// #pragma GCC target("avx2") // #pragma GCC optimize("O3") // #pragma GCC optimize("unroll-loops") #include using namespace std; namespace templates { // type using ll = long long; using ull = unsigned long long; template using pq = priority_queue; template using qp = priority_queue, greater>; #define vec(T, A, ...) vector A(__VA_ARGS__); #define vvec(T, A, h, ...) vector> A(h, vector(__VA_ARGS__)); #define vvvec(T, A, h1, h2, ...) vector>> A(h1, vector>(h2, vector(__VA_ARGS__))); // for loop #define fori1(a) for (ll _ = 0; _ < (a); _++) #define fori2(i, a) for (ll i = 0; i < (a); i++) #define fori3(i, a, b) for (ll i = (a); i < (b); i++) #define fori4(i, a, b, c) for (ll i = (a); ((c) > 0 || i > (b)) && ((c) < 0 || i < (b)); i += (c)) #define overload4(a, b, c, d, e, ...) e #define fori(...) overload4(__VA_ARGS__, fori4, fori3, fori2, fori1)(__VA_ARGS__) // declare and input // clang-format off #define INT(...) int __VA_ARGS__; inp(__VA_ARGS__); #define LL(...) ll __VA_ARGS__; inp(__VA_ARGS__); #define STRING(...) string __VA_ARGS__; inp(__VA_ARGS__); #define CHAR(...) char __VA_ARGS__; inp(__VA_ARGS__); #define DOUBLE(...) double __VA_ARGS__; STRING(str___); __VA_ARGS__ = stod(str___); #define VEC(T, A, n) vector A(n); inp(A); #define VVEC(T, A, n, m) vector> A(n, vector(m)); inp(A); // clang-format on // const value const ll MOD1 = 1000000007; const ll MOD9 = 998244353; const double PI = acos(-1); // other macro #ifndef RIN__LOCAL #define endl "\n" #endif #define spa ' ' #define len(A) ll(A.size()) #define all(A) begin(A), end(A) // function vector stoc(string &S) { int n = S.size(); vector ret(n); for (int i = 0; i < n; i++) ret[i] = S[i]; return ret; } string ctos(vector &S) { int n = S.size(); string ret = ""; for (int i = 0; i < n; i++) ret += S[i]; return ret; } template auto min(const T &a) { return *min_element(all(a)); } template auto max(const T &a) { return *max_element(all(a)); } template auto clamp(T &a, const S &l, const S &r) { return (a > r ? r : a < l ? l : a); } 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); } template inline bool chclamp(T &a, const S &l, const S &r) { auto b = clamp(a, l, r); return (a != b ? a = b, 1 : 0); } template T sum(vector &A) { T tot = 0; for (auto a : A) tot += a; return tot; } template vector compression(vector X) { sort(all(X)); X.erase(unique(all(X)), X.end()); return X; } // input and output namespace io { // vector template istream &operator>>(istream &is, vector &A) { for (auto &a : A) is >> a; return is; } template ostream &operator<<(ostream &os, vector &A) { for (size_t i = 0; i < A.size(); i++) { os << A[i]; if (i != A.size() - 1) os << ' '; } return os; } // vector> template istream &operator>>(istream &is, vector> &A) { for (auto &a : A) is >> a; return is; } template ostream &operator<<(ostream &os, vector> &A) { for (size_t i = 0; i < A.size(); i++) { os << A[i]; if (i != A.size() - 1) os << endl; } return os; } // pair template istream &operator>>(istream &is, pair &A) { is >> A.first >> A.second; return is; } template ostream &operator<<(ostream &os, pair &A) { os << A.first << ' ' << A.second; return os; } // vector> template istream &operator>>(istream &is, vector> &A) { for (size_t i = 0; i < A.size(); i++) { is >> A[i]; } return is; } template ostream &operator<<(ostream &os, vector> &A) { for (size_t i = 0; i < A.size(); i++) { os << A[i]; if (i != A.size() - 1) os << endl; } return os; } // tuple template struct TuplePrint { static ostream &print(ostream &os, const T &t) { TuplePrint::print(os, t); os << ' ' << get(t); return os; } }; template struct TuplePrint { static ostream &print(ostream &os, const T &t) { os << get<0>(t); return os; } }; template ostream &operator<<(ostream &os, const tuple &t) { TuplePrint::print(os, t); return os; } // io functions void FLUSH() { cout << flush; } void print() { cout << endl; } template void print(Head &&head, Tail &&...tail) { cout << head; if (sizeof...(Tail)) cout << spa; print(forward(tail)...); } template void prisep(vector &A, S sep) { int n = A.size(); for (int i = 0; i < n; i++) { cout << A[i]; if (i != n - 1) cout << sep; } cout << endl; } template void priend(T A, S end) { cout << A << end; } template void prispa(T A) { priend(A, spa); } template bool printif(bool f, T A, S B) { if (f) print(A); else print(B); return f; } template void inp(T &...a) { (cin >> ... >> a); } } // namespace io using namespace io; // read graph vector> read_edges(int n, int m, bool direct = false, int indexed = 1) { vector> edges(n, vector()); for (int i = 0; i < m; i++) { INT(u, v); u -= indexed; v -= indexed; edges[u].push_back(v); if (!direct) edges[v].push_back(u); } return edges; } vector> read_tree(int n, int indexed = 1) { return read_edges(n, n - 1, false, indexed); } template vector>> read_wedges(int n, int m, bool direct = false, int indexed = 1) { vector>> edges(n, vector>()); for (int i = 0; i < m; i++) { INT(u, v); T w; inp(w); u -= indexed; v -= indexed; edges[u].push_back({v, w}); if (!direct) edges[v].push_back({u, w}); } return edges; } template vector>> read_wtree(int n, int indexed = 1) { return read_wedges(n, n - 1, false, indexed); } // yes / no namespace yesno { // yes inline bool yes(bool f = true) { cout << (f ? "yes" : "no") << endl; return f; } inline bool Yes(bool f = true) { cout << (f ? "Yes" : "No") << endl; return f; } inline bool YES(bool f = true) { cout << (f ? "YES" : "NO") << endl; return f; } // no inline bool no(bool f = true) { cout << (!f ? "yes" : "no") << endl; return f; } inline bool No(bool f = true) { cout << (!f ? "Yes" : "No") << endl; return f; } inline bool NO(bool f = true) { cout << (!f ? "YES" : "NO") << endl; return f; } // possible inline bool possible(bool f = true) { cout << (f ? "possible" : "impossible") << endl; return f; } inline bool Possible(bool f = true) { cout << (f ? "Possible" : "Impossible") << endl; return f; } inline bool POSSIBLE(bool f = true) { cout << (f ? "POSSIBLE" : "IMPOSSIBLE") << endl; return f; } // impossible inline bool impossible(bool f = true) { cout << (!f ? "possible" : "impossible") << endl; return f; } inline bool Impossible(bool f = true) { cout << (!f ? "Possible" : "Impossible") << endl; return f; } inline bool IMPOSSIBLE(bool f = true) { cout << (!f ? "POSSIBLE" : "IMPOSSIBLE") << endl; return f; } // Alice Bob inline bool Alice(bool f = true) { cout << (f ? "Alice" : "Bob") << endl; return f; } inline bool Bob(bool f = true) { cout << (f ? "Bob" : "Alice") << endl; return f; } // Takahashi Aoki inline bool Takahashi(bool f = true) { cout << (f ? "Takahashi" : "Aoki") << endl; return f; } inline bool Aoki(bool f = true) { cout << (f ? "Aoki" : "Takahashi") << endl; return f; } } // namespace yesno using namespace yesno; } // namespace templates using namespace templates; template struct Matrix { int n, m; std::vector> A; Matrix() = default; Matrix(int n, int m) : n(n), m(m), A(n, std::vector(m, 0)) {} Matrix(int n) : n(n), m(n), A(n, std::vector(n, 0)) {} Matrix(std::vector> A) : n(A.size()), m(A[0].size()), A(A) {} inline const std::vector &operator[](int k) const { return (A.at(k)); } inline std::vector &operator[](int k) { return (A.at(k)); } Matrix T() { Matrix B(m, n); for (int i = 0; i < m; i++) for (int j = 0; j < n; j++) { B.A[i][j] = A[j][i]; } return B; } Matrix &operator+=(const Matrix &B) { assert(n == int(B.A.size())); assert(m == int(B.A[0].size())); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) { this->A[i][j] += B[i][j]; } return *this; } Matrix &operator-=(const Matrix &B) { assert(n == int(B.A.size())); assert(m == int(B.A[0].size())); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) { this->A[i][j] -= B[i][j]; } return *this; } Matrix &operator*=(const Matrix &B) { int k = B[0].size(); assert(m == int(B.A.size())); std::vector> C(n, std::vector(k, 0)); for (int i = 0; i < n; i++) for (int j = 0; j < k; j++) { for (int l = 0; l < m; l++) { C[i][j] += this->A[i][l] * B[l][j]; } } swap(this->A, C); return *this; } template Matrix &operator*=(const Ti x) { for (auto &row : A) { for (auto &e : row) { e *= x; } } return *this; } Matrix operator-() { return (Matrix(*this) *= -1); } Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); } Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); } Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); } type det() { auto arr = A; assert(n == m); type ret = 1; for (int i = 0; i < n; i++) { if (arr[i][i] == 0) { bool ng = true; for (int j = i + 1; j < n; j++) { if (arr[j][i] == 0) continue; swap(arr[i], arr[j]); ret *= -1; ng = false; break; } if (ng) return 0; } ret *= arr[i][i]; type inv = type(1) / arr[i][i]; for (int j = i; j < n; j++) arr[i][j] *= inv; for (int j = i + 1; j < n; j++) { type x = arr[j][i]; for (int k = i; k < n; k++) { arr[j][k] -= arr[i][k] * x; } } } return ret; } void I() { assert(n == m); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { if (i == j) A[i][j] = 1; else A[i][j] = 0; } } } Matrix inv() { assert(n == m); Matrix ret(n); ret.I(); auto &B = ret.A; auto arr = A; for (int j = 0; j < n; j++) { int ii = -1; for (int i = j; i < n; i++) { if (arr[i][j] != 0) { ii = i; break; } } if (ii == -1) { return {}; } swap(arr[j], arr[ii]); swap(B[j], B[ii]); ii = j; type inv = type(1) / arr[ii][j]; for (int jj = 0; jj < n; jj++) { B[ii][jj] *= inv; arr[ii][jj] *= inv; } for (int i = 0; i < n; i++) { if (i == ii) continue; type t = arr[i][j]; for (int jj = 0; jj < n; jj++) { arr[i][jj] -= arr[ii][jj] * t; B[i][jj] -= B[ii][jj] * t; } } } return ret; } int choose_pivot(int h, int c) const { for (int j = h; j < n; j++) { if (A[j][c] != type(0)) return j; } return -1; } int rank() const { auto arr = *this; int ret = 0; for (int i = 0; i < m; i++) { int j = arr.choose_pivot(ret, i); if (j == -1) continue; swap(arr[ret], arr[j]); type inv = type(1) / arr[ret][i]; for (int k = i; k < m; k++) { arr[ret][k] *= inv; } for (int j = ret + 1; j < n; j++) { type x = arr[j][i]; for (int k = i; k < m; k++) { arr[j][k] -= arr[ret][k] * x; } } ret++; } return ret; } friend std::ostream &operator<<(std::ostream &os, const Matrix &p) { for (int i = 0; i < p.n; i++) { for (auto &x : p.A[i]) { os << x << " "; } if (i != p.n - 1) { os << "\n"; } } return (os); } friend std::istream &operator>>(std::istream &is, Matrix &p) { for (auto &row : p.A) { for (auto &x : row) { is >> x; } } return (is); } }; template struct Modint { int x; Modint() : x(0) {} Modint(int64_t y) { if (y >= 0) x = y % MOD; else x = (y % MOD + MOD) % MOD; } Modint &operator+=(const Modint &p) { x += p.x; if (x >= MOD) x -= MOD; return *this; } Modint &operator-=(const Modint &p) { x -= p.x; if (x < 0) x += MOD; return *this; } Modint &operator*=(const Modint &p) { x = int(1LL * x * p.x % MOD); return *this; } Modint &operator/=(const Modint &p) { *this *= p.inverse(); return *this; } Modint &operator%=(const Modint &p) { assert(p.x == 0); return *this; } Modint operator-() const { return Modint(-x); } Modint &operator++() { x++; if (x == MOD) x = 0; return *this; } Modint &operator--() { if (x == 0) x = MOD; x--; return *this; } Modint operator++(int) { Modint result = *this; ++*this; return result; } Modint operator--(int) { Modint result = *this; --*this; return result; } friend Modint operator+(const Modint &lhs, const Modint &rhs) { return Modint(lhs) += rhs; } friend Modint operator-(const Modint &lhs, const Modint &rhs) { return Modint(lhs) -= rhs; } friend Modint operator*(const Modint &lhs, const Modint &rhs) { return Modint(lhs) *= rhs; } friend Modint operator/(const Modint &lhs, const Modint &rhs) { return Modint(lhs) /= rhs; } friend Modint operator%(const Modint &lhs, const Modint &rhs) { assert(rhs.x == 0); return Modint(lhs); } bool operator==(const Modint &p) const { return x == p.x; } bool operator!=(const Modint &p) const { return x != p.x; } bool operator<(const Modint &rhs) const { return x < rhs.x; } bool operator<=(const Modint &rhs) const { return x <= rhs.x; } bool operator>(const Modint &rhs) const { return x > rhs.x; } bool operator>=(const Modint &rhs) const { return x >= rhs.x; } Modint inverse() const { int a = x, b = MOD, u = 1, v = 0, t; while (b > 0) { t = a / b; a -= t * b; u -= t * v; std::swap(a, b); std::swap(u, v); } return Modint(u); } Modint pow(int64_t k) const { Modint ret(1); Modint y(x); while (k > 0) { if (k & 1) ret *= y; y *= y; k >>= 1; } return ret; } friend std::ostream &operator<<(std::ostream &os, const Modint &p) { return os << p.x; } friend std::istream &operator>>(std::istream &is, Modint &p) { int64_t y; is >> y; p = Modint(y); return (is); } static int get_mod() { return MOD; } }; struct Arbitrary_Modint { int x; static int MOD; static void set_mod(int mod) { MOD = mod; } Arbitrary_Modint() : x(0) {} Arbitrary_Modint(int64_t y) { if (y >= 0) x = y % MOD; else x = (y % MOD + MOD) % MOD; } Arbitrary_Modint &operator+=(const Arbitrary_Modint &p) { x += p.x; if (x >= MOD) x -= MOD; return *this; } Arbitrary_Modint &operator-=(const Arbitrary_Modint &p) { x -= p.x; if (x < 0) x += MOD; return *this; } Arbitrary_Modint &operator*=(const Arbitrary_Modint &p) { x = int(1LL * x * p.x % MOD); return *this; } Arbitrary_Modint &operator/=(const Arbitrary_Modint &p) { *this *= p.inverse(); return *this; } Arbitrary_Modint &operator%=(const