#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else #pragma GCC optimize("Ofast") #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) \ for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T ceil(T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } template pair divmod(T x, U y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sum = 0; for (auto &&a: A) sum += a; return sum; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { assert(!que.empty()); T a = que.top(); que.pop(); return a; } template T POP(vc &que) { assert(!que.empty()); T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x)); } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x)); } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io.hpp" // based on yosupo's fastio #include namespace fastio { #define FASTIO // クラスが read(), print() を持っているかを判定するメタ関数 struct has_write_impl { template static auto check(T &&x) -> decltype(x.write(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_write : public decltype(has_write_impl::check(std::declval())) { }; struct has_read_impl { template static auto check(T &&x) -> decltype(x.read(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_read : public decltype(has_read_impl::check(std::declval())) {}; struct Scanner { FILE *fp; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) { bool sep = false; for (size_t i = st; i < ed; i++) { if (isspace(line[i])) { sep = true; break; } } if (!sep) reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); } if (neg) ref = -ref; return true; } template ::value>::type * = nullptr> inline bool read_single(T &x) { x.read(); return true; } bool read_single(double &ref) { string s; if (!read_single(s)) return false; ref = std::stod(s); return true; } bool read_single(char &ref) { string s; if (!read_single(s) || s.size() != 1) return false; ref = s[0]; return true; } template bool read_single(vector &ref) { for (auto &d: ref) { if (!read_single(d)) return false; } return true; } template bool read_single(pair &p) { return (read_single(p.first) && read_single(p.second)); } template void read_single_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); read_single(x); read_single_tuple(t); } } template bool read_single(tuple &tpl) { read_single_tuple(tpl); return true; } void read() {} template void read(H &h, T &... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE *fp) : fp(fp) {} }; struct Printer { Printer(FILE *_fp) : fp(_fp) {} ~Printer() { flush(); } static constexpr size_t SIZE = 1 << 15; FILE *fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write(const char val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write('0'); return; } if (val < 0) { write('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char(0x30 | (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write(const string s) { for (char c: s) write(c); } void write(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write(s[i]); } void write(const double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } void write(const long double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } template ::value>::type * = nullptr> inline void write(T x) { x.write(); } template void write(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } template void write(const pair val) { write(val.first); write(' '); write(val.second); } template void write_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { write(' '); } const auto x = std::get(t); write(x); write_tuple(t); } } template bool write(tuple tpl) { write_tuple(tpl); return true; } template void write(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } void write(i128 val) { string s; bool negative = 0; if (val < 0) { negative = 1; val = -val; } while (val) { s += '0' + int(val % 10); val /= 10; } if (negative) s += "-"; reverse(all(s)); if (len(s) == 0) s = "0"; write(s); } }; Scanner scanner = Scanner(stdin); Printer printer = Printer(stdout); void flush() { printer.flush(); } void print() { printer.write('\n'); } template void print(Head &&head, Tail &&... tail) { printer.write(head); if (sizeof...