#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 "hld_test.cpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/tree/heavy_light_decomposition.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/tree/simple_tree.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);
    }
    size_t size() const {
        return n;
    }
    int n;
    std::vector<int> start;
    std::vector<E> elist;
    bool prepared = false;
};

} // namespace noya2::internal
#line 5 "/Users/noya2/Desktop/Noya2_library/tree/simple_tree.hpp"

namespace noya2 {

struct simple_tree {
    internal::csr<int> g;
    simple_tree () {}
    simple_tree (int _n) : g(_n, (_n - 1)*2) {
        if (_n == 1){
            g.build();
        }
    }
    void add_edge(int u, int v){
        g.add(u, v);
        int id = g.add(v, u);
        if (id + 1 == (g.n - 1)*2) g.build();
    }
    void input(int indexed = 1){
        for (int i = 0; i < g.n - 1; i++){
            int u, v; cin >> u >> v;
            u -= indexed, v -= indexed;
            add_edge(u, v);
        }
    }
    void input_parents(int indexed = 1){
        for (int i = 0; i < g.n - 1; i++){
            int v; cin >> v;
            v -= indexed;
            add_edge(i + 1, v);
        }
    }
    const auto operator[](int v) const {
        return g[v];
    }
    auto operator[](int v){
        return g[v];
    }
    size_t size() const {
        return g.size();
    }
};

} // namespace noya2
#line 5 "/Users/noya2/Desktop/Noya2_library/tree/heavy_light_decomposition.hpp"

namespace noya2 {

using namespace std;

struct hld_tree {
    internal::csr<int> g;
    hld_tree () {}
    hld_tree (int _n, int _root = 0) : g(_n,(_n - 1)*2), n(_n), root(_root) {
        if (_n == 1){
            build();
        }
    }
    hld_tree (simple_tree _g, int _root = 0) : g(_g.g), n(_g.g.n), root(_root){
        build();
    }

    size_t size() const {
        return g.n;
    }

    int root_v() const {
        return root;
    }

    void add_edge(int u, int v){
        g.add(u, v);
        int id = g.add(v, u);
        if (id + 1 == (n - 1)*2) build();
    }
    void input(int indexed = 1){
        for (int i = 0; i < n - 1; i++){
            int u, v; cin >> u >> v;
            u -= indexed, v -= indexed;
            add_edge(u, v);
        }
    }
    void input_parents(int indexed = 1){
        for (int i = 0; i < n - 1; i++){
            int v; cin >> v;
            v -= indexed;
            add_edge(i + 1, v);
        }
    }

    int depth(int v) const {
        return dep[v];
    }

    int parent(int v) const {
        if (v == root) return -1;
        return g[v].back();
    }

    int degree(int v) const {
        return g[v].size();
    }

    int subtree_size(int v) const {
        return sub[v];
    }

    // if d > dep[v], return -1
    int la(int v, int d) const {
        while (v != -1){
            int u = nxt[v];
            if (down[v] - d >= down[u]){
                v = tour[down[v] - d];
                break;
            }
            d -= down[v] - down[u] + 1;
            v = parent(u);
        }
        return v;
    }

    int lca(int u, int v) const {
        while (nxt[u] != nxt[v]){
            if (down[u] < down[v]) swap(u,v);
            u = parent(nxt[u]);
        }
        return dep[u] < dep[v] ? u : v;
    }

    int dist(int u, int v) const {
        return dep[u] + dep[v] - 2*dep[lca(u,v)];
    }

    // if d > dist(from, to), return -1
    int jump(int from, int to, int d) const {
        int l = lca(from,to);
        if (d <= dep[from] - dep[l]){
            return la(from, d);
        }
        d -= dep[from] - dep[l];
        if (d <= dep[to] - dep[l]){
            return la(to, dep[to] - dep[l] - d);
        }
        return -1;
    }

    // seg.set(index(v), X[v]);
    int index(int vertex) const {
        return down[vertex];
    }

    int index_from_edge(int u, int v) const {
        return (dep[u] < dep[v] ? down[v] : down[u]);
    }

    // X[vertex(i)] = seg.get(i);
    int vertex(int index) const {
        return tour[index];
    }

    int subtree_l(int v) const {
        return down[v];
    }

    int subtree_r(int v) const {
        return down[v] + sub[v];
    }

    // if r == v, return true
    bool is_in_subtree(int r, int v) const {
        return subtree_l(r) <= subtree_l(v) && subtree_l(v) < subtree_r(r);
    }

    bool is_in_path(int lv, int mv, int rv) const {
        return dist(lv,mv) + dist(mv,rv) == dist(lv,rv);
    }

