#line 2 "/home/nok0/documents/programming/library/template/header.hpp"
#include <bits/stdc++.h>
#line 3 "/home/nok0/documents/programming/library/graph/graph.hpp"
#pragma region graph

template <class cost_type = long long>
class graph {
   public:
	struct edge {
	   public:
		int from, to;
		cost_type cost;
		int id;
		edge() = default;
		edge(int from_, int to_, cost_type cost_ = 1, int id_ = -1) : from(from_), to(to_), cost(cost_), id(id_) {}
		bool operator<(const edge &a) const { return cost < a.cost; }
		bool operator>(const edge &a) const { return cost > a.cost; }
		friend std::ostream &operator<<(std::ostream &s, const edge &a) {
			s << '(' << a.from << " -> " << a.to << "), cost: " << a.cost << ", id: " << a.id;
			return s;
		}
	};

   private:
	std::vector<std::vector<edge>> edges;
	int next_edge_id = 0;

   public:
	inline const std::vector<edge> &operator[](int k) const { return edges[k]; }
	inline std::vector<edge> &operator[](int k) { return edges[k]; }

	int size() const { return int(edges.size()); }
	void resize(const int n) { edges.resize(n); }
	int edge_count() const { return next_edge_id; }

	friend std::ostream &operator<<(std::ostream &s, const graph<cost_type> &g) {
		for(const auto &adj : g.edges)
			for(const auto &ed : adj) s << ed << '\n';
		return s;
	}

	graph() = default;
	graph(int n) : edges(n) {}
	graph(int n, int e, bool weight = 0, bool directed = 0, int idx = 1) : edges(n) { input(e, weight, directed, idx); }
	const cost_type INF = std::numeric_limits<cost_type>::max() / 3;

	void input(int e = -1, bool weight = false, bool directed = false, int idx = 1) {
		if(e == -1) e = size() - 1;
		while(e--) {
			int u, v;
			std::cin >> u >> v;
			cost_type cost = 1;
			if(weight) std::cin >> cost;
			add_edge(u, v, cost, directed, idx);
		}
	}

	inline int add_edge(int u, int v, cost_type cost = 1, bool directed = false, int idx = 0) {
		u -= idx, v -= idx;
		edges[u].emplace_back(u, v, cost, next_edge_id);
		if(!directed && u != v) edges[v].emplace_back(v, u, cost, next_edge_id);
		return next_edge_id++;
	}

	// Ο(V+E)
	std::vector<cost_type> bfs(int s) const {
		std::vector<cost_type> dist(size(), INF);
		std::queue<int> que;
		dist[s] = 0;
		que.push(s);
		while(!que.empty()) {
			int v = que.front();
			que.pop();
			for(auto &e : edges[v]) {
				if(dist[e.to] != INF) continue;
				dist[e.to] = dist[v] + e.cost;
				que.push(e.to);
			}
		}
		return dist;
	}

	// Ο(V+E)
	// constraint: cost of each edge is zero or x (>= 0)
	std::vector<cost_type> zero_one_bfs(int s) const {
		std::vector<cost_type> dist(size(), INF);
		std::deque<int> deq;
		dist[s] = 0;
		deq.push_back(s);
		while(!deq.empty()) {
			int v = deq.front();
			deq.pop_front();
			for(auto &e : edges[v]) {
				if(dist[e.to] > dist[v] + e.cost) {
					dist[e.to] = dist[v] + e.cost;
					e.cost ? deq.push_back(e.to) : deq.push_front(e.to);
				}
			}
		}
		return dist;
	}

	// Ο((E+V) lg E)
	// unreachable: INF
	std::vector<cost_type> dijkstra(int s) const {
		std::vector<cost_type> dist(size(), INF);
		const auto compare = [](const std::pair<cost_type, int> &a, const std::pair<cost_type, int> &b) {
			return a.first > b.first;
		};
		std::priority_queue<std::pair<cost_type, int>, std::vector<std::pair<cost_type, int>>, decltype(compare)> que{compare};
		dist[s] = 0;
		que.emplace(0, s);
		while(!que.empty()) {
			std::pair<cost_type, int> p = que.top();
			que.pop();
			int v = p.second;
			if(dist[v] < p.first) continue;
			for(auto &e : edges[v]) {
				if(dist[e.to] > dist[v] + e.cost) {
					dist[e.to] = dist[v] + e.cost;
					que.emplace(dist[e.to], e.to);
				}
			}
		}
		return dist;
	}

