaive(*this).div_mod(g); return { q, r }; } FormalPowerSeries q = *this / g, r = *this - g * q; r.cut_trailing_zeros(); return { q, r }; } /* Shift Operations */ FormalPowerSeries& operator<<=(const int shamt) { return this->insert(this->begin(), shamt, 0), * this; } FormalPowerSeries& operator>>=(const int shamt) { return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this; } friend FormalPowerSeries operator<<(FormalPowerSeries f, const int shamt) { f <<= shamt; return f; } friend FormalPowerSeries operator>>(FormalPowerSeries f, const int shamt) { f >>= shamt; return f; } /* Compare */ friend bool operator==(const FormalPowerSeries& f, const FormalPowerSeries& g) { const int n = f.size(), m = g.size(); if (n < m) return g == f; for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false; for (int i = m; i < n; ++i) if (f[i] != 0) return false; return true; } friend bool operator!=(const FormalPowerSeries& f, const FormalPowerSeries& g) { return not (f == g); } /* Other Operations */ FormalPowerSeries& diff_inplace() { const int n = size(); for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i; return (*this)[n - 1] = 0, *this; } FormalPowerSeries diff() const { FormalPowerSeries res = *this; res.diff_inplace(); return res; } FormalPowerSeries& intg_inplace() { const int n = size(); inv_mods invs(n); this->resize(n + 1); for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i]; return (*this)[0] = 0, *this; } FormalPowerSeries intg() const { FormalPowerSeries res = *this; res.intg_inplace(); return res; } FormalPowerSeries& inv_inplace(int n = -1) { return *this = inv(n); } // reference: https://opt-cp.com/fps-fast-algorithms/ FormalPowerSeries inv(int n = -1) const { if (n < 0) n = size(); if (n < 60) return FPSNaive(cut_copy(n)).inv(); if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n); FormalPowerSeries f_fft, g_fft; FormalPowerSeries g{ (*this)[0].inv() }; for (int k = 1; k < n; k *= 2) { f_fft = cut_copy(2 * k), g_fft = g.cut_copy(2 * k); atcoder::internal::butterfly(f_fft); atcoder::internal::butterfly(g_fft); update_inv(k, f_fft, g_fft, g); } g.resize(n); return g; } FormalPowerSeries& log_inplace(int n = -1) { return *this = log(n); } FormalPowerSeries log(int n = -1) const { assert(safe_get(0) == 1); if (n < 0) n = size(); if (n < 60) return FPSNaive(cut_copy(n)).log(); if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n); FormalPowerSeries res = inv(n) * diff(); res.resize(n - 1); return res.intg(); } FormalPowerSeries& exp_inplace(int n = -1) { return *this = exp(n); } // https://arxiv.org/pdf/1301.5804.pdf FormalPowerSeries exp(int n = -1) const { assert(safe_get(0) == 0); if (n < 0) n = size(); if (n < 60) return FPSNaive(cut_copy(n)).exp(); if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n); // h = *this // f = exp(h) mod x ^ k // g = f^{-1} mod x ^ k FormalPowerSeries dh = diff(); FormalPowerSeries f{ 1 }, f_fft; FormalPowerSeries g{ 1 }, g_fft; for (int k = 1; k < n; k *= 2) { f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft); if (k > 1) update_inv(k / 2, f_fft, g_fft, g); FormalPowerSeries t = f.cut_copy(k); t.diff_inplace(); { FormalPowerSeries r = dh.cut_copy(k); r.back() = 