    // dist, v1, v2
    tuple<int,int,int> diameter(){
        int v1 = max_element(dep.begin(),dep.end()) - dep.begin();
        vector<int> dist_from_v1(n,numeric_limits<int>::max());
        queue<int> que;
        que.push(v1);
        dist_from_v1[v1] = 0;
        while (!que.empty()){
            int v = que.front(); que.pop();
            for (int u : g[v]){
                if (dist_from_v1[u] > dist_from_v1[v]+1){
                    dist_from_v1[u] = dist_from_v1[v]+1;
                    que.push(u);
                }
            }
        }
        int v2 = max_element(dist_from_v1.begin(),dist_from_v1.end()) - dist_from_v1.begin();
        return make_tuple(dist_from_v1[v2],v1,v2);
    }

    // vertex array : vector<int> {from, v1, v2, ... , to}
    vector<int> path(int from, int to){
        int l = lca(from,to);
        const int sizf = dep[from]-dep[l], sizt = dep[to]-dep[l];
        vector<int> pf = {from}, pt;
        pf.reserve(sizf+1); pt.reserve(sizt);
        for (int i = 0; i < sizf; i++){
            from = parent(from);
            pf.push_back(from);
        }
        for (int i = 0; i < sizt; i++){
            pt.push_back(to);
            to = parent(to);
        }
        pf.insert(pf.end(),pt.rbegin(),pt.rend());
        return pf;
    }

    template<typename F>
    void path_query(int u, int v, bool vertex, const F &f){
        int l = lca(u,v);
        for (auto [s, t] : ascend(u, l)){
            f(t, s + 1);
        }
        if (vertex) f(down[l], down[l] + 1);
        for (auto [s, t] : descend(l, v)){
            f(s, t + 1);
        }
    }

    template<typename F>
    void path_noncommutative_query(int u, int v, bool vertex, const F &f){
        int l = lca(u,v);
        for (auto [s, t] : ascend(u, l)){
            f(s + 1, t); // l > r ok
        }
        if (vertex) f(down[l],down[l] + 1);
        for (auto [s, t] : descend(l, v)){
            f(s, t + 1); // l > r ok
        }
    }

    template<typename F>
    void subtree_query(int v, bool vertex, const F &f){
        f(down[v] + (vertex ? 0 : 1), down[v] + sub[v]);
    }

    // adjacent to v
    const auto operator[](int v) const {
        return g[v];
    }
    auto operator[](int v){
        return g[v];
    }

    // only child
    const auto operator()(int v) const {
        return g(v, 0, degree(v) - (v == root ? 0 : 1));
    }
    auto operator()(int v){
        return g(v, 0, degree(v) - (v == root ? 0 : 1));
    }

  private:
    int n, root;
    vector<int> dep, sub, down, tour, nxt;

    // v is ancestor of u.
    // enumerate [closed] intervals of down ( interval [l, r] may hold l > r ).
    vector<pair<int,int>> ascend(int u, int v){
        vector<pair<int,int>> res;
        while (nxt[u] != nxt[v]){
            res.emplace_back(down[u], down[nxt[u]]);
            u = parent(nxt[u]);
        }
        if (u != v) res.emplace_back(down[u], down[v]+1);
        return res;
    }

    // u is ancestor of v.
    // enumerate [closed] intervals of down ( interval [l, r] may hold l > r ).
    vector<pair<int,int>> descend(int u, int v){
        if (u == v) return {};
        if (nxt[u] == nxt[v]){
            return {pair<int,int>(down[u]+1, down[v])};
        }
        vector<pair<int,int>> res = descend(u, parent(nxt[v]));
        res.emplace_back(down[nxt[v]], down[v]);
        return res;
    }

    void build(){
        g.build();
        init_sz();
        init_hld();
    }

    /*
        setup dep, sub
        if v is not root, g[v].back() is parent of v.
        if v is not leaf (i.e. v has child), g[v].front() is heavy child of v.
    */
    void init_sz(){
        dep.resize(n, 0);
        sub.resize(n, 1);
        auto dfs = [&](auto sfs, int v, int f) -> void {
            for (int &u : g[v]){
                // only one chance to take parent as u.
                if (u == f) swap(g[v].back(), u);
                // twice means u is the last element of g[v], i.e. parent of v.
                if (u == f) break;
                dep[u] = dep[v]+1;
                sfs(sfs, u, v);
                sub[v] += sub[u];
                if (sub[g[v].front()] < sub[u]){
                    swap(g[v].front(), u);
                }
            }
        };
        dfs(dfs, root, -1);
    }

