#ifndef ATCODER_INTERNAL_BITOP_HPP
#define ATCODER_INTERNAL_BITOP_HPP 1
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
	int x = 0;
	while ((1U << x) < (unsigned int)(n)) x++;
	return x;
}
// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
	unsigned long index;
	_BitScanForward(&index, n);
	return index;
#else
	return __builtin_ctz(n);
#endif
}
}  // namespace internal
}  // namespace atcoder
#endif  // ATCODER_INTERNAL_BITOP_HPP
#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1
#include <utility>
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 moduler 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`
	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;
	for (long long a : {2, 7, 61}) {
		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);
}  // namespace internal
}  // namespace atcoder
#endif  // ATCODER_INTERNAL_MATH_HPP
#ifndef ATCODER_INTERNAL_QUEUE_HPP
#define ATCODER_INTERNAL_QUEUE_HPP 1
#include <vector>
namespace atcoder {
namespace internal {
template <class T> struct simple_queue {
	std::vector<T> payload;
	int pos = 0;
	void reserve(int n) { payload.reserve(n); }
	int size() const { return int(payload.size()) - pos; }
	bool empty() const { return pos == int(payload.size()); }
	void push(const T& t) { payload.push_back(t); }
	T& front() { return payload[pos]; }
	void clear() {
		payload.clear();
		pos = 0;
	}
	void pop() { pos++; }
};
}  // namespace internal
}  // namespace atcoder
#endif  // ATCODER_INTERNAL_QUEUE_HPP
#ifndef ATCODER_INTERNAL_SCC_HPP
#define ATCODER_INTERNAL_SCC_HPP 1
#include <algorithm>
#include <utility>
#include <vector>
namespace atcoder {
namespace internal {
template <class E> struct csr {
	std::vector<int> start;
	std::vector<E> elist;
	csr(int n, const std::vector<std::pair<int, E>>& edges)
		: start(n + 1), elist(edges.size()) {
		for (auto e : edges) {
			start[e.first + 1]++;
		}
		for (int i = 1; i <= n; i++) {
			start[i] += start[i - 1];
		}
		auto counter = start;
		for (auto e : edges) {
			elist[counter[e.first]++] = e.second;
		}
	}
};
// Reference:
// R. Tarjan,
// Depth-First Search and Linear Graph Algorithms
struct scc_graph {
  public:
	scc_graph(int n) : _n(n) {}
	int num_vertices() { return _n; }
	void add_edge(int from, int to) { edges.push_back({from, {to}}); }
	// @return pair of (# of scc, scc id)
	std::pair<int, std::vector<int>> scc_ids() {
		auto g = csr<edge>(_n, edges);
		int now_ord = 0, group_num = 0;
		std::vector<int> visited, low(_n), ord(_n, -1), ids(_n);
		visited.reserve(_n);
		auto dfs = [&](auto self, int v) -> void {
			low[v] = ord[v] = now_ord++;
			visited.push_back(v);
			for (int i = g.start[v]; i < g.start[v + 1]; i++) {
				auto to = g.elist[i].to;
				if (ord[to] == -1) {
					self(self, to);
					low[v] = std::min(low[v], low[to]);
				} else {
					low[v] = std::min(low[v], ord[to]);
				}
			}
			if (low[v] == ord[v]) {
				while (true) {
					int u = visited.back();
					visited.pop_back();
					ord[u] = _n;
					ids[u] = group_num;
					if (u == v) break;
				}
				group_num++;
			}
		};
		for (int i = 0; i < _n; i++) {
			if (ord[i] == -1) dfs(dfs, i);
		}
		for (auto& x : ids) {
			x = group_num - 1 - x;
		}
		return {group_num, ids};
	}
	std::vector<std::vector<int>> scc() {
		auto ids = scc_ids();
		int group_num = ids.first;
		std::vector<int> counts(group_num);
		for (auto x : ids.second) counts[x]++;