0; atcoder::internal::butterfly(r); for (int i = 0; i < k; ++i) r[i] *= f_fft[i]; atcoder::internal::butterfly_inv(r); r /= -k; t += r; t <<= 1, t[0] = t[k], t.pop_back(); } t.resize(2 * k); atcoder::internal::butterfly(t); g_fft = g.cut_copy(2 * k); atcoder::internal::butterfly(g_fft); for (int i = 0; i < 2 * k; ++i) t[i] *= g_fft[i]; atcoder::internal::butterfly_inv(t); t.resize(k); t /= 2 * k; FormalPowerSeries v = cut_copy(2 * k) >>= k; t <<= k - 1; t.intg_inplace(); for (int i = 0; i < k; ++i) v[i] -= t[k + i]; v.resize(2 * k); atcoder::internal::butterfly(v); for (int i = 0; i < 2 * k; ++i) v[i] *= f_fft[i]; atcoder::internal::butterfly_inv(v); v.resize(k); v /= 2 * k; f.resize(2 * k); for (int i = 0; i < k; ++i) f[k + i] = v[i]; } f.cut(n); return f; } FormalPowerSeries& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); } FormalPowerSeries pow(const long long k, int n = -1) const { if (n < 0) n = size(); if (n < 60) return FPSNaive(cut_copy(n)).pow(k); if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n); if (k == 0) { FormalPowerSeries f{ 1 }; f.resize(n); return f; } int tlz = 0; while (tlz < size() and (*this)[tlz] == 0) ++tlz; if (tlz == size() or tlz > (n - 1) / k) return FormalPowerSeries(n, 0); const int m = n - tlz * k; FormalPowerSeries f = *this >> tlz; value_type base = f[0]; return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k)); } std::optional safe_sqrt(int n = -1) const { if (n < 0) n = size(); if (n < 60) return FPSNaive(cut_copy(n)).safe_sqrt(); if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n); int tlz = 0; while (tlz < size() and (*this)[tlz] == 0) ++tlz; if (tlz == size()) return FormalPowerSeries(n, 0); if (tlz & 1) return std::nullopt; const int m = n - tlz / 2; FormalPowerSeries h(this->begin() + tlz, this->end()); auto q0 = ::safe_sqrt(h[0]); if (not q0.has_value()) return std::nullopt; FormalPowerSeries f{ *q0 }, f_fft, g{ q0->inv() }, g_fft; for (int k = 1; k < m; k *= 2) { f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft); if (k > 1) update_inv(k / 2, f_fft, g_fft, g); g_fft = g.cut_copy(2 * k); atcoder::internal::butterfly(g_fft); FormalPowerSeries h_fft = h.cut_copy(2 * k); atcoder::internal::butterfly(h_fft); for (int i = 0; i < 2 * k; ++i) h_fft[i] = (h_fft[i] - f_fft[i] * f_fft[i]) * g_fft[i]; atcoder::internal::butterfly_inv(h_fft); f.resize(2 * k); const value_type iz = value_type(4 * k).inv(); for (int i = 0; i < k; ++i) f[k + i] = h_fft[k + i] * iz; } f.resize(m), f <<= (tlz / 2); return f; } FormalPowerSeries& sqrt_inplace(int n = -1) { return *this = sqrt(n); } FormalPowerSeries sqrt(int n = -1) const { return *safe_sqrt(n); } value_type eval(value_type x) const { value_type y = 0; for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i]; return y; } static FormalPowerSeries prod(const std::vector& fs) { if (fs.empty()) return { 1 }; std::deque dq(fs.begin(), fs.end()); std::sort(dq.begin(), dq.end(), [](auto& f, auto& g) { return f.size() < g.size(); }); while (dq.size() >= 2) { dq.push_back(dq[0] * dq[1]); dq.pop_front(); dq.pop_front(); } return dq.front(); } std::optional>> sparse_fps_format(int max_size) const { std::vector> res; for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v); if (int(res.size()) > max_size) return std::nullopt; return res; } private: static void update_inv(const int k, FormalPowerSeries& f_fft, FormalPowerSeries& g_fft, FormalPowerSeries& g) { FormalPowerSeries fg(2 * k); for (int i = 0; i < 2 * k; ++i) fg[i] = f_fft[i] * g_fft[i]; atcoder::internal::butterfly_inv(fg); fg >>= k, fg.resize(2 * k); atcoder::internal::butterfly(fg); for (int i = 0; i < 2 * k; ++i) fg[i] *= g_fft[i]; atcoder::internal::butterfly_inv(fg); const value_type iz = value_type(2 * k).inv(), c = -iz * iz; g.resize(2 * k); for (int i = 0; i < k; ++i) g[k + i] = fg[i] * c; } static FormalPowerSeries div_fps_sparse(const FormalPowerSeries& f, const std::vector>& g, int n) { const int siz = g.size(); assert(siz and g[0].first == 0); const value_type inv_g0 = g[0].second.inv(); FormalPowerSeries h(n); for (int i = 0; i < n; ++i) { value_type v = f.safe_get(i); for (int idx = 1; idx < siz; ++idx) { const auto& [j, gj] = g[idx]; if (j > i) break; v -= gj * h[i - j]; } h[i] = v * inv_g0; } return h; } static FormalPowerSeries inv_sparse(const std::vector>& g, const int n) { return div_fps_sparse(FormalPowerSeries{ 1 }, g, n); } static FormalPowerSeries exp_sparse(const std::vector>& f, const int n) { const int siz = f.size(); assert(not siz or f[0].first != 0); FormalPowerSeries g(n); g[0] = 1; inv_mods invs(n); for (int i = 1; i < n; ++i) { value_type v = 0; for (const auto& [j, fj] : f) { if (j > i) break; v += j * fj * g[i - j]; } v *= invs[i]; g[i] = v; } return g; } static FormalPowerSeries log_sparse(const std::vector>& f, const int n) { const int siz = f.size(); assert(siz and f[0].first == 0 and f[0].second == 1); FormalPowerSeries g(n); for (int idx = 1; idx < siz; ++idx) { const auto& [j, fj] = f[idx]; if (j >= n) break; g[j] = j * fj; } inv_mods invs(n); for (int i = 1; i < n; ++i) { value_type v = g[i]; for (int idx = 1; idx < siz; ++idx) { const auto& [j, fj] = f[idx]; if (j > i) break; v -= fj * g[i - j] * (i - j); } v *= invs[i]; g[i] = v; } return g; } static FormalPowerSeries pow_sparse(const std::vector>& f, const long long k, const int n) { if (k == 0) { FormalPowerSeries res(n, 0); res[0] = 1; return res; } const int siz = f.size(); if (not siz) return FormalPowerSeries(n, 0); const int p = f[0].first; if (p > (n - 1) / k) return FormalPowerSeries(n, 0); const value_type inv_f0 = f[0].second.inv(); const int lz = p * k; FormalPowerSeries g(n); g[lz] = f[0].second.pow(k); inv_mods invs(n); for (int i = 1; lz + i < n; ++i) { value_type v = 0; for (int idx = 1; idx < siz; ++idx) { auto [j, fj] = f[idx]; j -= p; if (j > i) break; v += fj * g[lz + i - j] * (value_type(k) * j - (i - j)); } v *= invs[i] * inv_f0; g[lz + i] = v; } return g; } static std::optional safe_sqrt_sparse(const std::vector>& f, const int n) { const int siz = f.size(); if (not siz) return FormalPowerSeries(n, 0); const int p = f[0].first; if (p % 2 == 1) return std::nullopt; if (p / 2 >= n) return FormalPowerSeries(n, 0); const value_type inv_f0 = f[0].second.inv(); const int lz = p / 2; FormalPowerSeries g(n); auto opt_g0 = ::safe_sqrt(f[0].second); if (not opt_g0.has_value()) return