    /*
        setup down, tour, nxt
        only heavy child c of v, nxt[c] = nxt[v]. for other child c, nxt[c] = c.
    */
    void init_hld(){
        down.resize(n);
        tour.resize(n);
        nxt.resize(n);
        nxt[root] = root;
        int clock = 0;
        auto dfs = [&](auto sfs, int v) -> void {
            down[v] = clock++;
            tour[down[v]] = v;
            // in case of no child, nothing to do
            if ((*this)(v).empty()) return ;
            // heavy child
            nxt[(*this)(v).front()] = nxt[v];
            sfs(sfs, (*this)(v).front());
            // other child
            for (int u : (*this)(v).next()){
                nxt[u] = u;
                sfs(sfs, u);
            }
        };
        dfs(dfs, root);
    }

  public:
    struct compressed_tree : public simple_tree {
        using simple_tree::simple_tree;
        using simple_tree::operator=;
        hld_tree &g;
        compressed_tree (hld_tree &g_, vector<int> vs) : g(g_){
            auto comp = [&](int lv, int rv){
                return g.index(lv) < g.index(rv);
            };
            sort(vs.begin(),vs.end(),comp);
            int sz = vs.size();
            for (int i = 0; i < sz-1; i++){
                vs.emplace_back(g.lca(vs[i],vs[i+1]));
            }
            sort(vs.begin(),vs.end(),comp);
            vs.erase(unique(vs.begin(),vs.end()),vs.end());
            sz = vs.size();
            (*this) = simple_tree(sz);
            real_vertex = vs;
            for (int i = 0; i < sz; i++){
                g.virtual_vertex[real_vertex[i]] = i;
            }
            stack<int> st;
            st.push(0);
            for (int i = 1; i < sz; i++){
                while (!g.is_in_subtree(real_vertex[st.top()], real_vertex[i])) st.pop();
                (*this).add_edge(st.top(),i);
                st.push(i);
            }
        }
        vector<int> real_vertex;
        int real_v(int virtual_v) const {
            return real_vertex[virtual_v];
        }
        int virtual_v(int real_v) const {
            return g.virtual_vertex[real_v];
        }
        size_t size() const {
            return real_vertex.size();
        }
    };
    compressed_tree compressed_tree_gen(const vector<int> &vs){
        if ((int)virtual_vertex.size() != n) virtual_vertex.resize(n);
        return compressed_tree(*this, vs);
    }
    vector<int> virtual_vertex;
};

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/misc/random_tree.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/misc/random_tree.hpp"

#line 2 "/Users/noya2/Desktop/Noya2_library/misc/rng.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/misc/rng.hpp"

namespace noya2 {

// [0, 2^64 - 1)
ull rng() {
  static ull _x = 88172645463325252UL;
  return _x ^= _x << 7, _x ^= _x >> 9;
}

// [l, r]
ll rng(ll l, ll r) {
  assert(l <= r);
  return l + rng() % ull(r - l + 1);
}

// [l, r)
ll randint(ll l, ll r) {
  assert(l < r);
  return l + rng() % ull(r - l);
}

// [0.0, 1.0)
ld rnd() { return rng() * 5.42101086242752217004e-20; }
// [l, r)
ld rnd(ld l, ld r) {
  assert(l < r);
  return l + rnd() * (r - l);
}

} // namespace noya2
#line 6 "/Users/noya2/Desktop/Noya2_library/misc/random_tree.hpp"

namespace noya2 {

// input: [c \in [0, n)] * (n-2), n >= 3
vector<vector<int>> pruefer_code(const vector<int>& code) {
    int n = code.size() + 2;
    assert(n > 2);
    vector<vector<int>> g(n);
    vector<int> deg(n, 1);
    int e = 0;
    for (auto& x : code) deg[x]++;
    set<int> ps;
    for (int j = 0; j < n; j++) {
        if (deg[j] == 1) ps.insert(j);
    }
    for (auto& i : code) {
        if (ps.empty()) break;
        int j = *begin(ps);
        ps.erase(j);
        g[i].push_back(j);
        g[j].push_back(i);
        if (deg[i] == 1) ps.erase(i);
        deg[i]--, deg[j]--;
        if (deg[i] == 1) ps.insert(i);
        e++;
    }
    int u = -1, v = -1;
    for (int i = 0; i < n; i++) {
        if (deg[i] == 1) (u == -1 ? u : v) = i;
    }
    assert(u != -1 and v != -1);
    g[u].push_back(v);
    g[v].push_back(u);
    e++;
    assert(e == n - 1);
    return g;
}

vector<vector<int>> random_tree(int n) {
    if (n <= 2) {
        vector<vector<int>> g(n);
        if (n == 2) {
            g[0].push_back(1);
            g[1].push_back(0);
        }
        return g;
    }
    vector<int> pruefer(n - 2);
    for (auto& x : pruefer) x = randint(0,n);
    return pruefer_code(pruefer);
}