		std::vector<std::vector<int>> groups(ids.first);
		for (int i = 0; i < group_num; i++) {
			groups[i].reserve(counts[i]);
		}
		for (int i = 0; i < _n; i++) {
			groups[ids.second[i]].push_back(i);
		}
		return groups;
	}
  private:
	int _n;
	struct edge {
		int to;
	};
	std::vector<std::pair<int, edge>> edges;
};
}  // namespace internal
}  // namespace atcoder
#endif  // ATCODER_INTERNAL_SCC_HPP
#ifndef ATCODER_INTERNAL_TYPE_TRAITS_HPP
#define ATCODER_INTERNAL_TYPE_TRAITS_HPP 1
#include <cassert>
#include <numeric>
#include <type_traits>
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
#endif  // ATCODER_INTERNAL_TYPE_TRAITS_HPP
#ifndef ATCODER_MODINT_HPP
#define ATCODER_MODINT_HPP 1
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
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());
	}
	static_modint(bool 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());
	}
	dynamic_modint(bool 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
#endif  // ATCODER_MODINT_HPP
#ifndef ATCODER_CONVOLUTION_HPP
#define ATCODER_CONVOLUTION_HPP 1
#include <algorithm>
#include <array>
#include <cassert>
#include <type_traits>
#include <vector>
namespace atcoder {
namespace internal {
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
	static constexpr int g = internal::primitive_root<mint::mod()>;
	int n = int(a.size());
	int h = internal::ceil_pow2(n);
	static bool first = true;
	static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
	if (first) {
		first = false;
		mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
		int cnt2 = bsf(mint::mod() - 1);
		mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
		for (int i = cnt2; i >= 2; i--) {
			// e^(2^i) == 1
			es[i - 2] = e;
			ies[i - 2] = ie;
			e *= e;
			ie *= ie;
		}
		mint now = 1;
		for (int i = 0; i < cnt2 - 2; i++) {
			sum_e[i] = es[i] * now;
			now *= ies[i];
		}
	}
	for (int ph = 1; ph <= h; ph++) {
		int w = 1 << (ph - 1), p = 1 << (h - ph);
		mint now = 1;
		for (int s = 0; s < w; s++) {
			int offset = s << (h - ph + 1);
			for (int i = 0; i < p; i++) {
				auto l = a[i + offset];
				auto r = a[i + offset + p] * now;
				a[i + offset] = l + r;
				a[i + offset + p] = l - r;
			}
			now *= sum_e[bsf(~(unsigned int)(s))];
		}
	}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
	static constexpr int g = internal::primitive_root<mint::mod()>;
	int n = int(a.size());
	int h = internal::ceil_pow2(n);
	static bool first = true;
	static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
	if (first) {
		first = false;
		mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
		int cnt2 = bsf(mint::mod() - 1);
		mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
		for (int i = cnt2; i >= 2; i--) {
			// e^(2^i) == 1
			es[i - 2] = e;
			ies[i - 2] = ie;
			e *= e;
			ie *= ie;
		}
		mint now = 1;
		for (int i = 0; i < cnt2 - 2; i++) {
			sum_ie[i] = ies[i] * now;
			now *= es[i];
		}
	}
	for (int ph = h; ph >= 1; ph--) {
		int w = 1 << (ph - 1), p = 1 << (h - ph);
		mint inow = 1;
		for (int s = 0; s < w; s++) {
			int offset = s << (h - ph + 1);
			for (int i = 0; i < p; i++) {
				auto l = a[i + offset];
				auto r = a[i + offset + p];
				a[i + offset] = l + r;
				a[i + offset + p] =
					(unsigned long long)(mint::mod() + l.val() - r.val()) *
					inow.val();
			}
			inow *= sum_ie[bsf(~(unsigned int)(s))];
		}
	}
}
}  // namespace internal