	// Ο(VE)
	// unreachable: INF
	// reachable via negative cycle: -INF
	std::vector<cost_type> bellman_ford(int s) const {
		int n = size();
		std::vector<cost_type> res(n, INF);
		res[s] = 0;
		for(int loop = 0; loop < n - 1; loop++) {
			for(int v = 0; v < n; v++) {
				if(res[v] == INF) continue;
				for(auto &e : edges[v]) {
					res[e.to] = std::min(res[e.to], res[v] + e.cost);
				}
			}
		}
		std::queue<int> que;
		std::vector<int> chk(n);
		for(int v = 0; v < n; v++) {
			if(res[v] == INF) continue;
			for(auto &e : edges[v]) {
				if(res[e.to] > res[v] + e.cost and !chk[e.to]) {
					que.push(e.to);
					chk[e.to] = 1;
				}
			}
		}
		while(!que.empty()) {
			int now = que.front();
			que.pop();
			for(auto &e : edges[now]) {
				if(!chk[e.to]) {
					chk[e.to] = 1;
					que.push(e.to);
				}
			}
		}
		for(int i = 0; i < n; i++)
			if(chk[i]) res[i] = -INF;
		return res;
	}

	// Ο(V^3)
	std::vector<std::vector<cost_type>> warshall_floyd() const {
		const int n = size();
		std::vector<std::vector<cost_type>> dist(n, std::vector<cost_type>(n, INF));
		for(int i = 0; i < n; i++) dist[i][i] = 0;
		for(int i = 0; i < n; i++)
			for(auto &e : edges[i]) dist[i][e.to] = std::min(dist[i][e.to], e.cost);
		for(int k = 0; k < n; k++)
			for(int i = 0; i < n; i++) {
				if(dist[i][k] == INF) continue;
				for(int j = 0; j < n; j++) {
					if(dist[k][j] == INF) continue;
					dist[i][j] = std::min(dist[i][j], dist[i][k] + dist[k][j]);
				}
			}
		return dist;
	}

	// Ο(V) (using DFS)
	// if a cycle exists, return {}
	std::vector<int> topological_sort() const {
		std::vector<int> res;
		std::vector<int> used(size(), 0);
		bool not_DAG = false;
		auto dfs = [&](auto self, int k) -> void {
			if(not_DAG) return;
			if(used[k]) {
				if(used[k] == 1) not_DAG = true;
				return;
			}
			used[k] = 1;
			for(auto &e : edges[k]) self(self, e.to);
			used[k] = 2;
			res.push_back(k);
		};
		for(int i = 0; i < size(); i++) dfs(dfs, i);
		if(not_DAG) return std::vector<int>{};
		std::reverse(res.begin(), res.end());
		return res;
	}

	bool is_dag() const { return !topological_sort().empty(); }

	// Ο(V)
	// array of the distance to the most distant vertex
	// constraint: the graph is a tree
	std::vector<cost_type> height() const {
		auto vec1 = bfs(0);
		int v1 = -1, v2 = -1;
		cost_type dia = -1;
		for(int i = 0; i < int(size()); i++)
			if(dia < vec1[i]) dia = vec1[i], v1 = i;
		vec1 = bfs(v1);
		dia = -1;
		for(int i = 0; i < int(size()); i++)
			if(dia < vec1[i]) dia = vec1[i], v2 = i;
		auto vec2 = bfs(v2);
		for(int i = 0; i < int(size()); i++) {
			if(vec1[i] < vec2[i]) vec1[i] = vec2[i];
		}
		return vec1;
	}

	// O(V+E)
	// vector<(int)(0 or 1)>
	// if it is not bipartite, return {}
	std::vector<int> bipartite_grouping() const {
		std::vector<int> colors(size(), -1);
		auto dfs = [&](auto self, int now, int col) -> bool {
			colors[now] = col;
			for(auto &e : edges[now]) {
				if(col == colors[e.to]) return false;
				if(colors[e.to] == -1 and !self(self, e.to, !col)) return false;
			}
			return true;
		};
		for(int i = 0; i < int(size()); i++)
			if(colors[i] == -1 and !dfs(dfs, i, 0)) return std::vector<int>{};
		return colors;
	}

	bool is_bipartite() const { return !bipartite_grouping().empty(); }

	// Ο(V+E)
	// (v1, v2, diameter)
	std::tuple<int, int, cost_type> diameter() {
		std::vector<cost_type> dist = bfs(0);
		auto it = std::max_element(dist.begin(), dist.end());
		const int v = it - dist.begin();
		dist = bfs(v);
		it = std::max_element(dist.begin(), dist.end());
		return std::make_tuple(v, int(it - dist.begin()), *it);
	}

