#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; using u128 = unsigned __int128; using f128 = __float128; 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); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(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 2 "/home/maspy/compro/library/nt/primetest.hpp" struct m64 { using i64 = int64_t; using u64 = uint64_t; using u128 = __uint128_t; inline static u64 m, r, n2; // r * m = -1 (mod 1<<64), n2 = 1<<128 (mod m) static void set_mod(u64 m) { assert((m & 1) == 1); m64::m = m; n2 = -u128(m) % m; r = m; FOR(_, 5) r *= 2 - m * r; r = -r; assert(r * m == -1ull); } static u64 reduce(u128 b) { return (b + u128(u64(b) * r) * m) >> 64; } u64 x; m64() : x(0) {} m64(u64 x) : x(reduce(u128(x) * n2)){}; u64 val() const { u64 y = reduce(x); return y >= m ? y - m : y; } m64 &operator+=(m64 y) { x += y.x - (m << 1); x = (i64(x) < 0 ? x + (m << 1) : x); return *this; } m64 &operator-=(m64 y) { x -= y.x; x = (i64(x) < 0 ? x + (m << 1) : x); return *this; } m64 &operator*=(m64 y) { x = reduce(u128(x) * y.x); return *this; } m64 operator+(m64 y) const { return m64(*this) += y; } m64 operator-(m64 y) const { return m64(*this) -= y; } m64 operator*(m64 y) const { return m64(*this) *= y; } bool operator==(m64 y) const { return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x); } bool operator!=(m64 y) const { return not operator==(y); } m64 pow(u64 n) const { m64 y = 1, z = *this; for (; n; n >>= 1, z *= z) if (n & 1) y *= z; return y; } }; bool primetest(const uint64_t x) { using u64 = uint64_t; if (x == 2 or x == 3 or x == 5 or x == 7) return true; if (x % 2 == 0 or x % 3 == 0 or x % 5 == 0 or x % 7 == 0) return false; if (x < 121) return x > 1; const u64 d = (x - 1) >> __builtin_ctzll(x - 1); m64::set_mod(x); const m64 one(1), minus_one(x - 1); auto ok = [&](u64 a) { auto y = m64(a).pow(d); u64 t = d; while (y != one and y != minus_one and t != x - 1) y *= y, t <<= 1; if (y != minus_one and t % 2 == 0) return false; return true; }; if (x < (1ull << 32)) { for (u64 a: {2, 7, 61}) if (not ok(a)) return false; } else { for (u64 a: {2, 325, 9375, 28178, 450775, 9780504, 1795265022}) { if (x <= a) return true; if (not ok(a)) return false; } } return true; } #line 3 "/home/maspy/compro/library/nt/factor.hpp" mt19937_64 rng_mt{random_device{}()}; ll rnd(ll n) { return uniform_int_distribution(0, n - 1)(rng_mt); } ll rho(ll n, ll c) { m64::set_mod(n); assert(n > 1); const m64 cc(c); auto f = [&](m64 x) { return x * x + cc; }; m64 x = 1, y = 2, z = 1, q = 1; ll g = 1; const ll m = 1LL << (__lg(n) / 5); // ? for (ll r = 1; g == 1; r <<= 1) { x = y; FOR(_, r) y = f(y); for (ll k = 0; k < r and g == 1; k += m) { z = y; FOR(_, min(m, r - k)) y = f(y), q *= x - y; g = gcd(q.val(), n); } } if (g == n) do { z = f(z); g = gcd((x - z).val(), n); } while (g == 1); return g; } ll find_prime_factor(ll n) { assert(n > 1); if (primetest(n)) return n; FOR(_, 100) { ll m = rho(n, rnd(n)); if (primetest(m)) return m; n = m; } cerr << "failed" << endl; assert(false); return -1; } // ソートしてくれる vc> factor(ll n) { assert(n >= 1); vc> pf; FOR3(p, 2, 100) { if (p * p > n) break; if (n % p == 0) { ll e = 0; do { n /= p, e += 1; } while (n % p == 0); pf.eb(p, e); } } while (n > 1) { ll p = find_prime_factor(n); ll e = 0; do { n /= p, e += 1; } while (n % p == 0); pf.eb(p, e); } sort(all(pf)); return pf; } vc> factor_by_lpf(ll n, vc& lpf) { vc> res; while (n > 1) { int p = lpf[n]; int