} // namespace noya2
#line 5 "hld_test.cpp"

#line 2 "hld_new.hpp"

#line 10 "hld_new.hpp"

namespace hld_new {

struct hld_tree {
    int n, root;
    std::vector<int> down, nxt, sub, tour;
	noya2::internal::csr<int> adjacent;

    // default constructor (nop)
    hld_tree () {}

    // tree with _n node
    // after construct, call input_edges / input_parents / add_edge _n - 1 times
    hld_tree (int _n, int _root = 0) : n(_n), root(_root), down(n), nxt(n), sub(n, 1), tour(n) {
        if (n == 1){
            nxt[0] = -1;
            down[0] = -1;
            build_from_parents();
        }
    }

    // par[i] < i, par[0] == -1
    hld_tree (const std::vector<int> &par) : n(par.size()), root(0), down(n, -1), nxt(par), sub(n, 1), tour(n){
        build_from_parents();
    }

    // par[i] < i, par[0] == -1
    hld_tree (std::vector<int> &&par) : n(par.size()), root(0), down(n, -1), sub(n, 1), tour(n) {
        nxt.swap(par);
        build_from_parents();
    }

    // distinct unweighted undirected n - 1 edges of tree 
    hld_tree (const std::vector<std::pair<int, int>> &es, int _root = 0) : n(es.size() + 1), root(_root), down(n), nxt(n), sub(n, 1), tour(n) {
        for (auto &[u, v] : es){
            down[u]++;
            down[v]++;
            nxt[u] ^= v;
            nxt[v] ^= u;
        }
        build_from_edges();
    }

    // input parents from cin
    template<int indexed = 1>
    void input_parents(){
        // using std::cin;
        nxt[0] = -1;
        for (int u = 1; u < n; u++){
            cin >> nxt[u];
            nxt[u] -= indexed;
        }
        build_from_parents();
    }

    // input n - 1 edges from cin
    template<int indexed = 1>
    void input_edges(){
        // using std::cin;
        for (int i = 1; i < n; i++){
            int u, v; cin >> u >> v;
            u -= indexed;
            v -= indexed;
            down[u]++;
            down[v]++;
            nxt[u] ^= v;
            nxt[v] ^= u;
        }
        build_from_edges();
    }

    void add_edge(int u, int v){
        down[u]++;
        down[v]++;
        nxt[u] ^= v;
        nxt[v] ^= u;
        // use tour[0] as counter
        if (++tour[0] == n - 1){
            build_from_edges();
        }
    }

    // top vertex of heavy path which contains v
    int leader(int v) const {
        return nxt[v] < 0 ? v : nxt[v];
    }

    // level ancestor
    // ret is ancestor of v, dist(ret, v) == d
    // if d > depth(v), return -1
    int la(int v, int d) const {
        while (v != -1){
            int u = leader(v);
            if (down[v] - d >= down[u]){
                v = tour[down[v] - d];
                break;
            }
            d -= down[v] - down[u] + 1;
            v = (u == root ? -1 : ~nxt[u]);
        }
        return v;
    }

    // lowest common ancestor of u and v
    int lca(int u, int v) const {
        int du = down[u], dv = down[v];
        if (du > dv){
            std::swap(du, dv);
            std::swap(u, v);
        }
        if (dv < du + sub[u]){
            return u;
        }
        while (du < dv){
            v = ~nxt[leader(v)];
            dv = down[v];
        }
        return v;
    }

    // distance from u to v
    int dist(int u, int v) const {
        int _dist = 0;
        while (leader(u) != leader(v)){
            if (down[u] > down[v]) std::swap(u, v);
            _dist += down[v] - down[leader(v)] + 1;
            v = ~nxt[leader(v)];
        }
        _dist += std::abs(down[u] - down[v]);
        return _dist;
    }

