#line 2 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"
using namespace std;

#include<bits/stdc++.h>
#line 1 "/Users/noya2/Desktop/Noya2_library/template/inout_old.hpp"
namespace noya2 {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p){
    os << p.first << " " << p.second;
    return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p){
    is >> p.first >> p.second;
    return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v){
    int s = (int)v.size();
    for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
    return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v){
    for (auto &x : v) is >> x;
    return is;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u){
    cin >> t;
    in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u){
    cout << t;
    if (sizeof...(u)) cout << sep;
    out(u...);
}

template<typename T>
void out(const vector<vector<T>> &vv){
    int s = (int)vv.size();
    for (int i = 0; i < s; i++) out(vv[i]);
}

struct IoSetup {
    IoSetup(){
        cin.tie(nullptr);
        ios::sync_with_stdio(false);
        cout << fixed << setprecision(15);
        cerr << fixed << setprecision(7);
    }
} iosetup_noya2;

} // namespace noya2
#line 1 "/Users/noya2/Desktop/Noya2_library/template/const.hpp"
namespace noya2{

const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 =  998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";

void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp"

#line 6 "/Users/noya2/Desktop/Noya2_library/template/utils.hpp"

namespace noya2{

unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
    if (a == 0 || b == 0) return a + b;
    int n = __builtin_ctzll(a); a >>= n;
    int m = __builtin_ctzll(b); b >>= m;
    while (a != b) {
        int mm = __builtin_ctzll(a - b);
        bool f = a > b;
        unsigned long long c = f ? a : b;
        b = f ? b : a;
        a = (c - b) >> mm;
    }
    return a << std::min(n, m);
}

template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(std::abs(a),std::abs(b))); }

long long sqrt_fast(long long n) {
    if (n <= 0) return 0;
    long long x = sqrt(n);
    while ((x + 1) * (x + 1) <= n) x++;
    while (x * x > n) x--;
    return x;
}

template<typename T> T floor_div(const T n, const T d) {
    assert(d != 0);
    return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}

template<typename T> T ceil_div(const T n, const T d) {
    assert(d != 0);
    return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}

template<typename T> void uniq(std::vector<T> &v){
    std::sort(v.begin(),v.end());
    v.erase(unique(v.begin(),v.end()),v.end());
}

template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }

template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }

template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }

} // namespace noya2
#line 8 "/Users/noya2/Desktop/Noya2_library/template/template.hpp"

#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()

using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;

namespace noya2{

/*　~ (. _________ . /)　*/

}

using namespace noya2;


#line 2 "c.cpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/graph/graph_query.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp"
#include<ranges>
#line 7 "/Users/noya2/Desktop/Noya2_library/data_structure/csr.hpp"

namespace noya2::internal {

template<class E>
struct csr {
    csr () {}
    csr (int _n) : n(_n) {}
    csr (int _n, int m) : n(_n){
        start.reserve(m);
        elist.reserve(m);
    }
    // ACL style constructor (do not have to call build)
    csr (int _n, const std::vector<std::pair<int,E>> &idx_elem) : n(_n), start(_n + 2), elist(idx_elem.size()) {
        for (auto &[i, e] : idx_elem){
            start[i + 2]++;
        }
        for (int i = 1; i < n; i++){
            start[i + 2] += start[i + 1];
        }
        for (auto &[i, e] : idx_elem){
            elist[start[i + 1]++] = e;
        }
        prepared = true;
    }
    int add(int idx, E elem){
        int eid = start.size();
        start.emplace_back(idx);
        elist.emplace_back(elem);
        return eid;
    }
    void build(){
        if (prepared) return ;
        int m = start.size();
        std::vector<E> nelist(m);
        std::vector<int> nstart(n + 2, 0);
        for (int i = 0; i < m; i++){
            nstart[start[i] + 2]++;
        }
        for (int i = 1; i < n; i++){
            nstart[i + 2] += nstart[i + 1];
        }
        for (int i = 0; i < m; i++){
            nelist[nstart[start[i] + 1]++] = elist[i];
        }
        swap(elist,nelist);
        swap(start,nstart);
        prepared = true;
    }
    const auto operator[](int idx) const {
        return std::ranges::subrange(elist.begin()+start[idx],elist.begin()+start[idx+1]);
    }
    auto operator[](int idx){
        return std::ranges::subrange(elist.begin()+start[idx],elist.begin()+start[idx+1]);
    }
    const auto operator()(int idx, int l, int r) const {
        return std::ranges::subrange(elist.begin()+start[idx]+l,elist.begin()+start[idx]+r);
    }
    auto operator()(int idx, int l, int r){
        return std::ranges::subrange(elist.begin()+start[idx]+l,elist.begin()+start[idx]+r);
    }
    int n;
    std::vector<int> start;
    std::vector<E> elist;
    bool prepared = false;
};