(Tail)) printer.write(' '); print(forward(tail)...); } void read() {} template void read(Head &head, Tail &... tail) { scanner.read(head); read(tail...); } } // namespace fastio using fastio::print; using fastio::flush; using fastio::read; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 3 "main.cpp" #line 2 "/home/maspy/compro/library/poly/multipoint.hpp" #line 2 "/home/maspy/compro/library/poly/count_terms.hpp" template int count_terms(const vc& f){ int t = 0; FOR(i, len(f)) if(f[i] != mint(0)) ++t; return t; } #line 2 "/home/maspy/compro/library/mod/modint_common.hpp" struct has_mod_impl { template static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_mod : public decltype(has_mod_impl::check(std::declval())) {}; template mint inv(int n) { static const int mod = mint::get_mod(); static vector dat = {0, 1}; assert(0 <= n); if (n >= mod) n %= mod; while (len(dat) <= n) { int k = len(dat); int q = (mod + k - 1) / k; dat.eb(dat[k * q - mod] * mint(q)); } return dat[n]; } template mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n); if (n >= mod) return 0; static vector dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint(len(dat))); return dat[n]; } template mint fact_inv(int n) { static const int mod = mint::get_mod(); assert(-1 <= n && n < mod); static vector dat = {1, 1}; if (n == -1) return mint(0); while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv(len(dat))); return dat[n]; } template mint fact_invs(Ts... xs) { return (mint(1) * ... * fact_inv(xs)); } template mint multinomial(Head &&head, Tail &&... tail) { return fact(head) * fact_invs(std::forward(tail)...); } template mint C_dense(int n, int k) { static vvc C; static int H = 0, W = 0; auto calc = [&](int i, int j) -> mint { if (i == 0) return (j == 0 ? mint(1) : mint(0)); return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0); }; if (W <= k) { FOR(i, H) { C[i].resize(k + 1); FOR(j, W, k + 1) { C[i][j] = calc(i, j); } } W = k + 1; } if (H <= n) { C.resize(n + 1); FOR(i, H, n + 1) { C[i].resize(W); FOR(j, W) { C[i][j] = calc(i, j); } } H = n + 1; } return C[n][k]; } template mint C(ll n, ll k) { assert(n >= 0); if (k < 0 || n < k) return 0; if (dense) return C_dense(n, k); if (!large) return multinomial(n, k, n - k); k = min(k, n - k); mint x(1); FOR(i, k) x *= mint(n - i); return x * fact_inv(k); } template mint C_inv(ll n, ll k) { assert(n >= 0); assert(0 <= k && k <= n); if (!large) return fact_inv(n) * fact(k) * fact(n - k); return mint(1) / C(n, k); } // [x^d] (1-x) ^ {-n} の計算 template mint C_negative(ll n, ll d) { assert(n >= 0); if (d < 0) return mint(0); if (n == 0) { return (d == 0 ? mint(1) : mint(0)); } return C(n + d - 1, d); } #line 3 "/home/maspy/compro/library/mod/modint.hpp" template struct modint { static_assert(mod < (1 << 30)); int val; constexpr modint(const ll val = 0) noexcept : val(val >= 0 ? val % mod : (mod - (-val) % mod) % mod) {} bool operator<(const modint &other) const { return val < other.val; } // To use std::map modint &operator+=(const modint &p) { if ((val += p.val) >= mod) val -= mod; return *this; } modint &operator-=(const modint &p) { if ((val += mod - p.val) >= mod) val -= mod; return *this; } modint &operator*=(const modint &p) { val = (int)(1LL * val * p.val % mod); return *this; } modint &operator/=(const modint &p) { *this *= p.inverse(); return *this; } modint operator-() const { return modint(-val); } modint operator+(const modint &p) const { return modint(*this) += p; } modint operator-(const modint &p) const { return modint(*this) -= p; } modint operator*(const modint &p) const { return modint(*this) *= p; } modint operator/(const modint &p) const { return modint(*this) /= p; } bool operator==(const modint &p) const { return val == p.val; } bool operator!