	// Ο(V+E)
	std::vector<int> subtree_size(const int root) {
		const int n = size();
		std::vector<int> ret(n, 1);
		auto dfs = [&](auto self, int now, int p = -1) -> void {
			for(const auto &e : (*this)[now]) {
				if(e.to == p) continue;
				self(self, e.to, now);
				ret[now] += ret[e.to];
			}
		};
		dfs(dfs, root);
		return ret;
	}

	// Ο(ElgE)
	cost_type prim() const {
		cost_type res = 0;
		std::priority_queue<edge, std::vector<edge>, std::greater<edge>> que;
		for(auto &e : edges[0]) que.push(e);
		std::vector<int> chk(size());
		chk[0] = 1;
		int cnt = 1;
		while(cnt < size()) {
			auto e = que.top();
			que.pop();
			if(chk[e.to]) continue;
			cnt++;
			res += e.cost;
			chk[e.to] = 1;
			for(auto &e2 : edges[e.to]) que.push(e2);
		}
		return res;
	}

	// Ο(ElgE)
	cost_type kruskal() const {
		std::vector<std::tuple<int, int, cost_type>> eds;
		for(const auto &adj : edges)
			for(const auto &ed : adj) eds.emplace_back(ed.from, ed.to, ed.cost);
		std::sort(eds.begin(), eds.end(), [](const std::tuple<int, int, cost_type> &a, const std::tuple<int, int, cost_type> &b) {
			return std::get<2>(a) < std::get<2>(b);
		});
		std::vector<int> uf_data(size(), -1);
		auto root = [&uf_data](auto self, int x) -> int {
			if(uf_data[x] < 0) return x;
			return uf_data[x] = self(self, uf_data[x]);
		};
		auto unite = [&uf_data, &root](int u, int v) -> bool {
			u = root(root, u), v = root(root, v);
			if(u == v) return false;
			if(uf_data[u] > uf_data[v]) std::swap(u, v);
			uf_data[u] += uf_data[v];
			uf_data[v] = u;
			return true;
		};
		cost_type ret = 0;
		for(auto &e : eds)
			if(unite(std::get<0>(e), std::get<1>(e))) ret += std::get<2>(e);
		return ret;
	}

	// O(V)
	std::vector<int> centroid() const {
		std::vector<int> centroid, sz(size());
		auto dfs = [&](auto self, int now, int per) -> void {
			sz[now] = 1;
			bool is_centroid = true;
			for(auto &e : edges[now]) {
				if(e.to != per) {
					self(self, e.to, now);
					sz[now] += sz[e.to];
					if(sz[e.to] > size() / 2) is_centroid = false;
				}
			}
			if(size() - sz[now] > size() / 2) is_centroid = false;
			if(is_centroid) centroid.push_back(now);
		};
		dfs(dfs, 0, -1);
		return centroid;
	}

	// O(V+E)
	// bridge: (s, t)  (s < t);
	std::pair<std::vector<std::pair<int, int>>, std::vector<int>> bridges_and_articulations() const {
		std::vector<int> order(size(), -1), low(size()), articulation;
		int order_next = 0;
		std::vector<std::pair<int, int>> bridge;
		auto dfs = [&](auto self, int now, int par = -1) -> void {
			low[now] = order[now] = order_next++;
			bool is_articulation = false;
			int cnt = 0;
			for(auto &ed : edges[now]) {
				int &nxt = ed.to;
				if(nxt == par) continue;
				if(order[nxt] == -1) {
					cnt++;
					self(self, nxt, now);
					if(order[now] < low[nxt]) bridge.push_back(std::minmax(now, nxt));
					if(order[now] <= low[nxt]) is_articulation = true;
					low[now] = std::min(low[now], low[nxt]);
				} else if(order[now] > order[nxt]) {
					low[now] = std::min(low[now], order[nxt]);
				}
			}
			if(par == -1 and cnt < 2) is_articulation = false;
			if(is_articulation) articulation.push_back(now);
			return;
		};
		for(int i = 0; i < (int)size(); i++)
			if(order[i] == -1) dfs(dfs, i);
		return std::make_pair(bridge, articulation);
	}

	// Ο(V+E)
	// directed graph from root to leaf
	graph root_to_leaf(int root = 0) const {
		graph res(size());
		std::vector<int> chk(size(), 0);
		chk[root] = 1;
		auto dfs = [&](auto self, int now) -> void {
			for(auto &e : edges[now]) {
				if(chk[e.to] == 1) continue;
				chk[e.to] = 1;
				res.add_edge(now, e.to, e.cost, 1, 0);
				self(self, e.to);
			}
		};
		dfs(dfs, root);
		return res;
	}