} // namespace noya2::internal
#line 2 "/Users/noya2/Desktop/Noya2_library/graph/unweighted_type.hpp"

namespace noya2 {

struct unweighted {};

} // namespace noya2
#line 6 "/Users/noya2/Desktop/Noya2_library/graph/graph_query.hpp"

#line 12 "/Users/noya2/Desktop/Noya2_library/graph/graph_query.hpp"

namespace noya2 {

template<typename Cost>
struct graph {
    int n;
    internal::csr<std::pair<int,Cost>> g;
    Cost dist_inf = std::numeric_limits<Cost>::max() / 2;
    graph (int _n = 0) : n(_n), g(_n) {}
    graph (int _n, int _m) : n(_n), g(_n,_m) {}
    // 有向辺を追加 (無向辺ではないことに注意！)
    int add_edge(int u, int v, Cost cost = 1){
        int id = g.add(u, {v,cost});
        return id;
    }
    template<bool directed>
    static graph input(int _n, int _m, int indexed = 1){
        if constexpr (directed){
            graph g(_n, _m*2);
            for (int i = 0; i < _m; i++){
                int u, v; std::cin >> u >> v;
                u -= indexed, v -= indexed;
                Cost c; std::cin >> c;
                g.add_edge(u, v, c);
                g.add_edge(v, u, c);
            }
            g.build();
            return g;
        }
        else {
            graph g(_n, _m);
            for (int i = 0; i < _m; i++){
                int u, v; std::cin >> u >> v;
                u -= indexed, v -= indexed;
                Cost c; std::cin >> c;
                g.add_edge(u, v, c);
            }
            g.build();
            return g;
        }
    }
    void build(){
        g.build();
    }
    void set_inf(Cost new_inf){
        dist_inf = new_inf;
    }
    std::vector<Cost> dijkstra(int s){
        g.build();
        std::vector<Cost> dist(n,dist_inf);
        dist[s] = 0;
        using P = std::pair<Cost,int>;
        std::priority_queue<P,std::vector<P>,std::greater<P>> pque;
        pque.push(P(0,s));
        while (!pque.empty()){
            auto [d, v] = pque.top(); pque.pop();
            if (dist[v] < d) continue;
            for (auto [u, c] : g[v]){
                if (chmin(dist[u],d+c)){
                    pque.push(P(dist[u],u));
                }
            }
        }
        return dist;
    }
    std::vector<int> reconstruct(int s, int t, const std::vector<Cost> &dist){
        if (dist[t] == dist_inf) return {};
        g.build();
        std::vector<int> from(n,-1);
        std::queue<int> que;
        que.push(s);
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (auto [u, c] : g[v]){
                if (from[u] == -1 && dist[u] == dist[v] + c){
                    from[u] = v;
                    que.push(u);
                }
            }
        }
        std::vector<int> ans = {t};
        while (t != s){
            t = from[t];
            ans.emplace_back(t);
        }
        std::reverse(ans.begin(),ans.end());
        return ans;
    }
    std::vector<Cost> bfs01(int s){
        g.build();
        std::vector<Cost> dist(n,dist_inf);
        dist[s] = 0;
        std::deque<int> que;
        que.push_back(s);
        while (!que.empty()){
            int v = que.front(); que.pop_front();
            for (auto [u, c] : g[v]){
                if (chmin(dist[u],dist[v]+c)){
                    if (c == 0) que.push_front(u);
                    else que.push_back(u);
                }
            }
        }
        return dist;
    }
    std::vector<Cost> bfs1(int s){
        g.build();
        std::vector<Cost> dist(n,dist_inf);
        dist[s] = 0;
        std::queue<int> que;
        que.push(s);
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (auto [u, c] : g[v]){
                if (chmin(dist[u],dist[v]+c)){
                    que.push(u);
                }
            }
        }
        return dist;
    }
    std::vector<Cost> bellman_ford(int s, bool &ng_cycle){
        g.build();
        std::vector<Cost> dist(n,dist_inf);
        std::vector<int> ng;
        dist[s] = 0;
        int tm = 0;
        while (tm < n){
            bool finish = true;
            for (int v = 0; v < n; v++){
                if (dist[v] == dist_inf) continue;
                for (auto [u, c] : g[v]){
                    if (chmin(dist[u],dist[v]+c)){
                        finish = false;
                        if (tm == n-1) ng.emplace_back(u);
                    }
                }
            }
            if (finish) break;
            tm++;
        }
        ng_cycle = (tm == n);
        if (ng_cycle){
            for (auto v : ng) dist[v] = -dist_inf;
            tm = n;
            while (tm--){
                for (int v = 0; v < n; v++){
                    if (dist[v] != -dist_inf) continue;
                    for (auto [u, c] : g[v]){
                        dist[u] = -dist_inf;
                    }
                }
            }
        }
        return dist;
    }
    std::vector<std::vector<Cost>> warshall_floyd(){
        g.build();
        std::vector<std::vector<Cost>> dist(n,std::vector<Cost>(n,dist_inf));
        for (int v = 0; v < n; v++){
            dist[v][v] = 0;
            for (auto [u, c] : g[v]){
                chmin(dist[v][u],c);
            }
        }
        for (int k = 0; k < n; k++){
            for (int i = 0; i < n; i++){
                for (int j = 0; j < n; j++){
                    chmin(dist[i][j],dist[i][k]+dist[k][j]);
                }
            }
        }
        return dist;
    }
    const auto operator[](int idx) const { return g[idx]; }
    auto operator[](int idx) { return g[idx]; }
};