=(const modint &p) const { return val != p.val; } modint inverse() const { int a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return modint(u); } modint pow(ll n) const { assert(n >= 0); modint ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } #ifdef FASTIO void write() { fastio::printer.write(val); } void read() { fastio::scanner.read(val); } #endif static constexpr int get_mod() { return mod; } // (n, r), r は 1 の 2^n 乗根 static constexpr pair ntt_info() { if (mod == 167772161) return {25, 17}; if (mod == 469762049) return {26, 30}; if (mod == 754974721) return {24, 362}; if (mod == 880803841) return {23, 211}; if (mod == 998244353) return {23, 31}; if (mod == 1045430273) return {20, 363}; if (mod == 1051721729) return {20, 330}; if (mod == 1053818881) return {20, 2789}; return {-1, -1}; } static constexpr bool can_ntt() { return ntt_info().fi != -1; } }; using modint107 = modint<1000000007>; using modint998 = modint<998244353>; #line 2 "/home/maspy/compro/library/mod/mod_inv.hpp" // long でも大丈夫 ll mod_inv(ll val, ll mod) { val %= mod; if (val < 0) val += mod; ll a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } if (u < 0) u += mod; return u; } #line 1 "/home/maspy/compro/library/poly/convolution_naive.hpp" template vector convolution_naive(const vector& a, const vector& b) { int n = int(a.size()), m = int(b.size()); vector ans(n + m - 1); if (n < m) { FOR(j, m) FOR(i, n) ans[i + j] += a[i] * b[j]; } else { FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j]; } return ans; } #line 2 "/home/maspy/compro/library/poly/ntt.hpp" template void ntt(vector& a, bool inverse) { assert(mint::can_ntt()); const int rank2 = mint::ntt_info().fi; const int mod = mint::get_mod(); static array root, iroot; static array rate2, irate2; static array rate3, irate3; static bool prepared = 0; if (!prepared) { prepared = 1; root[rank2] = mint::ntt_info().se; iroot[rank2] = mint(1) / root[rank2]; FOR_R(i, rank2) { 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]; } 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]; } } int n = int(a.size()); int h = topbit(n); assert(n == 1 << h); if (!inverse) { int len = 0; while (len < h) { if (h - len == 1) { int p = 1 << (h - len - 1); mint rot = 1; FOR(s, 1 << len) { int offset = s << (h - len); FOR(i, p) { auto l = a[i + offset]; auto r = a[i + offset + p] * rot; a[i + offset] = l + r; a[i + offset + p] = l - r; } rot *= rate2[topbit(~s & -~s)]; } len++; } else { int p = 1 << (h - len - 2); mint rot = 1, imag = root[2]; for (int s = 0; s < (1 << len); s++) { mint rot2 = rot * rot; mint rot3 = rot2 * rot; int offset = s << (h - len); for (int i = 0; i < p; i++) { u64 mod2 = u64(mod) * mod; u64 a0 = a[i + offset].val; u64 a1 = u64(a[i + offset + p].val) * rot.val; u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val; u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val; u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val; u64 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); } rot *= rate3[topbit(~s & -~s)]; } len += 2; } } } else { mint coef = mint(1) / mint(len(a)); FOR(i, len(a)) a[i] *= coef; int len = h; while (len) { if (len == 1) { int p = 1 << (h - len); mint irot = 1; FOR(s, 1 << (len - 1)) { int offset = s << (h - len + 1); FOR(i, p) { u64 l = a[i + offset].val; u64 r = a[i + offset + p].val; a[i + offset] = l + r; a[i + offset + p] = (mod + l - r) * irot.val; } irot *= irate2[topbit(~s & -~s)]; } len--; } else { int p = 1 << (h - len); mint irot = 1, iimag = iroot[2]; FOR(s, (1 << (len - 2))) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; int offset = s << (h - len + 2); for (int i = 0; i < p; i++) { u64 a0 = a[i + offset + 0 * p].val; u64 