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
	int n = int(a.size()), m = int(b.size());
	if (!n || !m) return {};
	if (std::min(n, m) <= 60) {
		if (n < m) {
			std::swap(n, m);
			std::swap(a, b);
		}
		std::vector<mint> ans(n + m - 1);
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				ans[i + j] += a[i] * b[j];
			}
		}
		return ans;
	}
	int z = 1 << internal::ceil_pow2(n + m - 1);
	a.resize(z);
	internal::butterfly(a);
	b.resize(z);
	internal::butterfly(b);
	for (int i = 0; i < z; i++) {
		a[i] *= b[i];
	}
	internal::butterfly_inv(a);
	a.resize(n + m - 1);
	mint iz = mint(z).inv();
	for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
	return a;
}
template <unsigned int mod = 998244353,
		  class T,
		  std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
	int n = int(a.size()), m = int(b.size());
	if (!n || !m) return {};
	using mint = static_modint<mod>;
	std::vector<mint> a2(n), b2(m);
	for (int i = 0; i < n; i++) {
		a2[i] = mint(a[i]);
	}
	for (int i = 0; i < m; i++) {
		b2[i] = mint(b[i]);
	}
	auto c2 = convolution(move(a2), move(b2));
	std::vector<T> c(n + m - 1);
	for (int i = 0; i < n + m - 1; i++) {
		c[i] = c2[i].val();
	}
	return c;
}
std::vector<long long> convolution_ll(const std::vector<long long>& a,
									  const std::vector<long long>& b) {
	int n = int(a.size()), m = int(b.size());
	if (!n || !m) return {};
	static constexpr unsigned long long MOD1 = 754974721;  // 2^24
	static constexpr unsigned long long MOD2 = 167772161;  // 2^25
	static constexpr unsigned long long MOD3 = 469762049;  // 2^26
	static constexpr unsigned long long M2M3 = MOD2 * MOD3;
	static constexpr unsigned long long M1M3 = MOD1 * MOD3;
	static constexpr unsigned long long M1M2 = MOD1 * MOD2;
	static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
	static constexpr unsigned long long i1 =
		internal::inv_gcd(MOD2 * MOD3, MOD1).second;
	static constexpr unsigned long long i2 =
		internal::inv_gcd(MOD1 * MOD3, MOD2).second;
	static constexpr unsigned long long i3 =
		internal::inv_gcd(MOD1 * MOD2, MOD3).second;
	auto c1 = convolution<MOD1>(a, b);
	auto c2 = convolution<MOD2>(a, b);
	auto c3 = convolution<MOD3>(a, b);
	std::vector<long long> c(n + m - 1);
	for (int i = 0; i < n + m - 1; i++) {
		unsigned long long x = 0;
		x += (c1[i] * i1) % MOD1 * M2M3;
		x += (c2[i] * i2) % MOD2 * M1M3;
		x += (c3[i] * i3) % MOD3 * M1M2;
		// B = 2^63, -B <= x, r(real value) < B
		// (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
		// r = c1[i] (mod MOD1)
		// focus on MOD1
		// r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
		// r = x,
		//     x - M' + (0 or 2B),
		//     x - 2M' + (0, 2B or 4B),
		//     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
		// (r - x) = 0, (0)
		//           - M' + (0 or 2B), (1)
		//           -2M' + (0 or 2B or 4B), (2)
		//           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
		// we checked that
		//   ((1) mod MOD1) mod 5 = 2
		//   ((2) mod MOD1) mod 5 = 3
		//   ((3) mod MOD1) mod 5 = 4
		long long diff =
			c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
		if (diff < 0) diff += MOD1;
		static constexpr unsigned long long offset[5] = {
			0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
		x -= offset[diff % 5];
		c[i] = x;
	}
	return c;
}
}  // namespace atcoder
#endif  // ATCODER_CONVOLUTION_HPP
#ifndef ATCODER_DSU_HPP
#define ATCODER_DSU_HPP 1
#include <algorithm>