template<>
struct graph<unweighted> {
    int n;
    internal::csr<int> g;
    int dist_inf = std::numeric_limits<int>::max() / 2;
    graph (int _n = 0) : n(_n), g(_n) {}
    graph (int _n, int _m) : n(_n), g(_n,_m) {}
    // 有向辺を追加 (無向辺ではないことに注意！)
    int add_edge(int u, int v){
        int id = g.add(u, v);
        return id;
    }
    template<bool directed>
    static graph input(int _n, int _m, int indexed = 1){
        if constexpr (directed){
            graph g(_n, _m*2);
            for (int i = 0; i < _m; i++){
                int u, v; std::cin >> u >> v;
                u -= indexed, v -= indexed;
                g.add_edge(u, v);
                g.add_edge(v, u);
            }
            g.build();
            return g;
        }
        else {
            graph g(_n, _m);
            for (int i = 0; i < _m; i++){
                int u, v; std::cin >> u >> v;
                u -= indexed, v -= indexed;
                g.add_edge(u, v);
            }
            g.build();
            return g;
        }
    }
    void build(){
        g.build();
    }
    void set_inf(int new_inf){
        dist_inf = new_inf;
    }
    std::vector<int> reconstruct(int s, int t, const std::vector<int> &dist){
        if (dist[t] == dist_inf) return {};
        g.build();
        std::vector<int> from(n,-1);
        std::queue<int> que;
        que.push(s);
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (auto u : g[v]){
                if (from[u] == -1 && dist[u] == dist[v] + 1){
                    from[u] = v;
                    que.push(u);
                }
            }
        }
        std::vector<int> ans = {t};
        while (t != s){
            t = from[t];
            ans.emplace_back(t);
        }
        std::reverse(ans.begin(),ans.end());
        return ans;
    }
    std::vector<int> bfs(int s){
        g.build();
        std::vector<int> dist(n,dist_inf);
        dist[s] = 0;
        std::queue<int> que;
        que.push(s);
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (auto u : g[v]){
                if (chmin(dist[u],dist[v]+1)){
                    que.push(u);
                }
            }
        }
        return dist;
    }
    const auto operator[](int idx) const { return g[idx]; }
    auto operator[](int idx) { return g[idx]; }
};