    // d times move from to its neighbor (direction of to)
    // if d > dist(from, to), return -1
    int jump(int from, int to, int d) const {
        int _from = from, _to = to;
        int dist_from_lca = 0, dist_to_lca = 0;
        while (leader(_from) != leader(_to)){
            if (down[_from] > down[_to]){
                dist_from_lca += down[_from] - down[leader(_from)] + 1;
                _from = ~nxt[leader(_from)];
            }
            else {
                dist_to_lca += down[_to] - down[leader(_to)] + 1;
                _to = ~nxt[leader(_to)];
            }
        }
        if (down[_from] > down[_to]){
            dist_from_lca += down[_from] - down[_to];
        }
        else {
            dist_to_lca += down[_to] - down[_from];
        }
        if (d <= dist_from_lca){
            return la(from, d);
        }
        d -= dist_from_lca;
        if (d <= dist_to_lca){
            return la(to, dist_to_lca - d);
        }
        return -1;
    }

    // parent of v (if v is root, return -1)
    int parent(int v) const {
        if (v == root) return -1;
        return (nxt[v] < 0 ? ~nxt[v] : tour[down[v] - 1]);
    }

    // visiting time in euler tour
    // usage : seg.set(index(v), X[v])
    int index(int vertex) const {
        return down[vertex];
    }

    // subtree size of v
    int subtree_size(int v) const {
        return sub[v];
    }

    // prod in subtree v : seg.prod(subtree_l(v), subtree_r(v))
    int subtree_l(int v) const {
        return down[v];
    }
    int subtree_r(int v) const {
        return down[v] + sub[v];
    }

    // v is in subtree r
    bool is_in_subtree(int r, int v) const {
        return subtree_l(r) <= subtree_l(v) && subtree_r(v) <= subtree_r(r);
    }
    // distance table from s
    std::vector<int> dist_table(int s) const {
        std::vector<int> table(n, -1);
        table[s] = 0;
        while (s != root){
            table[parent(s)] = table[s] + 1;
            s = parent(s);
        }
        for (int v : tour){
            if (table[v] == -1){
                table[v] = table[parent(v)] + 1;
            }
        }
        return table;
    }

    // dist, v1, v2
    std::tuple<int, int, int> diameter() const {
        std::vector<int> dep = dist_table(root);
        int v1 = std::ranges::max_element(dep) - dep.begin();
        std::vector<int> fromv1 = dist_table(v1);
        int v2 = std::ranges::max_element(fromv1) - fromv1.begin();
        return {fromv1[v2], v1, v2};
    }

    // vertex array {from, ..., to}
    std::vector<int> path(int from, int to) const {
        int d = dist(from, to);
        std::vector<int> _path(d + 1);
        int front = 0, back = d;
        while (from != to){
            if (down[from] > down[to]){
                _path[front++] = from;
                from = parent(from);
            }
            else {
                _path[back--] = to;
                to = parent(to);
            }
        }
        _path[front] = from;
        return _path;
    }

    // path decomposition and query (vertex weighted)
    // if l < r, decsending order tour[l, r)
    // if l > r, acsending order tour(l, r]
    void path_query(int u, int v, auto f) const {
        while (leader(u) != leader(v)){
            if (down[u] < down[v]){
                f(down[leader(v)], down[v] + 1);
                v = ~nxt[leader(v)];
            }
            else {
                f(down[u] + 1, down[leader(u)]);
                u = ~nxt[leader(u)];
            }
        }
        if (down[u] < down[v]){
            f(down[u], down[v] + 1);
        }
        else {
            f(down[u] + 1, down[v]);
        }
    }

    // {parent, mapping} : cptree i is correspond to tree mapping[i]. parent[i] is parent of i in cptree.
    // parent[i] < i, parent[0] == -1
	std::pair<std::vector<int>, std::vector<int>> compressed_tree(std::vector<int> vs) const {
        if (vs.empty()){
            return {{},{}};
        }
        auto comp = [&](int l, int r){
            return down[l] < down[r];
        };
		std::ranges::sort(vs, comp);
		int sz = vs.size(); vs.reserve(2*sz);
        for (int i = 0; i < sz-1; i++){
            vs.emplace_back(lca(vs[i], vs[i+1]));
        }
        std::sort(vs.begin() + sz, vs.end(), comp);
        std::ranges::inplace_merge(vs, vs.begin() + sz, comp);
        auto del = std::ranges::unique(vs);
        vs.erase(del.begin(), del.end());
        sz = vs.size();
        std::stack<int> st;
        std::vector<int> par(sz);
        par[0] = -1;
        st.push(0);
        for (int i = 1; i < sz; i++){
            while (!is_in_subtree(vs[st.top()], vs[i])) st.pop();
            par[i] = st.top();
            st.push(i);
        }
        return {par, vs};
	}

