// #pragma GCC target("avx2")
// #pragma GCC optimize("O3")
// #pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
// #define INTERACTIVE

namespace templates {
// type
using ll  = long long;
using ull = unsigned long long;
using Pii = pair<int, int>;
using Pil = pair<int, ll>;
using Pli = pair<ll, int>;
using Pll = pair<ll, ll>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using qp = priority_queue<T, vector<T>, greater<T>>;
// clang-format off
#define vec(T, A, ...) vector<T> A(__VA_ARGS__);
#define vvec(T, A, h, ...) vector<vector<T>> A(h, vector<T>(__VA_ARGS__));
#define vvvec(T, A, h1, h2, ...) vector<vector<vector<T>>> A(h1, vector<vector<T>>(h2, vector<T>(__VA_ARGS__)));
// clang-format on

// 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<T> A(n); inp(A);
#define VVEC(T, A, n, m) vector<vector<T>> A(n, vector<T>(m)); inp(A);
// clang-format on

// const value
const ll MOD1   = 1000000007;
const ll MOD9   = 998244353;
const double PI = acos(-1);

// other macro
#if !defined(RIN__LOCAL) && !defined(INTERACTIVE)
#define endl "\n"
#endif
#define spa ' '
#define len(A) ll(A.size())
#define all(A) begin(A), end(A)

// function
vector<char> stoc(string &S) {
    int n = S.size();
    vector<char> ret(n);
    for (int i = 0; i < n; i++) ret[i] = S[i];
    return ret;
}
string ctos(vector<char> &S) {
    int n      = S.size();
    string ret = "";
    for (int i = 0; i < n; i++) ret += S[i];
    return ret;
}

template <class T>
auto min(const T &a) {
    return *min_element(all(a));
}
template <class T>
auto max(const T &a) {
    return *max_element(all(a));
}
template <class T, class S>
auto clamp(T &a, const S &l, const S &r) {
    return (a > r ? r : a < l ? l : a);
}
template <class T, class S>
inline bool chmax(T &a, const S &b) {
    return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
    return (a > b ? a = b, 1 : 0);
}
template <class T, class S>
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 <typename T>
T sum(vector<T> &A) {
    T tot = 0;
    for (auto a : A) tot += a;
    return tot;
}

template <typename T>
vector<T> compression(vector<T> X) {
    sort(all(X));
    X.erase(unique(all(X)), X.end());
    return X;
}

// input and output
namespace io {
// __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;
}

// vector<T>
template <typename T>
istream &operator>>(istream &is, vector<T> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<T> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << ' ';
    }
    return os;
}

// vector<vector<T>>
template <typename T>
istream &operator>>(istream &is, vector<vector<T>> &A) {
    for (auto &a : A) is >> a;
    return is;
}
template <typename T>
ostream &operator<<(ostream &os, vector<vector<T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// pair<S, T>
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &A) {
    is >> A.first >> A.second;
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, pair<S, T> &A) {
    os << A.first << ' ' << A.second;
    return os;
}

// vector<pair<S, T>>
template <typename S, typename T>
istream &operator>>(istream &is, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        is >> A[i];
    }
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, vector<pair<S, T>> &A) {
    for (size_t i = 0; i < A.size(); i++) {
        os << A[i];
        if (i != A.size() - 1) os << endl;
    }
    return os;
}

// tuple
template <typename T, size_t N>
struct TuplePrint {
    static ostream &print(ostream &os, const T &t) {
        TuplePrint<T, N - 1>::print(os, t);
        os << ' ' << get<N - 1>(t);
        return os;
    }
};
template <typename T>
struct TuplePrint<T, 1> {
    static ostream &print(ostream &os, const T &t) {
        os << get<0>(t);
        return os;
    }
};
template <typename... Args>
ostream &operator<<(ostream &os, const tuple<Args...> &t) {
    TuplePrint<decltype(t), sizeof...(Args)>::print(os, t);
    return os;
}

// io functions
void FLUSH() {
    cout << flush;
}

void print() {
    cout << endl;
}
template <class Head, class... Tail>
void print(Head &&head, Tail &&...tail) {
    cout << head;
    if (sizeof...(Tail)) cout << spa;
    print(std::forward<Tail>(tail)...);
}

template <typename T, typename S>
void prisep(vector<T> &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 <typename T, typename S>
void priend(T A, S end) {
    cout << A << end;
}
template <typename T>
void prispa(T A) {
    priend(A, spa);
}
template <typename T, typename S>
bool printif(bool f, T A, S B) {
    if (f)
        print(A);
    else
        print(B);
    return f;
}

template <class... T>
void inp(T &...a) {
    (cin >> ... >> a);
}

} // namespace io
using namespace io;