e = 0; while (n % p == 0) { n /= p; ++e; } res.eb(p, e); } return res; } #line 2 "/home/maspy/compro/library/random/base.hpp" u64 RNG_64() { static uint64_t x_ = uint64_t(chrono::duration_cast( chrono::high_resolution_clock::now().time_since_epoch()) .count()) * 10150724397891781847ULL; x_ ^= x_ << 7; return x_ ^= x_ >> 9; } u64 RNG(u64 lim) { return RNG_64() % lim; } ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); } #line 2 "/home/maspy/compro/library/mod/mod_sqrt.hpp" template mint mod_sqrt(mint a) { int p = mint::get_mod(); if (p == 2) return a; if (a == 0) return 0; int k = (p - 1) / 2; if (a.pow(k) != 1) return 0; auto find = [&]() -> pair { while (1) { mint b = RNG(2, p); mint D = b * b - a; if (D == 0) return {b, D}; if (D.pow(k) != mint(1)) return {b, D}; } }; auto [b, D] = find(); if (D == 0) return b; ++k; // (b + sqrt(D))^k mint f0 = b, f1 = 1; mint g0 = 1, g1 = 0; while (k) { if (k & 1) { tie(g0, g1) = mp(f0 * g0 + D * f1 * g1, f1 * g0 + f0 * g1); } tie(f0, f1) = mp(f0 * f0 + D * f1 * f1, mint(2) * f0 * f1); k >>= 1; } return g0; } #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::raw(q)); } return dat[n]; } template mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n && n < mod); static vector dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat))); return dat[n]; } template mint fact_inv(int n) { static vector dat = {1, 1}; if (n < 0) 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 constexpr (dense) return C_dense(n, k); if constexpr (!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 2 "/home/maspy/compro/library/mod/primitive_root.hpp" #line 2 "/home/maspy/compro/library/mod/barrett.hpp" // https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp struct Barrett { u32 m; u64 im; explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {} u32 umod() const { return m; } u32 modulo(u64 z) { if (m == 1) return 0; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z - y + (z < y ? m : 0)); } u64 floor(u64 z) { if (m == 1) return z; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; return (z < y ? x - 1 : x); } pair divmod(u64 z) { if (m == 1) return {z, 0}; u64 x = (u64)(((unsigned __int128)(z)*im) >> 64); u64 y = x * m; if (z < y) return {x - 1, z - y + m}; return {x, z - y}; } u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); } }; #line 3 "/home/maspy/compro/library/mod/mod_pow.hpp" // int ll mod_pow(ll a, ll n, int mod) { assert(n >= 0); a %= mod; if (a < 0) a += mod; Barrett bt(mod); ll p = a, v = bt.modulo(1); while (n) { if (n & 1) v = bt.mul(v, p); p = bt.mul(p, p); n >>= 1; } return v; } ll mod_pow_long(ll a, ll n, ll mod) { assert(n >= 0); a %= mod; if (a < 0) a += mod; ll p = a, v = 1 % mod; while (n) { if (n & 1) v = i128(v) * p % mod; p = i128(p) * p % mod; n >>= 1; } return v; } #line 6 "/home/maspy/compro/library/mod/primitive_root.hpp" // int int primitive_root(int p) { auto pf = factor(p - 1); auto is_ok = [&](int g) -> bool { for (auto&& [q, e]: pf) if (mod_pow(g, (p - 1) / q, p) == 1) return false; return true; }; while (1) { int x = RNG(1, p); if (is_ok(x)) return x; } return -1; } ll primitive_root_long(ll p) { auto pf = factor(p - 1); auto is_ok = [&](ll g) -> bool { for (auto&& [q, e]: pf) if (mod_pow_long(g, (p - 1) / q, p) == 1) return false; return true; }; while (1) { ll x = RNG(1, p); if (is_ok(x)) return x; } return -1; } #line 5 "/home/maspy/compro/library/mod/dynamic_modint.hpp" struct Dynamic_Modint { static constexpr bool is_modint = true; using mint = Dynamic_Modint; u32 val; static Barrett bt; static u32 umod() { return bt.umod(); } static int get_mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = Barrett(m); } static Dynamic_Modint raw(u32 v) { Dynamic_Modint x; x.val = v; return x; } Dynamic_Modint() : val(0) {} Dynamic_Modint(u32 x) : val(bt.modulo(x)) {} Dynamic_Modint(u64 x) : val(bt.modulo(x)) {} Dynamic_Modint(int x) : val((x %= get_mod()) < 0 ? x + get_mod() : x) {} Dynamic_Modint(ll x) : val((x %= get_mod()) < 0 ? x + get_mod() : x) {} mint& operator+=(const mint& rhs) { val = (val += rhs.val) < umod() ? val : val - umod(); return *this; } mint& operator-=(const mint& rhs) { val = (val += umod() - rhs.val) < umod() ? val : val - umod(); return *this; } mint& operator*=(const mint& rhs) { val = bt.mul(val, rhs.val); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inverse(); } mint operator-() const { return mint() - *this; } mint pow(ll n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x, n >>= 1; } return r; } mint inverse() const { int x = val, mod = get_mod(); int a = x, 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; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs.val == rhs.val; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs.val != rhs.val; } #ifdef FASTIO void write() { fastio::printer.write(val); } void read() { fastio::scanner.read(val); val = bt.modulo(val); } #endif static pair& get_ntt() { static pair p = {-1, -1}; return p; } static void set_ntt_info() { int mod = get_mod(); int k = lowbit(mod - 1); int r = primitive_root(mod); r = mod_pow(r, (mod - 1) >> k, mod); get_ntt() = {k, r}; } static pair ntt_info() { return get_ntt(); } static bool can_ntt() { return ntt_info().fi != -1; } }; using dmint = Dynamic_Modint; Barrett dmint::bt(1); #line 5 "/home/maspy/compro/library/nt/four_square.hpp" // N = a^2+b^2+c^2+d^2 となる (a,b,c,d) をひとつ返す tuple four_square(ll N) { if (N == 0) return {0, 0, 0, 0}; using T4 = tuple; auto mul = [&](T4 x, T4 y) -> T4 { auto [x1, x2, x3, x4] = x; auto [y1, y2, y3, y4] = y; ll z1 = abs(x1 * y1 + x2 * y2 + x3 * y3 + x4 * y4); ll z2 = abs(x1 * y2 - x2 * y1 + x3 * y4 - x4 * y3); ll z3 = abs(x1 * y3 - x2 * y4 - x3 * y1 + x4 * y2); ll z4 = abs(x1 * y4 + x2 * y3 - x3 * y2 - x4 * y1); return {z1, z2, z3, z4}; }; auto solve_p = [&](ll p) -> T4 { if (p == 2) return {1, 1, 0, 0}; using mint = dmint; mint::set_mod(p); auto [a, b] = [&]() -> pair { while (1) { ll a = RNG(0, p); ll bb = (p - 1 - a * a) % p; if (bb < 0) bb += p; ll b = mod_sqrt(bb).val; if ((a * a + b * b + 1) % p == 0) return {a, b}; } return {0, 0}; }(); T4 x = {a, b, 1, 0}; while (1) { auto& [x1, x2, x3, x4] = x; chmin(x1, p - x1), chmin(x2, p - x2), chmin(x3, p - x3), chmin(x4, p - x4); ll m = (x1 * x1 + x2 * x2 + x3 * x3 + x4 * x4) / p; if (m == 1) break; ll y1 = x1 % m, y2 = x2 % m, y3 = x3 % m, y4 = x4 % m; if (y1 > m / 2) y1 -= m; if (y2 > m / 2) y2 -= m; if (y3 > m / 2) y3 -= m; if (y4 > m / 2) y4 -= m; auto [z1, z2, z3, z4] = mul(x, {y1, y2, y3, y4}); x = mt(z1 / m, z2 / m, z3 / m, z4 / m); } return x; }; T4 x = {1, 0, 0, 0}; for (auto&& [p, e]: factor(N)) { FOR(e) x = mul(x, solve_p(p)); } return x; } #line 4 "main.cpp" void solve() { LL(N); auto [a, b, c, d] = four_square(N); vi ANS; FOR(4) { tie(a, b, c, d) = mt(b, c, d, a); if (a > 0) ANS.eb(a * a); } print(len(ANS)); print(ANS); } signed main() { int T = 1; // INT(T); FOR(T) solve(); return 0; }