a1 = a[i + offset + 1 * p].val; u64 a2 = a[i + offset + 2 * p].val; u64 a3 = a[i + offset + 3 * p].val; u64 x = (mod + a2 - a3) * iimag.val % mod; a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val; a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val; a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val; } irot *= irate3[topbit(~s & -~s)]; } len -= 2; } } } } #line 1 "/home/maspy/compro/library/poly/fft.hpp" namespace CFFT { using real = double; struct C { real x, y; C() : x(0), y(0) {} C(real x, real y) : x(x), y(y) {} inline C operator+(const C& c) const { return C(x + c.x, y + c.y); } inline C operator-(const C& c) const { return C(x - c.x, y - c.y); } inline C operator*(const C& c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); } inline C conj() const { return C(x, -y); } }; const real PI = acosl(-1); int base = 1; vector rts = {{0, 0}, {1, 0}}; vector rev = {0, 1}; void ensure_base(int nbase) { if (nbase <= base) return; rev.resize(1 << nbase); rts.resize(1 << nbase); for (int i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while (base < nbase) { real angle = PI * 2.0 / (1 << (base + 1)); for (int i = 1 << (base - 1); i < (1 << base); i++) { rts[i << 1] = rts[i]; real angle_i = angle * (2 * i + 1 - (1 << base)); rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i)); } ++base; } } void fft(vector& a, int n) { assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = base - zeros; for (int i = 0; i < n; i++) { if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for (int k = 1; k < n; k <<= 1) { for (int i = 0; i < n; i += 2 * k) { for (int j = 0; j < k; j++) { C z = a[i + j + k] * rts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } } // namespace CFFT #line 7 "/home/maspy/compro/library/poly/convolution.hpp" template vector convolution_ntt(vector a, vector b) { if (a.empty() || b.empty()) return {}; int n = int(a.size()), m = int(b.size()); int sz = 1; while (sz < n + m - 1) sz *= 2; // sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。 if ((n + m - 3) <= sz / 2) { auto a_last = a.back(), b_last = b.back(); a.pop_back(), b.pop_back(); auto c = convolution(a, b); c.resize(n + m - 1); c[n + m - 2] = a_last * b_last; FOR(i, len(a)) c[i + len(b)] += a[i] * b_last; FOR(i, len(b)) c[i + len(a)] += b[i] * a_last; return c; } a.resize(sz), b.resize(sz); bool same = a == b; ntt(a, 0); if (same) { b = a; } else { ntt(b, 0); } FOR(i, sz) a[i] *= b[i]; ntt(a, 1); a.resize(n + m - 1); return a; } template vector convolution_garner(const vector& a, const vector& b) { int n = len(a), m = len(b); if (!n || !m) return {}; static const long long nttprimes[] = {754974721, 167772161, 469762049}; using mint0 = modint<754974721>; using mint1 = modint<167772161>; using mint2 = modint<469762049>; vc a0(n), b0(m); vc a1(n), b1(m); vc a2(n), b2(m); FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val; FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val; auto c0 = convolution_ntt(a0, b0); auto c1 = convolution_ntt(a1, b1); auto c2 = convolution_ntt(a2, b2); static const long long m01 = 1LL * nttprimes[0] * nttprimes[1]; static const long long m0_inv_m1 = mint1(nttprimes[0]).inverse().val; static const long long m01_inv_m2 = mint2(m01).inverse().val; const int mod = mint::get_mod(); auto garner = [&](mint0 x0, mint1 x1, mint2 x2) -> mint { int r0 = x0.val, r1 = x1.val, r2 = x2.val; int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1]; auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * mint2(m01_inv_m2); return mint(r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val); }; vc c(len(c0)); FOR(i, len(c)) c[i] = garner(c0[i], c1[i], c2[i]); return c; } template vc convolution_fft(const vc& a, const vc& b) { using C = CFFT::C; int need = (int)a.size() + (int)b.size() - 1; int nbase = 1; while ((1 << nbase) < need) nbase++; CFFT::ensure_base(nbase); int