	// Ο(V+E)
	// directed graph from leaf to root
	graph leaf_to_root(int root = 0) const {
		graph res(size());
		std::vector<int> chk(size(), 0);
		chk[root] = 1;
		auto dfs = [&](auto self, int now) -> void {
			for(auto &e : edges[now]) {
				if(chk[e.to] == 1) continue;
				chk[e.to] = 1;
				res.add_edge(e.to, now, e.cost, 1, 0);
				self(self, e.to);
			}
		};
		dfs(dfs, root);
		return res;
	}

	// cost_type Chu_Liu_Edmonds(int root = 0) {}
};
#pragma endregion
#line 3 "/home/nok0/documents/programming/library/graph/namori.hpp"

template <class cost_type>
struct namori {
   public:
	std::vector<std::vector<int>> cycles;
	std::vector<int> cycle_number, root;
	namori(graph<cost_type> &g) : n(g.size()), g(g) { build(); }

   private:
	int n;
	std::vector<bool> dfs_used, dfs_now;
	std::vector<int> dfs_ver;
	graph<cost_type> &g;
	graph<cost_type> forest;
	void build() {
		forest.resize(n);
		dfs_used.assign(n, false);
		dfs_now.assign(n, false);
		dfs_ver.clear();
		cycle_number.assign(n, -1);
		root.assign(n, -1);
		for(int i = 0; i < n; i++) {
			if(dfs_used[i]) continue;
			cycles.emplace_back();
			build_dfs(i, -1, cycles.back());
			if(cycles.back().empty()) cycles.back().push_back(i);
		}
		for(int i = 0; i < (int)cycles.size(); i++)
			for(const auto &v : cycles[i]) cycle_number[v] = i;
		dfs_used.assign(n, false);
		for(int i = 0; i < n; i++)
			if(cycle_number[i] != -1) tree_dfs(i, i);
	}

	void build_dfs(int now, int par, std::vector<int> &cycle) {
		dfs_used[now] = true;
		dfs_now[now] = true;
		dfs_ver.push_back(now);
		bool multiple_par = false;
		for(const auto &e : g[now]) {
			if(dfs_now[e.to] and cycle.empty() and (e.to != par or multiple_par)) {
				for(const auto &v : dfs_ver) {
					if(!cycle.empty() or v == e.to) cycle.push_back(v);
				}
			}
			if(e.to == par) multiple_par = true;
			if(dfs_used[e.to]) continue;
			build_dfs(e.to, now, cycle);
		}
		dfs_now[now] = false;
		dfs_ver.pop_back();
	}

	void tree_dfs(int now, int r) {
		dfs_used[now] = true;
		root[now] = r;
		for(const auto &e : g[now]) {
			if(dfs_used[e.to] or cycle_number[e.to] != -1) continue;
			forest.add_edge(now, e.to, e.cost);
			tree_dfs(e.to, r);
		}
	}
};
#line 1 "/home/nok0/documents/programming/library/atcoder/modint.hpp"



#line 6 "/home/nok0/documents/programming/library/atcoder/modint.hpp"
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "/home/nok0/documents/programming/library/atcoder/internal_math.hpp"



#line 5 "/home/nok0/documents/programming/library/atcoder/internal_math.hpp"

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m < 2^31`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
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;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
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 = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
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};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    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;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
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 = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 1 "/home/nok0/documents/programming/library/atcoder/internal_type_traits.hpp"



#line 7 "/home/nok0/documents/programming/library/atcoder/internal_type_traits.hpp"

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


#line 14 "/home/nok0/documents/programming/library/atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    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;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        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 {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            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;
    }

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

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    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;
    }

    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 = internal::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;
    }

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

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

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 4 "/home/nok0/documents/programming/library/math/factorial.hpp"

template <class T>
struct factorial {
   private:
	int _n;
	std::vector<T> _fac, _inv, _finv;

	void extend(int n) {
		n++;
		if(n <= _n) return;
		_fac.resize(n);
		_inv.resize(n);
		_finv.resize(n);
		for(int i = _n; i < n; i++) {
			_fac[i] = _fac[i - 1] * i;
			_inv[i] = (-_inv[T::mod() % i]) * (T::mod() / i);
			_finv[i] = _finv[i - 1] * _inv[i];
		}
		_n = n;
	}

   public:
	explicit factorial(int n = 2) {
		_n = 2;
		_fac = {1, 1};
		_inv = {1, 1};
		_finv = {1, 1};
		extend(n);
	}

	T fac(int k) {
		extend(k);
		return _fac[k];
	}
	T inv(int k) {
		extend(k);
		return _inv[k];
	}
	T finv(int k) {
		extend(k);
		return _finv[k];
	}