// read graph
vector<vector<int>> read_edges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<int>> edges(n, vector<int>());
    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<vector<int>> read_tree(int n, int indexed = 1) {
    return read_edges(n, n - 1, false, indexed);
}

template <typename T = long long>
vector<vector<pair<int, T>>> read_wedges(int n, int m, bool direct = false, int indexed = 1) {
    vector<vector<pair<int, T>>> edges(n, vector<pair<int, T>>());
    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 <typename T = long long>
vector<vector<pair<int, T>>> read_wtree(int n, int indexed = 1) {
    return read_wedges<T>(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;

struct UnionFind {
    int n;
    std::vector<int> par;
    int group;

    UnionFind() = default;
    UnionFind(int n) : n(n) {
        par.assign(n, -1);
        group = n;
    }

    int find(int x) {
        if (par[x] < 0) return x;
        par[x] = find(par[x]);
        return par[x];
    }

    bool unite(int x, int y) {
        x = find(x);
        y = find(y);
        if (x == y) return false;
        if (par[x] > par[y]) std::swap(x, y);
        group--;
        par[x] += par[y];
        par[y] = x;
        return true;
    }

    bool same(int x, int y) {
        return find(x) == find(y);
    }

    int size(int x) {
        return -par[find(x)];
    }

    std::vector<int> roots() {
        std::vector<int> ret;
        for (int i = 0; i < n; i++) {
            if (i == find(i)) ret.push_back(i);
        }
        return ret;
    }

    bool isroot(int x) {
        return x == find(x);
    }
};

template <typename type>
struct Matrix {
    int n, m;
    std::vector<std::vector<type>> A;
    Matrix() = default;
    Matrix(int n, int m) : n(n), m(m), A(n, std::vector<type>(m, 0)) {}
    Matrix(int n) : n(n), m(n), A(n, std::vector<type>(n, 0)) {}
    Matrix(std::vector<std::vector<type>> A) : n(A.size()), m(A[0].size()), A(A) {}

    inline const std::vector<type> &operator[](int k) const {
        return (A.at(k));
    }

    inline std::vector<type> &operator[](int k) {
        return (A.at(k));
    }

    Matrix T() {
        Matrix<type> 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 std::vector<std::vector<type>> &B) {
        n = B.size();
        m = B[0].size();
        A = B;
        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) {
        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<std::vector<type>> C(n, std::vector<type>(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;
    }

    std::vector<type> operator*(const std::vector<type> &x) {
        assert(m == int(x.size()));
        std::vector<type> ret(n, 0);
        for (int i = 0; i < n; i++)
            for (int j = 0; j < m; j++) ret[i] += this->A[i][j] * x[j];
        return ret;
    }

    template <typename Ti>
    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<type> inv() {
        assert(n == m);
        Matrix<type> 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;
        if (arr.n < arr.m) {
            arr = arr.T();
        }
        int ret = 0;
        for (int i = 0; i < arr.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 < arr.m; k++) {
                arr[ret][k] *= inv;
            }
            for (int j = ret + 1; j < arr.n; j++) {
                type x = arr[j][i];
                for (int k = i; k < arr.m; k++) {
                    arr[j][k] -= arr[ret][k] * x;
                }
            }
            ret++;
        }
        return ret;
    }

    Matrix<type> pow(long long k) {
        assert(n == m);
        Matrix<type> B(n);
        B.I();
        Matrix<type> A(*this);
        while (k) {
            if (k & 1) B *= A;
            A *= A;
            k >>= 1;
        }
        return B;
    }

    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 <typename T>
void hessenberg_reduction(Matrix<T> &A) {
    assert(A.n == A.m);
    for (int r = 0; r < A.n - 2; r++) {
        int piv = A.choose_pivot(r + 1, r);
        if (piv < 0) continue;

        for (int i = 0; i < A.n; i++) {
            std::swap(A[r + 1][i], A[piv][i]);
        }
        for (int i = 0; i < A.n; i++) {
            std::swap(A[i][r + 1], A[i][piv]);
        }

        T inv = T(1) / A[r + 1][r];
        for (int i = r + 2; i < A.n; i++) {
            T t = A[i][r] * inv;
            for (int j = 0; j < A.n; j++) {
                A[i][j] -= A[r + 1][j] * t;
            }
            for (int j = 0; j < A.n; j++) {
                A[j][r + 1] += A[j][i] * t;
            }
        }
    }
}

template <typename T>
std::vector<T> characteristic_polynomial(Matrix<T> A) {
    hessenberg_reduction(A);
    if (A.n == 0) {
        return std::vector<T>{T(1)};
    }
    std::vector<std::vector<T>> P(A.n + 1);
    P[0] = {T(1)};
    for (int i = 0; i < A.n; i++) {
        P[i + 1].assign(i + 2, T(0));
        for (int j = 0; j < i + 1; j++) {
            P[i + 1][j + 1] += P[i][j];
        }
        for (int j = 0; j < i + 1; j++) {
            P[i + 1][j] -= P[i][j] * A[i][i];
        }
        T x = T(1);
        for (int j = i - 1; j >= 0; j--) {
            x *= A[j + 1][j];
            T h = -A[j][i] * x;
            for (int k = 0; k < j + 1; k++) {
                P[i + 1][k] += h * P[j][k];
            }
        }
    }
    return P.back();
}