template<>
struct graph<bool> {
    int n;
    internal::csr<std::pair<int,bool>> g;
    int dist_inf = std::numeric_limits<int>::max() / 2;
    graph (int _n = 0) : n(_n), g(_n) {}
    graph (int _n, int _m) : n(_n), g(_n,_m) {}
    // 有向辺を追加 (無向辺ではないことに注意！)
    int add_edge(int u, int v, bool cost){
        int id = g.add(u, {v, cost});
        return id;
    }
    void build(){
        g.build();
    }
    void set_inf(int new_inf){
        dist_inf = new_inf;
    }
    std::vector<int> reconstruct(int s, int t, const std::vector<int> &dist){
        if (dist[t] == dist_inf) return {};
        g.build();
        std::vector<int> from(n,-1);
        std::queue<int> que;
        que.push(s);
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (auto [u, b] : g[v]){
                int c = (int)b;
                if (from[u] == -1 && dist[u] == dist[v] + c){
                    from[u] = v;
                    que.push(u);
                }
            }
        }
        std::vector<int> ans = {t};
        while (t != s){
            t = from[t];
            ans.emplace_back(t);
        }
        std::reverse(ans.begin(),ans.end());
        return ans;
    }
    std::vector<int> bfs01(int s){
        g.build();
        std::vector<int> dist(n,dist_inf);
        dist[s] = 0;
        std::deque<int> que;
        que.push_back(s);
        while (!que.empty()){
            int v = que.front(); que.pop_front();
            for (auto [u, b] : g[v]){
                int c = (int)b;
                if (chmin(dist[u],dist[v]+c)){
                    if (c == 0) que.push_front(u);
                    else que.push_back(u);
                }
            }
        }
        return dist;
    }
    const auto operator[](int idx) const { return g[idx]; }
    auto operator[](int idx) { return g[idx]; }
};

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/math/matrix.hpp"

#line 8 "/Users/noya2/Desktop/Noya2_library/math/matrix.hpp"

namespace noya2 {

template<typename T, size_t hw = -1uz>
struct matrix {
    static constexpr int h = hw, w = hw;
    std::array<T, hw*hw> m;
    matrix () : m({}) {}
    matrix (const std::array<T, hw*hw> &_m) : m(_m) {}
    matrix (const std::array<std::array<T, hw>, hw> &_m){
        for (int i = 0; i < h; i++) for (int j = 0; j < w; j++){
            m[idx(i,j)] = _m[i][j];
        }
    }
    matrix (const std::vector<std::vector<T>> &_m){
        for (int i = 0; i < h; i++) for (int j = 0; j < w; j++){
            m[idx(i,j)] = _m[i][j];
        }
    }
    auto operator[](int i) const {
        return std::ranges::subrange(m.begin()+i*w,m.begin()+(i+1)*w);
    }
    auto operator[](int i){
        return std::ranges::subrange(m.begin()+i*w,m.begin()+(i+1)*w);
    }
    matrix &operator+= (const matrix &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] += r.m[idx(i,j)];
            }
        }
        return *this;
    }
    matrix &operator-= (const matrix &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] -= r.m[idx(i,j)];
            }
        }
        return *this;
    }
    matrix &operator*= (const matrix &r){
        matrix ret;
        for (int i = 0; i < h; i++){
            for (int k = 0; k < w; k++){
                for (int j = 0; j < r.w; j++){
                    ret.m[idx(i,j)] += m[idx(i,k)] * r.m[idx(k,j)];
                }
            }
        }
        return *this = ret;
    }
    matrix operator+ (const matrix &r) const { return matrix(*this) += r; }
    matrix operator- (const matrix &r) const { return matrix(*this) -= r; }
    matrix operator* (const matrix &r) const { return matrix(*this) *= r; }
    matrix& operator*=(const T &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] *= r;
            }
        }
        return *this;
    }
    friend matrix operator* (const T &r, const matrix &mat){
        return matrix(mat) *= r;
    }
    friend matrix operator* (const matrix &mat, const T &r){
        return matrix(mat) *= r;
    }
    matrix pow(long long n){
        if (n == 0) return e();
        matrix f = pow(n / 2);
        matrix ret = f * f;
        if (n & 1) ret *= (*this);
        return ret;
    }
    int idx(int i, int j){
        return i * w + j;
    }
    static matrix e(){
        matrix ret;
        for (int i = 0; i < h; i++){
            ret[i][i] = T(1);
        }
        return ret;
    }
    friend std::ostream &operator<<(std::ostream &os, const matrix &mat){
        for (int i = 0; i < mat.h; i++){
            if (i != 0) os << '\n';
            for (int j = 0; j < mat.w; j++){
                if (j != 0) os << ' ';
                os << mat[i][j];
            }
        }
        return os;
    }
    friend std::istream &operator>>(std::istream &is, matrix &mat){
        for (int i = 0; i < mat.h; i++){
            for (int j = 0; j < mat.w; j++){
                is >> mat[i][j];
            }
        }
        return is;
    }
};