	T binom(int n, int k) {
		if(k < 0 or n < k) return 0;
		extend(n);
		return _fac[n] * _finv[k] * _finv[n - k];
	}

	T large_binom(long long n, int k) {
		if(k < 0 or n < k) return 0;
		extend(k);
		T ret = _finv[k];
		for(int i = n; i > n - k; i--) ret *= i;
		return ret;
	}

	T catalan(int n) {
		extend(n * 2);
		return binom(n * 2, n) * _inv[n + 1];
	}

	T perm(int n, int k) {
		if(k < 0 or n < k) return 0;
		extend(n);
		return _fac[n] * _finv[n - k];
	}
};
#line 3 "/home/nok0/documents/programming/library/math/modint_iostream.hpp"

#line 5 "/home/nok0/documents/programming/library/math/modint_iostream.hpp"
template<int m>
std::istream &std::operator>>(std::istream &is, atcoder::static_modint<m> &a) {
    long long v;
    is >> v;
    a = v;
    return is;
}
template<int m>
std::istream &std::operator>>(std::istream &is, atcoder::dynamic_modint<m> &a) {
    long long v;
    is >> v;
    a = v;
    return is;
}
template<int m>
std::ostream &std::operator<<(std::ostream &os, const atcoder::static_modint<m> &a) { return os << a.val(); }
template<int m>
std::ostream &std::operator<<(std::ostream &os, const atcoder::dynamic_modint<m> &a) { return os << a.val(); }
#line 3 "/home/nok0/documents/programming/library/template/def_const.hpp"

const int inf = 1000000000;
const long long INF = 1000000000000000000ll;
#line 4 "/home/nok0/documents/programming/library/template/debug.hpp"

namespace viewer {
void view(const long long &e) {
	if(e == INF)
		std::cerr << "INF";
	else if(e == -INF)
		std::cerr << "-INF";
	else
		std::cerr << e;
}

void view(const int &e) {
	if(e == inf)
		std::cerr << "inf";
	else if(e == -inf)
		std::cerr << "-inf";
	else
		std::cerr << e;
}

template <typename T>
void view(const T &e) {
	std::cerr << e;
}

template <typename T, typename U>
void view(const std::pair<T, U> &p) {
	std::cerr << "(";
	view(p.first);
	std::cerr << ", ";
	view(p.second);
	std::cerr << ")";
}

template <class T0, class T1, class T2>
void view(const std::tuple<T0, T1, T2> &p) {
	std::cerr << "(";
	view(std::get<0>(p));
	std::cerr << ", ";
	view(std::get<1>(p));
	std::cerr << ", ";
	view(std::get<2>(p));
	std::cerr << ")";
}

template <class T0, class T1, class T2, class T3>
void view(const std::tuple<T0, T1, T2, T3> &p) {
	std::cerr << "(";
	view(std::get<0>(p));
	std::cerr << ", ";
	view(std::get<1>(p));
	std::cerr << ", ";
	view(std::get<2>(p));
	std::cerr << ", ";
	view(std::get<3>(p));
	std::cerr << ")";
}

template <typename T>
void view(const std::set<T> &s) {
	if(s.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(auto &t : s) {
		view(t);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T>
void view(const std::unordered_set<T> &s) {
	if(s.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(auto &t : s) {
		view(t);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T>
void view(const std::multiset<T> &s) {
	if(s.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(auto &t : s) {
		view(t);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T>
void view(const std::unordered_multiset<T> &s) {
	if(s.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(auto &t : s) {
		view(t);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T>
void view(const std::vector<T> &v) {
	if(v.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(const auto &e : v) {
		view(e);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T, std::size_t ary_size>
void view(const std::array<T, ary_size> &v) {
	if(v.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << "{ ";
	for(const auto &e : v) {
		view(e);
		std::cerr << ", ";
	}
	std::cerr << "\b\b }";
}

template <typename T>
void view(const std::vector<std::vector<T>> &vv) {
	std::cerr << "{\n";
	for(const auto &v : vv) {
		std::cerr << "\t";
		view(v);
		std::cerr << '\n';
	}
	std::cerr << "}";
}

template <typename T, typename U>
void view(const std::vector<std::pair<T, U>> &v) {
	std::cerr << "{\n";
	for(const auto &c : v) {
		std::cerr << "\t(";
		view(c.first);
		std::cerr << ", ";
		view(c.second);
		std::cerr << ")\n";
	}
	std::cerr << "}";
}