#include <cassert>
#include <vector>
namespace atcoder {
// Implement (union by size) + (path compression)
// Reference:
// Zvi Galil and Giuseppe F. Italiano,
// Data structures and algorithms for disjoint set union problems
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 atcoder
#endif  // ATCODER_DSU_HPP
#ifndef ATCODER_FENWICKTREE_HPP
#define ATCODER_FENWICKTREE_HPP 1
#include <cassert>
#include <vector>
namespace atcoder {
// Reference: https://en.wikipedia.org/wiki/Fenwick_tree
template <class T> struct fenwick_tree {
	using U = internal::to_unsigned_t<T>;
  public:
	fenwick_tree() : _n(0) {}
	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] += U(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;
	std::vector<U> data;
	U sum(int r) {
		U s = 0;
		while (r > 0) {
			s += data[r - 1];
			r -= r & -r;
		}
		return s;
	}
};
}  // namespace atcoder
#endif  // ATCODER_FENWICKTREE_HPP
#ifndef ATCODER_LAZYSEGTREE_HPP
#define ATCODER_LAZYSEGTREE_HPP 1
#include <algorithm>
#include <cassert>
#include <iostream>
#include <vector>
namespace atcoder {
template <class S,
		  S (*op)(S, S),
		  S (*e)(),
		  class F,
		  S (*mapping)(F, S),
		  F (*composition)(F, F),
		  F (*id)()>
struct lazy_segtree {
  public:
	lazy_segtree() : lazy_segtree(0) {}
	lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
	lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
		log = internal::ceil_pow2(_n);
		size = 1 << log;
		d = std::vector<S>(2 * size, e());
		lz = std::vector<F>(size, id());
		for (int i = 0; i < _n; i++) d[size + i] = v[i];
		for (int i = size - 1; i >= 1; i--) {
			update(i);
		}
	}
	void set(int p, S x) {
		assert(0 <= p && p < _n);
		p += size;
		for (int i = log; i >= 1; i--) push(p >> i);
		d[p] = x;
		for (int i = 1; i <= log; i++) update(p >> i);
	}
	S get(int p) {
		assert(0 <= p && p < _n);
		p += size;
		for (int i = log; i >= 1; i--) push(p >> i);
		return d[p];
	}
	S prod(int l, int r) {
		assert(0 <= l && l <= r && r <= _n);
		if (l == r) return e();
		l += size;
		r += size;
		for (int i = log; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push(r >> i);
		}
		S sml = e(), smr = e();
		while (l < r) {
			if (l & 1) sml = op(sml, d[l++]);
			if (r & 1) smr = op(d[--r], smr);
			l >>= 1;
			r >>= 1;
		}
		return op(sml, smr);
	}
	S all_prod() { return d[1]; }
	void apply(int p, F f) {
		assert(0 <= p && p < _n);
		p += size;
		for (int i = log; i >= 1; i--) push(p >> i);
		d[p] = mapping(f, d[p]);
		for (int i = 1; i <= log; i++) update(p >> i);
	}
	void apply(int l, int r, F f) {
		assert(0 <= l && l <= r && r <= _n);
		if (l == r) return;
		l += size;
		r += size;
		for (int i = log; i >= 1; i--) {
			if (((l >> i) << i) != l) push(l >> i);
			if (((r >> i) << i) != r) push((r - 1) >> i);
		}
		{
			int l2 = l, r2 = r;
			while (l < r) {
				if (l & 1) all_apply(l++, f);
				if (r & 1) all_apply(--r, f);
				l >>= 1;
				r >>= 1;
			}
			l = l2;
			r = r2;
		}
		for (int i = 1; i <= log; i++) {
			if (((l >> i) << i) != l) update(l >> i);
			if (((r >> i) << i) != r) update((r - 1) >> i);
		}
	}
	template <bool (*g)(S)> int max_right(int l) {
		return max_right(l, [](S x) { return g(x); });
	}
	template <class G> int max_right(int l, G g) {
		assert(0 <= l && l <= _n);
		assert(g(e()));
		if (l == _n) return _n;
		l += size;
		for (int i = log; i >= 1; i--) push(l >> i);
		S sm = e();
		do {