	// build csr for using operator[], operator()
	void build_csr(){
		adjacent = noya2::internal::csr<int>(n, n - 1);
		for (int v = 0; v < n; v++){
			if (v == root) continue;
			adjacent.add(parent(v), v);
		}
		adjacent.build();
	}
	const auto operator[](int v) const {
		return adjacent[v];
	}
	auto operator[](int v){
		return adjacent[v];
	}

    // nxt[v] : parent of v, nxt[0] == -1
    void build_from_parents(){
        for (int u = n - 1; u >= 1; u--){
            int v = nxt[u];
            sub[v] += sub[u];
            down[v] = std::max(down[v], sub[u]);
        }
        for (int u = n - 1; u >= 1; u--){
            int v = nxt[u];
            if (down[v] == sub[u]){
                sub[u] = ~sub[u];
                down[v] = ~down[v];
            }
        }

        sub[0] = ~down[0] + 1;
        down[0] = 0;
        for (int u = 1; u < n; u++){
            int v = nxt[u];
            int nsub = ~down[u] + 1;
            if (sub[u] < 0){
                down[u] = down[v] + 1;
                nxt[u] = (nxt[v] < 0 ? v : nxt[v]);
            }
            else {
                down[u] = down[v] + sub[v];
                sub[v] += sub[u];
                nxt[u] = ~v;
            }
            sub[u] = nsub;
        }

        for (int u = 0; u < n; u++){
            tour[down[u]] = u;
        }
    }

    // down[v] : degree of v
    // nxt[v] : xor prod of neighbor of v
    void build_from_edges(){
        // use tour as queue
        int back = 0;
        for (int u = 0; u < n; u++){
            if (u != root && down[u] == 1){
                tour[back++] = u;
            }
        }
        for (int front = 0; front < n - 1; front++){
            int u = tour[front];
            down[u] = -1;
            int v = nxt[u]; // parent of v
            nxt[v] ^= u;
            if (--down[v] == 1 && v != root){
                tour[back++] = v;
            }
        }
        // check : now, tour is reverse of topological order

        tour.pop_back();

        // check : now, down[*] <= 1
        for (int u : tour){
            int v = nxt[u];
            // subtree size (initialized (1,1,...,1))
            sub[v] += sub[u];
            // heaviest subtree of its child
            down[v] = std::max(down[v], sub[u]);
        }
        for (int u : tour){
            int v = nxt[u];
            // whether u is not the top of heavy path
            if (down[v] == sub[u]){
                sub[u] = ~sub[u];
                down[v] = ~down[v];
            }
        }

        // after appearing v as u (or v == root), 
        // down[v] is the visiting time of euler tour
        // nxt[v] is the lowest vertex of heavy path which contains v
        //   (if v itself, nxt[v] is ~(parent of v))
        // sub[v] + down[v] is the light child's starting time of euler tour
        // note : heavy child's visiting time of euler tour is (the time of its parent) + 1
        sub[root] = ~down[root] + 1;
        down[root] = 0;
        nxt[root] = -1;
        for (int u : tour | std::views::reverse){
            int v = nxt[u];
            int nsub = ~down[u] + 1;
            // heavy child
            if (sub[u] < 0){
                down[u] = down[v] + 1;
                nxt[u] = (nxt[v] < 0 ? v : nxt[v]);
            }
            // light child
            else {
                down[u] = down[v] + sub[v];
                sub[v] += sub[u];
                nxt[u] = ~v;
            }
            sub[u] = nsub;
        }

        // tour is inverse permutation of down
        tour.push_back(0);
        for (int u = 0; u < n; u++){
            tour[down[u]] = u;
        }
    }
};

} // namespace hld_new
#line 7 "hld_test.cpp"

void jikken1(int n){
    auto g = random_tree(n);
    int root = randint(0,n);
    noya2::hld_tree tr1(n,root);
    hld_new::hld_tree tr2(n,root);
    rep(i,n){
        for (int j : g[i]){
            if (i < j){
                tr1.add_edge(i, j);
                tr2.add_edge(i, j);
            }
        }
    }
    // subtree size
    rep(i,n){
        assert(tr1.subtree_size(i) == tr2.subtree_size(i));
    }
    cout << "subtree ok" << endl;
    // parent
    rep(i,n){
        assert(tr1.parent(i) == tr2.parent(i));
    }
    cout << "parent ok" << endl;
    // diameter
    {
        assert(get<0>(tr1.diameter()) == get<0>(tr2.diameter()));
    }
    cout << "diameter ok" << endl;
    // lca
    rep(i,n) rep(j,n){
        assert(tr1.lca(i,j) == tr2.lca(i,j));
    }
    cout << "lca ok" << endl;
    // la
    rep(i,n) rep(j,n){
        assert(tr1.la(i,j) == tr2.la(i,j));
    }
    cout << "la ok" << endl;
    // path
    rep(i,n) rep(j,n){
        assert(tr1.path(i,j) == tr2.path(i,j));
    }
    cout << "path ok" << endl;
    // dist
    rep(i,n) rep(j,n){
        assert(tr1.dist(i,j) == tr2.dist(i,j));
    }
    cout << "dist ok" << endl;
}