template <class T0, class T1, class T2>
void view(const std::vector<std::tuple<T0, T1, T2>> &v) {
	if(v.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << '{';
	for(const auto &t : v) {
		std::cerr << "\n\t";
		view(t);
		std::cerr << ",";
	}
	std::cerr << "\n}";
}

template <class T0, class T1, class T2, class T3>
void view(const std::vector<std::tuple<T0, T1, T2, T3>> &v) {
	if(v.empty()) {
		std::cerr << "{ }";
		return;
	}
	std::cerr << '{';
	for(const auto &t : v) {
		std::cerr << "\n\t";
		view(t);
		std::cerr << ",";
	}
	std::cerr << "\n}";
}

template <typename T, typename U>
void view(const std::map<T, U> &m) {
	std::cerr << "{\n";
	for(const auto &t : m) {
		std::cerr << "\t[";
		view(t.first);
		std::cerr << "] : ";
		view(t.second);
		std::cerr << '\n';
	}
	std::cerr << "}";
}

template <typename T, typename U>
void view(const std::unordered_map<T, U> &m) {
	std::cerr << "{\n";
	for(const auto &t : m) {
		std::cerr << "\t[";
		view(t.first);
		std::cerr << "] : ";
		view(t.second);
		std::cerr << '\n';
	}
	std::cerr << "}";
}
}  // namespace viewer

// when compiling : g++ foo.cpp -DLOCAL
#ifdef LOCAL
void debug_out() {}
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
	viewer::view(H);
	std::cerr << ", ";
	debug_out(T...);
}
#define debug(...)                                                \
	do {                                                          \
		std::cerr << __LINE__ << " [" << #__VA_ARGS__ << "] : ["; \
		debug_out(__VA_ARGS__);                                   \
		std::cerr << "\b\b]\n";                                   \
	} while(0)
#define dump(x)                                      \
	do {                                             \
		std::cerr << __LINE__ << " " << #x << " : "; \
		viewer::view(x);                             \
		std::cerr << '\n';                           \
	} while(0)

#else
#define debug(...) (void(0))
#define dump(x)    (void(0))
#endif
#line 3 "/home/nok0/documents/programming/library/template/def_name.hpp"

#define pb        push_back
#define eb        emplace_back
#define fi        first
#define se        second
#define SZ(x)     ((int)(x).size())
#define all(x)    (x).begin(), (x).end()
#define rall(x)   (x).rbegin(), (x).rend()
#define popcnt(x) __builtin_popcountll(x)
template <class T = int>
using V = std::vector<T>;
template <class T = int>
using VV = std::vector<std::vector<T>>;
template <class T>
using pqup = std::priority_queue<T, std::vector<T>, std::greater<T>>;
using ll = long long;
using ld = long double;
using int128 = __int128_t;
using pii = std::pair<int, int>;
using pll = std::pair<long long, long long>;
#line 3 "/home/nok0/documents/programming/library/template/fast_io.hpp"

struct fast_io {
	fast_io() {
		std::ios::sync_with_stdio(false);
		std::cin.tie(nullptr);
		std::cout << std::fixed << std::setprecision(15);
	}
} fast_io_;
#line 3 "/home/nok0/documents/programming/library/template/input.hpp"

template <class T, class U>
std::istream &operator>>(std::istream &is, std::pair<T, U> &p) {
	is >> p.first >> p.second;
	return is;
}
template <class T>
std::istream &operator>>(std::istream &is, std::vector<T> &v) {
	for(T &i : v) is >> i;
	return is;
}
std::istream &operator>>(std::istream &is, __int128_t &a) {
	std::string s;
	is >> s;
	__int128_t ret = 0;
	for(int i = 0; i < (int)s.length(); i++)
		if('0' <= s[i] and s[i] <= '9')
			ret = 10 * ret + s[i] - '0';
	a = ret * (s[0] == '-' ? -1 : 1);
	return is;
}
namespace scanner {
void scan(int &a) { std::cin >> a; }
void scan(long long &a) { std::cin >> a; }
void scan(std::string &a) { std::cin >> a; }
void scan(char &a) { std::cin >> a; }
void scan(char a[]) { std::ignore = std::scanf("%s", a); }
void scan(double &a) { std::cin >> a; }
void scan(long double &a) { std::cin >> a; }
template <class T, class U>
void scan(std::pair<T, U> &p) { std::cin >> p; }
template <class T>
void scan(std::vector<T> &a) { std::cin >> a; }
void INPUT() {}
template <class Head, class... Tail>
void INPUT(Head &head, Tail &...tail) {
	scan(head);
	INPUT(tail...);
}
}  // namespace scanner
#define VEC(type, name, size)     \
	std::vector<type> name(size); \
	scanner::INPUT(name)
#define VVEC(type, name, h, w)                                    \
	std::vector<std::vector<type>> name(h, std::vector<type>(w)); \
	scanner::INPUT(name)
#define INT(...)     \
	int __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define LL(...)            \
	long long __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define STR(...)             \
	std::string __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define CHAR(...)     \
	char __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define DOUBLE(...)     \
	double __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#define LD(...)              \
	long double __VA_ARGS__; \
	scanner::INPUT(__VA_ARGS__)
#line 3 "/home/nok0/documents/programming/library/template/math.hpp"