sz = 1 << nbase; vector fa(sz); for (int i = 0; i < sz; i++) { int x = (i < (int)a.size() ? a[i] : 0); int y = (i < (int)b.size() ? b[i] : 0); fa[i] = C(x, y); } CFFT::fft(fa, sz); C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r; fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r; fa[i] = z; } for (int i = 0; i < (sz >> 1); i++) { C A0 = (fa[i] + fa[i + (sz >> 1)]) * t; C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * CFFT::rts[(sz >> 1) + i]; fa[i] = A0 + A1 * s; } CFFT::fft(fa, sz >> 1); vector ret(need); for (int i = 0; i < need; i++) { ret[i] = (i & 1 ? fa[i >> 1].y : fa[i >> 1].x); } return ret; } vector convolution(const vector& a, const vector& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (min(n, m) <= 60) return convolution_naive(a, b); ll abs_sum_a = 0, abs_sum_b = 0; ll LIM = 1e15; FOR(i, n) abs_sum_a = min(LIM, abs_sum_a + abs(a[i])); FOR(i, n) abs_sum_b = min(LIM, abs_sum_b + abs(b[i])); if (i128(abs_sum_a) * abs_sum_b < 1e15) { vc c = convolution_fft(a, b); vc res(len(c)); FOR(i, len(c)) res[i] = ll(floor(c[i] + .5)); return res; } static constexpr unsigned long long MOD1 = 754974721; // 2^24 static constexpr unsigned long long MOD2 = 167772161; // 2^25 static constexpr unsigned long long MOD3 = 469762049; // 2^26 static constexpr unsigned long long M2M3 = MOD2 * MOD3; static constexpr unsigned long long M1M3 = MOD1 * MOD3; static constexpr unsigned long long M1M2 = MOD1 * MOD2; static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3; static const unsigned long long i1 = mod_inv(MOD2 * MOD3, MOD1); static const unsigned long long i2 = mod_inv(MOD1 * MOD3, MOD2); static const unsigned long long i3 = mod_inv(MOD1 * MOD2, MOD3); using mint1 = modint; using mint2 = modint; using mint3 = modint; vc a1(n), b1(m); vc a2(n), b2(m); vc a3(n), b3(m); FOR(i, n) a1[i] = a[i], a2[i] = a[i], a3[i] = a[i]; FOR(i, m) b1[i] = b[i], b2[i] = b[i], b3[i] = b[i]; auto c1 = convolution_ntt(a1, b1); auto c2 = convolution_ntt(a2, b2); auto c3 = convolution_ntt(a3, b3); vc c(n + m - 1); FOR(i, n + m - 1) { u64 x = 0; x += (c1[i].val * i1) % MOD1 * M2M3; x += (c2[i].val * i2) % MOD2 * M1M3; x += (c3[i].val * i3) % MOD3 * M1M2; ll diff = c1[i].val - ((long long)(x) % (long long)(MOD1)); if (diff < 0) diff += MOD1; static constexpr unsigned long long offset[5] = {0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3}; x -= offset[diff % 5]; c[i] = x; } return c; } template vc convolution(const vc& a, const vc& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (mint::can_ntt()) { if (min(n, m) <= 50) return convolution_naive(a, b); return convolution_ntt(a, b); } if (min(n, m) <= 200) return convolution_naive(a, b); return convolution_garner(a, b); } #line 4 "/home/maspy/compro/library/poly/fps_inv.hpp" template vc fps_inv_sparse(const vc& f) { int N = len(f); vc> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc g(N); mint g0 = mint(1) / f[0]; g[0] = g0; FOR(n, 1, N) { mint rhs = 0; for (auto&& [k, fk]: dat) { if (k > n) break; rhs -= fk * g[n - k]; } g[n] = rhs * g0; } return g; } template vc fps_inv_dense_ntt(const vc& F) { vc G = {mint(1) / F[0]}; ll N = len(F), n = 1; G.reserve(N); while (n < N) { vc f(2 * n), g(2 * n); FOR(i, min(N, 2 * n)) f[i] = F[i]; FOR(i, n) g[i] = G[i]; ntt(f, false), ntt(g, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n) f[i] = 0; ntt(f, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n, min(N, 2 * n)) G.eb(-f[i]); n *= 2; } return G; } template vc fps_inv_dense(const vc& F) { if (mint::can_ntt()) return fps_inv_dense_ntt(F); const int N = len(F); vc R = {mint(1) / F[0]}; vc p; int m = 1; while (m < N) { p = convolution(R, R); p.resize(m + m); vc f = {F.begin(), F.begin() + min(m + m, N)}; p = convolution(p, f); R.resize(m + m); FOR(i, m + m) R[i] = R[i] + R[i] - p[i]; m += m; } R.resize(N); return R; } template vc fps_inv(const vc& f) { assert(f[0] != mint(0)); int n = count_terms(f); int t = (mint::can_ntt() ? 