template<typename T>
struct matrix<T,-1uz> {
    int h, w;
    std::vector<T> m;
    matrix () {}
    matrix (int _h) : matrix(_h,_h) {}
    matrix (int _h, int _w) : h(_h), w(_w), m(_h*_w) {}
    matrix (int _h, int _w, const std::vector<T> &_m) : h(_h), w(_w), m(_m) {
        assert((int)_m.size() == _h*_w);
    }
    matrix (const std::vector<std::vector<T>> &_m){
        h = _m.size();
        assert(h >= 1);
        w = _m[0].size();
        for (int i = 0; i < h; i++) for (int j = 0; j < w; j++){
            m[idx(i,j)] = _m[i][j];
        }
    }
    auto operator[](int i) const {
        return std::ranges::subrange(m.begin()+i*w,m.begin()+(i+1)*w);
    }
    auto operator[](int i){
        return std::ranges::subrange(m.begin()+i*w,m.begin()+(i+1)*w);
    }
    matrix &operator+= (const matrix &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] += r.m[idx(i,j)];
            }
        }
        return *this;
    }
    matrix &operator-= (const matrix &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] -= r.m[idx(i,j)];
            }
        }
        return *this;
    }
    matrix &operator*= (const matrix &r){
        matrix ret(h, r.w);
        for (int i = 0; i < h; i++){
            for (int k = 0; k < w; k++){
                for (int j = 0; j < r.w; j++){
                    ret.m[idx(i,j)] += m[idx(i,k)] * r.m[idx(k,j)];
                }
            }
        }
        return *this = ret;
    }
    matrix operator+ (const matrix &r) const { return matrix(*this) += r; }
    matrix operator- (const matrix &r) const { return matrix(*this) -= r; }
    matrix operator* (const matrix &r) const { return matrix(*this) *= r; }
    matrix& operator*=(const T &r){
        for (int i = 0; i < h; ++i){
            for (int j = 0; j < w; ++j){
                m[idx(i,j)] *= r;
            }
        }
        return *this;
    }
    friend matrix operator* (const T &r, const matrix &mat){
        return matrix(mat) *= r;
    }
    friend matrix operator* (const matrix &mat, const T &r){
        return matrix(mat) *= r;
    }
    matrix pow(long long n){
        if (n == 0) return e(h);
        matrix f = pow(n / 2);
        matrix ret = f * f;
        if (n & 1) ret *= (*this);
        return ret;
    }
    int idx(int i, int j){
        return i * w + j;
    }
    static matrix e(int _h){
        auto ret = matrix(_h, _h);
        for (int i = 0; i < _h; i++){
            ret[i][i] = T(1);
        }
        return ret;
    }
    friend std::ostream &operator<<(std::ostream &os, const matrix &mat){
        for (int i = 0; i < mat.h; i++){
            if (i != 0) os << '\n';
            for (int j = 0; j < mat.w; j++){
                if (j != 0) os << ' ';
                os << mat[i][j];
            }
        }
        return os;
    }
    friend std::istream &operator>>(std::istream &is, matrix &mat){
        for (int i = 0; i < mat.h; i++){
            for (int j = 0; j < mat.w; j++){
                is >> mat[i][j];
            }
        }
        return is;
    }
};