std::nullopt; g[lz] = *opt_g0; value_type k = mint(2).inv(); inv_mods invs(n); for (int i = 1; lz + i < n; ++i) { value_type v = 0; for (int idx = 1; idx < siz; ++idx) { auto [j, fj] = f[idx]; j -= p; if (j > i) break; v += fj * g[lz + i - j] * (k * j - (i - j)); } v *= invs[i] * inv_f0; g[lz + i] = v; } return g; } static FormalPowerSeries sqrt_sparse(const std::vector>& f, const int n) { return *safe_sqrt(f, n); } }; } // namespace suisen template suisen::FormalPowerSeries sqrt(suisen::FormalPowerSeries a) { return a.sqrt(); } template suisen::FormalPowerSeries log(suisen::FormalPowerSeries a) { return a.log(); } template suisen::FormalPowerSeries exp(suisen::FormalPowerSeries a) { return a.exp(); } template suisen::FormalPowerSeries pow(suisen::FormalPowerSeries a, T b) { return a.pow(b); } template suisen::FormalPowerSeries inv(suisen::FormalPowerSeries a) { return a.inv(); } namespace suisen { template typename FPSType::value_type bostan_mori(FPSType P, FPSType Q, unsigned long long n) { auto alternate = [](FPSType&& a, bool odd) -> FPSType&& { int i = 0; for (int j = odd; j < a.size(); j += 2) a[i++] = a[j]; a.erase(a.begin() + i, a.end()); return std::move(a); }; for (; n; n >>= 1) { if (n < int(P.size())) P.resize(n + 1); if (n < int(Q.size())) Q.resize(n + 1); FPSType mQ = Q; for (int i = 1; i < Q.size(); i += 2) mQ[i] = -mQ[i]; P = alternate(P * mQ, n & 1); Q = alternate(Q * mQ, 0); } return P[0]; } template typename FPSType::value_type nth_term_of_linearly_recurrent_sequence(const FPSType& a, const FPSType& c, const unsigned long long n) { const int K = c.size(); assert(K <= a.size()); FPSType Q(K + 1); Q[0] = 1; for (int i = 0; i < K; ++i) { Q[i + 1] = -c[i]; } FPSType P = a * Q; P.cut(K); return bostan_mori(P, Q, n); } } // namespace suisen using fps = FormalPowerSeries; int main() { int n; read(n); vector> g(n + 1); REP(i, 1, n + 1) { int p; read(p); g[p].push_back(i); } vector w(n + 1); REP(i, 1, n + 1) { read(w[i]); } debug(w); vector>> paths; auto dfs = [&](auto dfs, int u, mint p, vector>& path) -> void { path.emplace_back(u, p); if (g[u].empty()) { paths.push_back(path); } else { mint sumw = 0; for (int v : g[u]) { sumw += w[v]; } mint inv_sumw = sumw.inv(); for (int v : g[u]) { mint pv = p * w[v] * inv_sumw; dfs(dfs, v, pv, path); } } path.pop_back(); }; vector> path{}; dfs(dfs, 0, 1, path); for (auto path : paths) { debug(path); } fps f(n + 2); for (const auto& path : paths) { f[path.size()] += path.back().second; } int q; read(q); LOOP(q) { int a, k; read(a, k); fps f0(n + 2), f1(n + 2); for (const auto& path : paths) { auto it = find_if(ALL(path), [&](pair x) { return x.first == a; }); (it != path.end() ? f1 : f0)[path.size()] += path.back().second; } fps g0(n + 2), g1(n + 2); auto dfs = [&](auto dfs, int u, mint p, int dep, bool found) -> void { (found ? g1 : g0)[dep] += p; if (g[u].size()) { mint sumw = 0; for (int v : g[u]) { sumw += w[v]; } mint inv_sumw = sumw.inv(); for (int v : g[u]) { mint pv = p * w[v] * inv_sumw; dfs(dfs, v, pv, dep + 1, found | (a == v)); } } }; dfs(dfs, 0, 1, 0, false); fps P = g1 - g1 * f0 + f1 * g0; fps Q = (fps{1} - f) * (fps{1} - f); print(bostan_mori(P, Q, k)); } return 0; }