160 : 820); return (n <= t ? fps_inv_sparse(f) : fps_inv_dense(f)); } #line 2 "/home/maspy/compro/library/poly/middle_product.hpp" #line 4 "/home/maspy/compro/library/poly/middle_product.hpp" // n, m 次多項式 (n>=m) a, b → n-m 次多項式 c // c[i] = sum_j b[j]a[i+j] template vc middle_product(vc& a, vc& b) { assert(len(a) >= len(b)); if (b.empty()) return vc(len(a) - len(b) + 1); if (min(len(b), len(a) - len(b) + 1) <= 60) { return middle_product_naive(a, b); } if (!(mint::can_ntt())) { return middle_product_garner(a, b); } else { int n = 1 << __lg(2 * len(a) - 1); vc fa(n), fb(n); copy(a.begin(), a.end(), fa.begin()); copy(b.rbegin(), b.rend(), fb.begin()); ntt(fa, 0), ntt(fb, 0); FOR(i, n) fa[i] *= fb[i]; ntt(fa, 1); fa.resize(len(a)); fa.erase(fa.begin(), fa.begin() + len(b) - 1); return fa; } } template vc middle_product_garner(vc& a, vc b) { int n = len(a), m = len(b); if (!n || !m) return {}; static const long long nttprimes[] = {754974721, 167772161, 469762049}; using mint0 = modint<754974721>; using mint1 = modint<167772161>; using mint2 = modint<469762049>; vc a0(n), b0(m); vc a1(n), b1(m); vc a2(n), b2(m); FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val; FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val; auto c0 = middle_product(a0, b0); auto c1 = middle_product(a1, b1); auto c2 = middle_product(a2, b2); static const long long m01 = 1LL * nttprimes[0] * nttprimes[1]; static const long long m0_inv_m1 = mint1(nttprimes[0]).inverse().val; static const long long m01_inv_m2 = mint2(m01).inverse().val; static const int mod = mint::get_mod(); auto garner = [&](mint0 x0, mint1 x1, mint2 x2) -> mint { int r0 = x0.val, r1 = x1.val, r2 = x2.val; int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1]; auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * mint2(m01_inv_m2); return mint(r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val); }; vc c(len(c0)); FOR(i, len(c)) c[i] = garner(c0[i], c1[i], c2[i]); return c; } template vc middle_product_naive(vc& a, vc& b) { vc res(len(a) - len(b) + 1); FOR(i, len(res)) FOR(j, len(b)) res[i] += b[j] * a[i + j]; return res; } #line 2 "/home/maspy/compro/library/mod/all_inverse.hpp" template vc all_inverse(vc& X) { for (auto&& x: X) assert(x != mint(0)); int N = len(X); vc res(N + 1); res[0] = mint(1); FOR(i, N) res[i + 1] = res[i] * X[i]; mint t = res.back().inverse(); res.pop_back(); FOR_R(i, N) { res[i] *= t; t *= X[i]; } return res; } #line 6 "/home/maspy/compro/library/poly/multipoint.hpp" template struct SubproductTree { int m; int sz; vc> T; SubproductTree(const vc& x) { m = len(x); sz = 1; while (sz < m) sz *= 2; T.resize(2 * sz); FOR(i, sz) T[sz + i] = {1, (i < m ? -x[i] : 0)}; FOR3_R(i, 1, sz) T[i] = convolution(T[2 * i], T[2 * i + 1]); } vc evaluation(vc f) { int n = len(f); if (n == 0) return vc(m, mint(0)); f.resize(2 * n - 1); vc> g(2 * sz); g[1] = T[1]; g[1].resize(n); g[1] = fps_inv(g[1]); g[1] = middle_product(f, g[1]); g[1].resize(sz); FOR3(i, 1, sz) { g[2 * i] = middle_product(g[i], T[2 * i + 1]); g[2 * i + 1] = middle_product(g[i], T[2 * i]); } vc vals(m); FOR(i, m) vals[i] = g[sz + i][0]; return vals; } vc interpolation(vc& y) { assert(len(y) == m); vc a(m); FOR(i, m) a[i] = T[1][m - i - 1] * (i + 1); a = evaluation(a); vc> t(2 * sz); FOR(i, sz) t[sz + i] = {(i < m ? y[i] / a[i] : 0)}; FOR3_R(i, 1, sz) { t[i] = convolution(t[2 * i], T[2 * i + 1]); auto tt = convolution(t[2 * i + 1], T[2 * i]); FOR(k, len(t[i])) t[i][k] += tt[k]; } t[1].resize(m); reverse(all(t[1])); return t[1]; } }; template vc multipoint_eval(vc& f, vc& x) { if (x.empty()) return {}; SubproductTree F(x); return F.evaluation(f); } template vc multipoint_interpolate(vc& x, vc& y) { if (x.empty()) return {}; SubproductTree F(x); return F.interpolation(y); } // calculate f(ar^k) for 0 <= k < m template vc multipoint_eval_on_geom_seq(vc f, mint a, mint r, int m) { const int n = len(f); if (m == 0) return {}; auto eval = [&](mint x) -> mint { mint fx = 0; mint pow = 1; FOR(i, n) fx += f[i] * pow, pow *= x; return fx; }; if (r == mint(0)) { vc res(m); FOR(i, 1, m) res[i] = f[0]; res[0] = eval(a); return res; } if (n < 60 || m < 60) { vc res(m); FOR(i, m) res[i] = eval(a), a *= r; return res; } assert(r != mint(0)); // a == 1 に帰着 mint pow_a = 1; FOR(i, n) f[i] *= pow_a, pow_a *= a; auto calc = [&](mint r, int m) -> vc { // r^{t_i} の計算 vc res(m); mint pow = 1; res[0] = 1; FOR(i, m - 1) { res[i + 1] = res[i] * pow; pow *= r; } return res; }; vc A = calc(r, n + m - 1), B = calc(r.inverse(), max(n, m)); FOR(i, n) f[i] *= B[i]; f = middle_product(A, f); FOR(i, m) f[i] *= B[i]; return f; } // Y[i] = f(ar^i) template vc multipoint_interpolate_on_geom_seq(vc Y, mint a, mint r) { const int n = len(Y); if (n == 0) return {}; if (n == 1) return {Y[0]}; assert(r != mint(0)); mint ir = r.inverse(); vc POW(n + n - 1), tPOW(n + n - 1); POW[0] = tPOW[0] = mint(1); FOR(i, n + n - 2) POW[i + 1] = POW[i] * r, tPOW[i + 1] = tPOW[i] * POW[i]; vc iPOW(n + n - 1), itPOW(n + n - 1); iPOW[0] = itPOW[0] = mint(1); FOR(i, n) iPOW[i + 1] = iPOW[i] * ir, itPOW[i + 1] = itPOW[i] * iPOW[i]; // prod_[1,i] 1-r^k vc S(n); S[0] = mint(1); FOR(i, 1, n) S[i] = S[i - 1] * (mint(1) - POW[i]); vc iS = all_inverse(S); mint sn = S[n - 1] * (mint(1) - POW[n]); FOR(i, n) { Y[i] = Y[i] * tPOW[n - 1 - i] * itPOW[n - 1] * iS[i] * iS[n - 1 - i]; if (i % 2 == 1) Y[i] = -Y[i]; } // sum_i Y[i] / 1-r^ix FOR(i, n) Y[i] *= itPOW[i]; vc f = middle_product(tPOW, Y); FOR(i, n) f[i] *= itPOW[i]; // prod 1-r^ix vc g(n); g[0] = mint(1); FOR(i, 1, n) { g[i] = tPOW[i] * sn * iS[i] * iS[n - i]; if (i % 2 == 1) g[i] = -g[i]; } f = convolution(f, g); f.resize(n); reverse(all(f)); mint ia = a.inverse(); mint pow = 1; FOR(i, n) f[i] *= pow, pow *= ia; return f; } #line 2 "/home/maspy/compro/library/alg/monoid/mul.hpp" template struct Monoid_Mul { using value_type = T; using X = T; static constexpr X op(const X &x, const X &y) noexcept { return x * y; } static constexpr X inverse(const X &x) noexcept { return X(1) / x; } static constexpr X unit() { return X(1); } static constexpr bool commute = true; }; #line 2 "/home/maspy/compro/library/ds/power_query.hpp" // 定数をべき乗するクエリ。 B 乗分ずつ前計算。 template struct Power_Query { using X = typename Mono::value_type; vvc dat; Power_Query(X a) { dat.eb(make_pow(a)); } X operator()(ll n) { X res = Mono::unit(); int k = 0; while (n) { int r = n % B; n /= B; if (len(dat) == k) { dat.eb(make_pow(dat[k - 1].back())); } res = Mono::op(res, dat[k][r]); ++k; } return res; } X operator[](ll n) { return (*this)(n); } private: vc make_pow(X a) { vc res = {Mono::unit()}; FOR(B) { res.eb(Mono::op(res.back(), a)); } return res; } }; #line 7 "main.cpp" using mint = modint998; void solve() { mint A; read(A); LL(N); Power_Query, 1 << 15> PQ(A); auto pow = [&](ll x) -> mint { return PQ[x % 998244352]; }; mint ANS = 0; const int X = 119 << 11; while (N % X) { --N; ANS += pow(N * N); } N /= X; /* sum_{i,j} pow((iX+j)^2) = sum_{i,j} pow(iiXX + 2Xij + jj) = sum_j pow(jj) sum_i pow(iXX + 2Xji) = sum_j pow(jj) sum_i pow(XX+2Xj) ^i = sum_j pow(jj) * geometric sequence sum */ // geometric sequence sum は 2^{11} 種類しかない vc num(1 << 11), den(1 << 11); FOR(j, 1 << 11) { int x = ll(X) * (X + j + j) % 998244352; mint a = pow(x) - mint(1); if (a == mint(0)) { num[j] = mint(N); den[j] = mint(1); continue; } num[j] = pow(x * N) - mint(1); den[j] = a; } den = all_inverse(den); vc coef(1 << 11); FOR(j, X) { coef[j & 2047] += pow(j * j); } FOR(j, 1 << 11) ANS += num[j] * den[j] * coef[j]; print(ANS); } signed main() { INT(T); FOR(T) solve(); return 0; }