template <typename T>
std::vector<T> determinant_of_first_degree_poly_mat(Matrix<T> M0, Matrix<T> M1) {
    int n         = M0.n;
    int cx        = 0;
    T detAdetBinv = T(1);
    for (int p = 0; p < n; p++) {
        int pivot = M1.choose_pivot(p, p);
        if (pivot < 0) {
            if (++cx > n) {
                return std::vector<T>(n + 1);
            }

            for (int i = 0; i < p; i++) {
                T v      = M1[i][p];
                M1[i][p] = 0;
                for (int i2 = 0; i2 < n; i2++) {
                    M0[i2][p] -= v * M0[i2][i];
                }
            }

            for (int i = 0; i < n; i++) {
                std::swap(M0[i][p], M1[i][p]);
            }

            p--;
            continue;
        }

        if (pivot != p) {
            std::swap(M1[pivot], M1[p]);
            std::swap(M0[pivot], M0[p]);
            detAdetBinv *= -1;
        }

        T v    = M1[p][p];
        T vinv = T(1) / v;
        detAdetBinv *= v;

        for (int j = 0; j < n; j++) {
            M0[p][j] *= vinv;
            M1[p][j] *= vinv;
        }

        for (int i = 0; i < n; i++) {
            if (i == p) continue;
            T v = M1[i][p];
            for (int j = 0; j < n; j++) {
                M0[i][j] -= v * M0[p][j];
                M1[i][j] -= v * M1[p][j];
            }
        }
    }

    M0 *= -1;

    auto poly = characteristic_polynomial(M0);
    poly.erase(poly.begin(), poly.begin() + cx);
    for (size_t i = 0; i < poly.size(); i++) {
        poly[i] *= detAdetBinv;
    }
    poly.resize(n + 1);

    return poly;
}

template <int MOD>
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<MOD>(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;
using mint    = modint9;

void solve() {
    LL(n, m);
    UnionFind UF(n);
    vvec(bool, E, n, n, false);
    fori(m) {
        INT(u, v);
        u--;
        v--;
        E[u][v] = true;
        UF.unite(u, v);
    }
    if (UF.group == 1) {
        print(0);
        Matrix<mint> A(n - 1), B(n - 1);
        fori(i, n) fori(j, i + 1, n) {
            if (E[i][j]) {
                A[i][i]++;
                if (j != n - 1) {
                    A[i][j]--;
                    A[j][i]--;
                    A[j][j]++;
                }
            } else {
                B[i][i]++;
                if (j != n - 1) {
                    B[i][j]--;
                    B[j][i]--;
                    B[j][j]++;
                }
            }
        }
        auto res = determinant_of_first_degree_poly_mat(A, B);
        print(res[0] + res[1]);

    } else {
        vvec(int, group, n);
        fori(i, n) {
            group[UF.find(i)].push_back(i);
        }
        vec(ll, S, n, 0);
        fori(i, n) S[i] = group[i].size();
        sort(all(S));
        reverse(all(S));
        {
            mint c = n * (n - 1);
            fori(i, 2, n) {
                c -= S[i] * (S[i] - 1);
            }
            c -= (S[0] + S[1]) * (S[0] + S[1] - 1);
            print(c);
        }

        mint ans = 1;
        for (auto &g : group) {
            ll nn = g.size();
            if (nn <= 1) continue;
            Matrix<mint> A(nn - 1);
            fori(i, nn) fori(j, i + 1, nn) {
                if (E[g[i]][g[j]]) {
                    A[i][i]++;
                    if (j != nn - 1) {
                        A[i][j]--;
                        A[j][i]--;
                        A[j][j]++;
                    }
                }
            }
            ans *= A.det();
        }

        ll c = 0;
        for (auto s : S) {
            if (s == S[0]) c++;
        }
        if (c >= 2) {
            ans *= mint(c) * (c - 1) / 2 * S[0] * S[0];
        } else {
            ll c2 = 0;
            for (auto s : S) {
                if (s == S[1]) c2++;
            }
            ans *= mint(c) * c2 * S[0] * S[1];
        }
        print(ans);
    }
}

int main() {
#ifndef INTERACTIVE
    cin.tie(0)->sync_with_stdio(0);
#endif
    // cout << fixed << setprecision(12);
    int t;
    t = 1;
    // cin >> t;
    while (t--) solve();
    return 0;
}