			while (l % 2 == 0) l >>= 1;
			if (!g(op(sm, d[l]))) {
				while (l < size) {
					push(l);
					l = (2 * l);
					if (g(op(sm, d[l]))) {
						sm = op(sm, d[l]);
						l++;
					}
				}
				return l - size;
			}
			sm = op(sm, d[l]);
			l++;
		} while ((l & -l) != l);
		return _n;
	}
	template <bool (*g)(S)> int min_left(int r) {
		return min_left(r, [](S x) { return g(x); });
	}
	template <class G> int min_left(int r, G g) {
		assert(0 <= r && r <= _n);
		assert(g(e()));
		if (r == 0) return 0;
		r += size;
		for (int i = log; i >= 1; i--) push((r - 1) >> i);
		S sm = e();
		do {
			r--;
			while (r > 1 && (r % 2)) r >>= 1;
			if (!g(op(d[r], sm))) {
				while (r < size) {
					push(r);
					r = (2 * r + 1);
					if (g(op(d[r], sm))) {
						sm = op(d[r], sm);
						r--;
					}
				}
				return r + 1 - size;
			}
			sm = op(d[r], sm);
		} while ((r & -r) != r);
		return 0;
	}
  private:
	int _n, size, log;
	std::vector<S> d;
	std::vector<F> lz;
	void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
	void all_apply(int k, F f) {
		d[k] = mapping(f, d[k]);
		if (k < size) lz[k] = composition(f, lz[k]);
	}
	void push(int k) {
		all_apply(2 * k, lz[k]);
		all_apply(2 * k + 1, lz[k]);
		lz[k] = id();
	}
};
}  // namespace atcoder
#endif  // ATCODER_LAZYSEGTREE_HPP
#ifndef ATCODER_MATH_HPP
#define ATCODER_MATH_HPP 1
#include <algorithm>
#include <cassert>
#include <tuple>
#include <vector>
namespace atcoder {
long long pow_mod(long long x, long long n, int m) {
	assert(0 <= n && 1 <= m);
	if (m == 1) return 0;
	internal::barrett bt((unsigned int)(m));
	unsigned int r = 1, y = (unsigned int)(internal::safe_mod(x, m));
	while (n) {
		if (n & 1) r = bt.mul(r, y);
		y = bt.mul(y, y);
		n >>= 1;
	}
	return r;
}
long long inv_mod(long long x, long long m) {
	assert(1 <= m);
	auto z = internal::inv_gcd(x, m);
	assert(z.first == 1);
	return z.second;
}
// (rem, mod)
std::pair<long long, long long> crt(const std::vector<long long>& r,
									const std::vector<long long>& m) {
	assert(r.size() == m.size());
	int n = int(r.size());
	// Contracts: 0 <= r0 < m0
	long long r0 = 0, m0 = 1;
	for (int i = 0; i < n; i++) {
		assert(1 <= m[i]);
		long long r1 = internal::safe_mod(r[i], m[i]), m1 = m[i];
		if (m0 < m1) {
			std::swap(r0, r1);
			std::swap(m0, m1);
		}
		if (m0 % m1 == 0) {
			if (r0 % m1 != r1) return {0, 0};
			continue;
		}
		// assume: m0 > m1, lcm(m0, m1) >= 2 * max(m0, m1)
		// (r0, m0), (r1, m1) -> (r2, m2 = lcm(m0, m1));
		// r2 % m0 = r0
		// r2 % m1 = r1
		// -> (r0 + x*m0) % m1 = r1
		// -> x*u0*g % (u1*g) = (r1 - r0) (u0*g = m0, u1*g = m1)
		// -> x = (r1 - r0) / g * inv(u0) (mod u1)
		// im = inv(u0) (mod u1) (0 <= im < u1)
		long long g, im;
		std::tie(g, im) = internal::inv_gcd(m0, m1);
		long long u1 = (m1 / g);
		// |r1 - r0| < (m0 + m1) <= lcm(m0, m1)
		if ((r1 - r0) % g) return {0, 0};
		// u1 * u1 <= m1 * m1 / g / g <= m0 * m1 / g = lcm(m0, m1)
		long long x = (r1 - r0) / g % u1 * im % u1;
		// |r0| + |m0 * x|
		// < m0 + m0 * (u1 - 1)
		// = m0 + m0 * m1 / g - m0
		// = lcm(m0, m1)
		r0 += x * m0;
		m0 *= u1;  // -> lcm(m0, m1)
		if (r0 < 0) r0 += m0;
	}
	return {r0, m0};
}
long long floor_sum(long long n, long long m, long long a, long long b) {
	long long ans = 0;  
	if (a >= m) {
		ans += (n - 1) * n * (a / m) / 2;
		a %= m;
	}
	if (b >= m) {
		ans += n * (b / m);
		b %= m;
	}
	long long y_max = (a * n + b) / m, x_max = (y_max * m - b);