void jikken2(){
    int n; in(n);
    int t; in(t);
    while (t--){
        jikken1(n);
    }
    cout << "jikken2 done" << endl;
}

void jikken3(int n){
    auto g = random_tree(n);
    vector<int> par = [&]{
        vector<int> ret(n);
        noya2::hld_tree tr(n);
        rep(i,n) for (int j : g[i]){
            if (i < j){
                tr.add_edge(i, j);
            }
        }
        ret[0] = -1;
        repp(i,1,n){
            ret[tr.index(i)] = tr.index(tr.parent(i));
        }
        return ret;
    }();
    noya2::hld_tree tr1(n);
    hld_new::hld_tree tr2(par);
    repp(i,1,n){
        tr1.add_edge(i, par[i]);
    }
    // subtree size
    rep(i,n){
        assert(tr1.subtree_size(i) == tr2.subtree_size(i));
    }
    cout << "subtree ok" << endl;
    // parent
    rep(i,n){
        assert(tr1.parent(i) == tr2.parent(i));
    }
    cout << "parent ok" << endl;
    // diameter
    {
        assert(get<0>(tr1.diameter()) == get<0>(tr2.diameter()));
    }
    cout << "diameter ok" << endl;
    // lca
    rep(i,n) rep(j,n){
        assert(tr1.lca(i,j) == tr2.lca(i,j));
    }
    cout << "lca ok" << endl;
    // la
    rep(i,n) rep(j,n){
        assert(tr1.la(i,j) == tr2.la(i,j));
    }
    cout << "la ok" << endl;
    // path
    rep(i,n) rep(j,n){
        assert(tr1.path(i,j) == tr2.path(i,j));
    }
    cout << "path ok" << endl;
    // dist
    rep(i,n) rep(j,n){
        assert(tr1.dist(i,j) == tr2.dist(i,j));
    }
    cout << "dist ok" << endl;
}

void jikken4(){
    int n; in(n);
    int t; in(t);
    while (t--){
        jikken3(n);
    }
    cout << "jikken4 done" << endl;
}

#line 2 "/Users/noya2/Desktop/Noya2_library/misc/timer.hpp"

#line 5 "/Users/noya2/Desktop/Noya2_library/misc/timer.hpp"

namespace noya2{

struct Timer {
  private:
    std::chrono::high_resolution_clock::time_point start, end;
 
  public:
    Timer() { set(); }
    void set() { start = std::chrono::high_resolution_clock::now(); }
    int time() {
        end = std::chrono::high_resolution_clock::now();
        return std::chrono::duration_cast<std::chrono::milliseconds>(end - start).count();
    }
    double dtime(){
        return (double)(time()) / 1000;
    }
    bool before(double T) { return time() < (int)(T * 1000); }
    void print() { std::cerr << "elapsed time: " << (double)time() / 1000 << " sec" << std::endl; }
};

} // namespace noya2
#line 135 "hld_test.cpp"

void jikken5(int n){
    auto g = random_tree(n);
    vector<pii> es; es.reserve(n-1);
    rep(i,n) for (int j : g[i]){
        if (i < j){
            es.emplace_back(i, j);
        }
    }
    vector<pii> a(n);
    rep(i,n){
        a[i] = {randint(0,n),randint(0,n)};
    }
    Timer tm;
    tm.set();
    noya2::hld_tree tr1(n);
    for (auto [u, v] : es){
        tr1.add_edge(u, v);
    }
    tm.print();
    tm.set();
    int val = 0;
    for (auto [u, v] : a){
        val ^= tr1.lca(u, v);
    }
    tm.print();
    tm.set();
    hld_new::hld_tree tr2(n);
    for (auto [u, v] : es){
        tr2.add_edge(u, v);
    }
    tm.print();
    tm.set();
    for (auto [u, v] : a){
        val ^= tr2.lca(u, v);
    }
    tm.print();
    assert(val == 0);
}

void jikken6(){
    int n; in(n);
    int t; in(t);
    while (t--){
        jikken5(n);
    }
}