Arbitrary_Modint &p) { assert(p.x == 0); return *this; } Arbitrary_Modint operator-() const { return Arbitrary_Modint(-x); } Arbitrary_Modint &operator++() { x++; if (x == MOD) x = 0; return *this; } Arbitrary_Modint &operator--() { if (x == 0) x = MOD; x--; return *this; } Arbitrary_Modint operator++(int) { Arbitrary_Modint result = *this; ++*this; return result; } Arbitrary_Modint operator--(int) { Arbitrary_Modint result = *this; --*this; return result; } friend Arbitrary_Modint operator+(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) { return Arbitrary_Modint(lhs) += rhs; } friend Arbitrary_Modint operator-(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) { return Arbitrary_Modint(lhs) -= rhs; } friend Arbitrary_Modint operator*(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) { return Arbitrary_Modint(lhs) *= rhs; } friend Arbitrary_Modint operator/(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) { return Arbitrary_Modint(lhs) /= rhs; } friend Arbitrary_Modint operator%(const Arbitrary_Modint &lhs, const Arbitrary_Modint &rhs) { assert(rhs.x == 0); return Arbitrary_Modint(lhs); } bool operator==(const Arbitrary_Modint &p) const { return x == p.x; } bool operator!=(const Arbitrary_Modint &p) const { return x != p.x; } bool operator<(const Arbitrary_Modint &rhs) { return x < rhs.x; } bool operator<=(const Arbitrary_Modint &rhs) { return x <= rhs.x; } bool operator>(const Arbitrary_Modint &rhs) { return x > rhs.x; } bool operator>=(const Arbitrary_Modint &rhs) { return x >= rhs.x; } Arbitrary_Modint inverse() const { int a = x, b = MOD, u = 1, v = 0, t; while (b > 0) { t = a / b; a -= t * b; u -= t * v; std::swap(a, b); std::swap(u, v); } return Arbitrary_Modint(u); } Arbitrary_Modint pow(int64_t k) const { Arbitrary_Modint ret(1); Arbitrary_Modint y(x); while (k > 0) { if (k & 1) ret *= y; y *= y; k >>= 1; } return ret; } friend std::ostream &operator<<(std::ostream &os, const Arbitrary_Modint &p) { return os << p.x; } friend std::istream &operator>>(std::istream &is, Arbitrary_Modint &p) { int64_t y; is >> y; p = Arbitrary_Modint(y); return (is); } static int get_mod() { return MOD; } }; int Arbitrary_Modint::MOD = 998244353; using modint9 = Modint<998244353>; using modint1 = Modint<1000000007>; using modint = Arbitrary_Modint; struct RandomNumberGenerator { std::mt19937 mt; RandomNumberGenerator() : mt(std::chrono::steady_clock::now().time_since_epoch().count()) {} RandomNumberGenerator(int seed) : mt(seed) {} int operator()(int a, int b) { std::uniform_int_distribution dist(a, b - 1); return dist(mt); } int operator()(int b) { return (*this)(0, b); } template void shuffle(std::vector &v) { std::shuffle(v.begin(), v.end(), mt); } }; using mint = modint9; void solve() { LL(n, m); RandomNumberGenerator rng(42); vec(ll, U, m); vec(ll, V, m); vec(ll, W, m); fori(i, m) { inp(U[i], V[i], W[i]); U[i]--; V[i]--; W[i]--; } set se; while (1) { Matrix A(n, n); fori(i, m) { ll x = rng(1 << 30); ll u = U[i]; ll v = V[i]; ll w = W[i]; A[u][v] += x; A[v][u] -= x; A[v][w] += x; A[w][v] -= x; A[w][u] += x; A[u][w] -= x; } auto res = A.rank(); if (se.count(res)) { print(res / 2); return; } se.insert(res); } } int main() { cin.tie(0)->sync_with_stdio(0); cout << fixed << setprecision(12); int t; t = 1; // cin >> t; while (t--) solve(); return 0; }