template <class T, class U>
inline bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; }
template <class T, class U>
inline bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; }
template <class T>
T divup(T x, T y) { return (x + y - 1) / y; }
template <class T>
T POW(T a, long long n) {
	T ret = 1;
	while(n) {
		if(n & 1) ret *= a;
		a *= a;
		n >>= 1;
	}
	return ret;
}
long long POW(long long a, long long n, const int mod) {
	long long ret = 1;
	a = (a % mod + mod) % mod;
	while(n) {
		if(n & 1) (ret *= a) %= mod;
		(a *= a) %= mod;
		n >>= 1;
	}
	return ret;
}
template <class T, class F>
T bin_search(T ok, T ng, const F &f) {
	while(abs(ok - ng) > 1) {
		T mid = (ok + ng) >> 1;
		(f(mid) ? ok : ng) = mid;
	}
	return ok;
}
template <class T, class F>
T bin_search(T ok, T ng, const F &f, int loop) {
	for(int i = 0; i < loop; i++) {
		T mid = (ok + ng) / 2;
		(f(mid) ? ok : ng) = mid;
	}
	return ok;
}
#line 3 "/home/nok0/documents/programming/library/template/output.hpp"


template <class T, class U>
std::ostream &operator<<(std::ostream &os, const std::pair<T, U> &p) {
	os << p.first << " " << p.second;
	return os;
}
template <class T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &a) {
	for(int i = 0; i < int(a.size()); ++i) {
		if(i) os << " ";
		os << a[i];
	}
	return os;
}
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;
}
template <class T>
void print(const T a) { std::cout << a << '\n'; }
template <class Head, class... Tail>
void print(Head H, Tail... T) {
	std::cout << H << ' ';
	print(T...);
}
template <class T>
void println(const T a) { std::cout << a << '\n'; }
template <class T>
void println(const std::vector<T> &a) {
	for(const auto &v : a)
		std::cout << v << '\n';
}
template <class Head, class... Tail>
void println(Head H, Tail... T) {
	std::cout << H << '\n';
	println(T...);
}
void Yes(const bool b = true) { std::cout << (b ? "Yes\n" : "No\n"); }
void No() { std::cout << "No\n"; }
void YES(const bool b = true) { std::cout << (b ? "YES\n" : "NO\n"); }
void NO() { std::cout << "NO\n"; }
#line 2 "/home/nok0/documents/programming/library/template/rep.hpp"

#define foa(v, a)                   for (auto &v : a)
#define repname(a, b, c, d, e, ...) e
#define rep(...)                    repname(__VA_ARGS__, rep3, rep2, rep1, rep0)(__VA_ARGS__)
#define rep0(x)                     for (int rep_counter = 0; rep_counter < (x); ++rep_counter)
#define rep1(i, x)                  for (int i = 0; i < (x); ++i)
#define rep2(i, l, r)               for (int i = (l); i < (r); ++i)
#define rep3(i, l, r, c)            for (int i = (l); i < (r); i += (c))

#define repsname(a, b, c, ...) c
#define reps(...)              repsname(__VA_ARGS__, reps1, reps0)(__VA_ARGS__)
#define reps0(x)               for (int reps_counter = 1; reps_counter <= (x); ++reps_counter)
#define reps1(i, x)            for (int i = 1; i <= (x); ++i)

#define rrepname(a, b, c, ...) c
#define rrep(...)              rrepname(__VA_ARGS__, rrep1, rrep0)(__VA_ARGS__)
#define rrep0(x)               for (int rrep_counter = (x)-1; rrep_counter >= 0; --rrep_counter)
#define rrep1(i, x)            for (int i = (x)-1; i >= 0; --i)
#line 3 "/home/nok0/documents/programming/library/template/string_converter.hpp"