void jikken7(int n){
    auto g = random_tree(n);
    vector<int> par = [&]{
        vector<int> ret(n);
        noya2::hld_tree tr(n);
        rep(i,n) for (int j : g[i]){
            if (i < j){
                tr.add_edge(i, j);
            }
        }
        ret[0] = -1;
        repp(i,1,n){
            ret[tr.index(i)] = tr.index(tr.parent(i));
        }
        return ret;
    }();
    vector<pii> a(n);
    rep(i,n){
        a[i] = {randint(0,n),randint(0,n)};
    }
    Timer tm;
    tm.set();
    noya2::hld_tree tr1(n);
    repp(i,1,n){
        tr1.add_edge(i, par[i]);
    }
    tm.print();
    tm.set();
    int val = 0;
    for (auto [u, v] : a){
        val ^= tr1.dist(u, v);
    }
    tm.print();
    tm.set();
    hld_new::hld_tree tr2(par);
    tm.print();
    tm.set();
    for (auto [u, v] : a){
        val ^= tr2.dist(u, v);
    }
    tm.print();
    assert(val == 0);
}

void jikken8(){
    int n; in(n);
    int t; in(t);
    while (t--){
        jikken7(n);
    }
}

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

#line 6 "/Users/noya2/Desktop/Noya2_library/data_structure/dsu.hpp"

namespace noya2{

struct dsu {
  public:
    dsu() : _n(0) {}
    dsu(int n) : _n(n), parent_or_size(n, -1) {}

    int merge(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        int x = leader(a), y = leader(b);
        if (x == y) return x;
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return x;
    }

    bool same(int a, int b) {
        assert(0 <= a && a < _n);
        assert(0 <= b && b < _n);
        return leader(a) == leader(b);
    }

    int leader(int a) {
        assert(0 <= a && a < _n);
        if (parent_or_size[a] < 0) return a;
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    int size(int a) {
        assert(0 <= a && a < _n);
        return -parent_or_size[leader(a)];
    }

    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(
            std::remove_if(result.begin(), result.end(),
                           [&](const std::vector<int>& v) { return v.empty(); }),
            result.end());
        return result;
    }

  private:
    int _n;
    // root node: -1 * component size
    // otherwise: parent
    std::vector<int> parent_or_size;
};

} // namespace noya2
#line 2 "/Users/noya2/Desktop/Noya2_library/data_structure/fenwick_tree.hpp"

#line 4 "/Users/noya2/Desktop/Noya2_library/data_structure/fenwick_tree.hpp"

namespace noya2{

template <class T> struct fenwick_tree {
  public:
    fenwick_tree() : _n(0) {}
    explicit fenwick_tree(int n_) : _n(n_), data(n_) {}

    void add(int p, T x) {
        assert(0 <= p && p < _n);
        p++;
        while (p <= _n) {
            data[p - 1] += x;
            p += p & -p;
        }
    }

    T sum(int l, int r) {
        assert(0 <= l && l <= r && r <= _n);
        return sum(r) - sum(l);
    }

  private:
    int _n;
    vector<T> data;

    T sum(int r) {
        T s = 0;
        while (r > 0) {
            s += data[r - 1];
            r -= r & -r;
        }
        return s;
    }
};

} // 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 238 "hld_test.cpp"
using mint = modint998244353;


void jikken9(){
    int n, m; in(n,m);
    vector<vector<int>> to(n);
    rep(i,m){
        int u, v; in(u,v); u--, v--;
        to[v].emplace_back(u);
    }
    dsu d(n);
    vector<int> top(n); iota(all(top),0);
    hld_new::hld_tree g(n,n-1);
    rep(v,n){
        for (int u : to[v]){
            if (d.same(u,v)) continue;
            g.add_edge(top[d.leader(u)],v);
            top[d.merge(u,v)] = v;
        }
    }
    fenwick_tree<mint> fen(n);
    mint ans = 0;
    rep(v,n){
        auto [par, mapping] = g.compressed_tree(to[v]);
        mint dp = 1;
        if (!par.empty()){
            dp += fen.sum(g.index(mapping[0]),g.index(mapping[0])+1);
        }
        repp(i,1,par.size()){
            g.path_query(mapping[i],g.jump(mapping[par[i]],mapping[i],1),[&](int l, int r){
                if (l > r) swap(l,r);
                dp += fen.sum(l,r);
            });
        }
        fen.add(g.index(v),dp);
        // out(v,dp);
        ans += dp;
    }
    out(ans);
}

void solve(){
    // jikken2();
    // jikken4();
    // jikken6();
    // jikken8();
    jikken9();
}

int main(){
    std::cin.tie(0)->sync_with_stdio(0);
    int t = 1; //in(t);
    while (t--) { solve(); }
}