template<typename T, size_t _hw = -1uz>
T determinant(matrix<T, _hw> mat){
    int hw = mat.h;
    T ret = 1;
    for (int i = 0; i < hw; i++) {
        int idx = -1;
        for (int j = i; j < hw; j++) {
            if (mat[j][i] != 0) {
                idx = j;
                break;
            }
        }
        if (idx == -1) return 0;
        if (i != idx) {
            ret *= T(-1);
            for (int j = 0; j < hw; j++){
                std::swap(mat[i][j],mat[idx][j]);
            }
        }
        ret *= mat[i][i];
        T inv = T(1) / mat[i][i];
        for (int j = 0; j < hw; j++) {
            mat[i][j] *= inv;
        }
        for (int j = i + 1; j < hw; j++) {
            T a = mat[j][i];
            if (a == 0) continue;
            for (int k = i; k < hw; k++) {
                mat[j][k] -= mat[i][k] * a;
            }
        }
    }
    return ret;
}

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/math/prime.hpp"
namespace noya2 {

constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime_flag = is_prime_constexpr(n);

constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;
    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u; 
        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root_flag = primitive_root_constexpr(m);

constexpr long long primitive_root_constexpr(long long m){
    if (m == (1LL << 47) - (1LL << 24) + 1) return 3;
    return primitive_root_constexpr(static_cast<int>(m));
}

} // namespace noya2
#line 6 "/Users/noya2/Desktop/Noya2_library/utility/modint.hpp"

namespace noya2{

struct barrett {
    unsigned int _m;
    unsigned long long im;
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
    unsigned int umod() const { return _m; }
    unsigned int mul(unsigned int a, unsigned int b) const {
        unsigned long long z = a;
        z *= b;
        unsigned long long x = (unsigned long long)((__uint128_t(z) * im) >> 64);
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

template <int m>
struct static_modint {
    using mint = static_modint;
  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }
    constexpr static_modint() : _v(0) {}
    template<std::signed_integral T>
    constexpr static_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    constexpr static_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    constexpr unsigned int val() const { return _v; }
    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }
    constexpr mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    constexpr mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    constexpr mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (uint)(z % umod());
        return *this;
    }
    constexpr mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint() - *this; }
    constexpr mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    constexpr mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = is_prime_flag<m>;
};


template <int id> struct dynamic_modint {
    using mint = dynamic_modint;
  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template<std::signed_integral T>
    dynamic_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    dynamic_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    uint val() const { return _v; }
    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }
    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }
    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = noya2::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }
    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._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> noya2::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

template<typename T>
concept Modint = requires (T &a){
    T::mod();
    a.inv();
    a.val();
    a.pow(declval<int>());
};

} // namespace noya2
#line 6 "c.cpp"
using mint = modint998244353;

void solve(){
    int n, m, x; in(n,m,x);
    graph<unweighted> g(n);
    rep(t,m){
        int u, v; in(u,v); u--, v--;
        g.add_edge(u,v);
        if (u == v) continue;
        g.add_edge(v,u);
    }
    g.build();
    matrix<mint> mat(n*n+n,n*n+n);
    auto idx = [&](int i, int j){
        return i * n + j;
    };
    vector<vector<bool>> done(n,vector<bool>(n,false));
    rep(i,n){
        for (int j : g[i]){
            if (done[i][j]) continue;
            if (g[j].size() == 1u){
                mat[idx(i,j)][idx(n,j)] += 1;
                done[i][j] = true;
                continue;
            }
            mint jsz = mint(g[j].size()-1u).inv();
            mint sum = 0;
            for (int k : g[j]){
                mat[idx(i,j)][idx(j,k)] += jsz;
                sum += jsz;
            }
            mat[idx(i,j)][idx(j,i)] -= jsz;
            sum -= jsz;
            done[i][j] = true;
        }
    }
    rep(j,n){
        if (g[j].empty()){
            mat[idx(n,j)][idx(n,j)] = 1;
            continue;
        }
        mint jsz = mint(g[j].size()).inv();
        for (int k : g[j]){
            mat[idx(n,j)][idx(j,k)] += jsz;
        }
    }
    matrix<mint> ini(1,n*n+n);
    ini[0][idx(n,0)] = 1;
    ini *= mat.pow(x);
    rep(j,n){
        mint ans = 0;
        rep(i,n+1){
            ans += ini[0][idx(i,j)];
        }
        out(ans);
    }
}

int main(){
    int t = 1; //in(t);
    while (t--) { solve(); }
}