struct string_converter {
	char start = 0;
	char type(const char &c) const { return (islower(c) ? 'a' : isupper(c) ? 'A' :
		                                                    isdigit(c)     ? '0' :
		                                                                     0); }
	int convert(const char &c) {
		if(!start) start = type(c);
		return c - start;
	}
	int convert(const char &c, const std::string &chars) { return chars.find(c); }
	template <typename T>
	auto convert(const T &v) {
		std::vector<decltype(convert(v[0]))> ret;
		ret.reserve(size(v));
		for(auto &&e : v) ret.emplace_back(convert(e));
		return ret;
	}
	template <typename T>
	auto convert(const T &v, const std::string &chars) {
		std::vector<decltype(convert(v[0], chars))> ret;
		ret.reserve(size(v));
		for(auto &&e : v) ret.emplace_back(convert(e, chars));
		return ret;
	}
	int operator()(const char &v, char s = 0) {
		start = s;
		return convert(v);
	}
	int operator()(const char &v, const std::string &chars) { return convert(v, chars); }
	template <typename T>
	auto operator()(const T &v, char s = 0) {
		start = s;
		return convert(v);
	}
	template <typename T>
	auto operator()(const T &v, const std::string &chars) { return convert(v, chars); }
} toint;
#line 3 "/home/nok0/documents/programming/library/template/vector.hpp"

template <class T>
int lb(const std::vector<T> &a, const T x) { return std::distance((a).begin(), std::lower_bound((a).begin(), (a).end(), (x))); }
template <class T>
int ub(const std::vector<T> &a, const T x) { return std::distance((a).begin(), std::upper_bound((a).begin(), (a).end(), (x))); }
template <class T>
void UNIQUE(std::vector<T> &a) {
	std::sort(a.begin(), a.end());
	a.erase(std::unique(a.begin(), a.end()), a.end());
}
template <class T>
std::vector<T> press(std::vector<T> &a) {
	auto res = a;
	UNIQUE(res);
	for(auto &v : a)
		v = lb(res, v);
	return res;
}
#define SORTname(a, b, c, ...) c
#define SORT(...)              SORTname(__VA_ARGS__, SORT1, SORT0, ...)(__VA_ARGS__)
#define SORT0(a)               std::sort((a).begin(), (a).end())
#define SORT1(a, c)            std::sort((a).begin(), (a).end(), [](const auto x, const auto y) { return x c y; })
template <class T>
void ADD(std::vector<T> &a, const T x = 1) {
	for(auto &v : a) v += x;
}
template <class T>
void SUB(std::vector<T> &a, const T x = 1) {
	for(auto &v : a) v -= x;
}
template <class T>
struct cum_vector {
   public:
	cum_vector() = default;
	template <class U>
	cum_vector(const std::vector<U> &vec) : cum((int)vec.size() + 1) {
		for(int i = 0; i < (int)vec.size(); i++)
			cum[i + 1] = cum[i] + vec[i];
	}
	T prod(int l, int r) {
		return cum[r] - cum[l];
	}

   private:
	std::vector<T> cum;
};
std::vector<std::pair<char, int>> rle(const std::string &s) {
	const int n = s.size();
	std::vector<std::pair<char, int>> ret;
	ret.reserve(n);
	for(int l = 0; l < n;) {
		int r = l + 1;
		for(; r < n and s[l] == s[r]; r++) {}
		ret.emplace_back(s[l], r - l);
		l = r;
	}
	return ret;
}
template <class T>
std::vector<std::pair<T, int>> rle(const std::vector<T> &v) {
	const int n = v.size();
	std::vector<std::pair<T, int>> ret;
	ret.reserve(n);
	for(int l = 0; l < n;) {
		int r = l + 1;
		for(; r < n and v[l] == v[r]; r++) {}
		ret.emplace_back(v[l], r - l);
		l = r;
	}
	return ret;
}
std::vector<int> iota(int n) {
	std::vector<int> p(n);
	std::iota(p.begin(), p.end(), 0);
	return p;
}
#line 12 "/home/nok0/documents/programming/library/template/all"
using namespace std;
#line 5 "g.cpp"

using mint = atcoder::modint998244353;
void main_();
int main() {
	int t = 1;
	while(t--) main_();
}

void main_() {
	INT(n, k);
	graph g(n, n);
	namori nm(g);
	auto cy = nm.cycles[0];
	int clen = SZ(cy);
	//長さ 3 のサイクルを
	mint res = mint(k - 1).pow(n - clen);
	vector dp(clen, vector(2, mint(0)));
	dp[0][0] = 1;
	rep(i, 1, clen) {
		dp[i][1] = dp[i - 1][0] * (k - 1) + dp[i - 1][1] * (k - 2);
		dp[i][0] = dp[i - 1][1];
	}
	debug(dp);
	mint tmp = dp.back()[1] * k;
	print(res * tmp);
}