	if (y_max == 0) return ans;
	ans += (n - (x_max + a - 1) / a) * y_max;
	ans += floor_sum(y_max, a, m, (a - x_max % a) % a);
	return ans;
}
}  // namespace atcoder
#endif  // ATCODER_MATH_HPP
#ifndef ATCODER_MAXFLOW_HPP
#define ATCODER_MAXFLOW_HPP 1
#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>
namespace atcoder {
template <class Cap> struct mf_graph {
  public:
	mf_graph() : _n(0) {}
	mf_graph(int n) : _n(n), g(n) {}
	int add_edge(int from, int to, Cap cap) {
		assert(0 <= from && from < _n);
		assert(0 <= to && to < _n);
		assert(0 <= cap);
		int m = int(pos.size());
		pos.push_back({from, int(g[from].size())});
		g[from].push_back(_edge{to, int(g[to].size()), cap});
		g[to].push_back(_edge{from, int(g[from].size()) - 1, 0});
		return m;
	}
	struct edge {
		int from, to;
		Cap cap, flow;
	};
	edge get_edge(int i) {
		int m = int(pos.size());
		assert(0 <= i && i < m);
		auto _e = g[pos[i].first][pos[i].second];
		auto _re = g[_e.to][_e.rev];
		return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
	}
	std::vector<edge> edges() {
		int m = int(pos.size());
		std::vector<edge> result;
		for (int i = 0; i < m; i++) {
			result.push_back(get_edge(i));
		}
		return result;
	}
	void change_edge(int i, Cap new_cap, Cap new_flow) {
		int m = int(pos.size());
		assert(0 <= i && i < m);
		assert(0 <= new_flow && new_flow <= new_cap);
		auto& _e = g[pos[i].first][pos[i].second];
		auto& _re = g[_e.to][_e.rev];
		_e.cap = new_cap - new_flow;
		_re.cap = new_flow;
	}
	Cap flow(int s, int t) {
		return flow(s, t, std::numeric_limits<Cap>::max());
	}
	Cap flow(int s, int t, Cap flow_limit) {
		assert(0 <= s && s < _n);
		assert(0 <= t && t < _n);
		std::vector<int> level(_n), iter(_n);
		internal::simple_queue<int> que;
		auto bfs = [&]() {
			std::fill(level.begin(), level.end(), -1);
			level[s] = 0;
			que.clear();
			que.push(s);
			while (!que.empty()) {
				int v = que.front();
				que.pop();
				for (auto e : g[v]) {
					if (e.cap == 0 || level[e.to] >= 0) continue;
					level[e.to] = level[v] + 1;
					if (e.to == t) return;
					que.push(e.to);
				}
			}
		};
		auto dfs = [&](auto self, int v, Cap up) {
			if (v == s) return up;
			Cap res = 0;
			int level_v = level[v];
			for (int& i = iter[v]; i < int(g[v].size()); i++) {
				_edge& e = g[v][i];
				if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
				Cap d =
					self(self, e.to, std::min(up - res, g[e.to][e.rev].cap));
				if (d <= 0) continue;
				g[v][i].cap += d;
				g[e.to][e.rev].cap -= d;
				res += d;
				if (res == up) break;
			}
			return res;
		};
		Cap flow = 0;
		while (flow < flow_limit) {
			bfs();
			if (level[t] == -1) break;
			std::fill(iter.begin(), iter.end(), 0);
			while (flow < flow_limit) {
				Cap f = dfs(dfs, t, flow_limit - flow);
				if (!f) break;
				flow += f;
			}
		}
		return flow;
	}
	std::vector<bool> min_cut(int s) {
		std::vector<bool> visited(_n);
		internal::simple_queue<int> que;
		que.push(s);
		while (!que.empty()) {
			int p = que.front();
			que.pop();
			visited[p] = true;
			for (auto e : g[p]) {
				if (e.cap && !visited[e.to]) {
					visited[e.to] = true;
					que.push(e.to);
				}
			}
		}
		return visited;
	}
  private:
	int _n;
	struct _edge {
		int to, rev;
		Cap cap;
	};
	std::vector<std::pair<int, int>> pos;
	std::vector<std::vector<_edge>> g;
};
}  // namespace atcoder
#endif  // ATCODER_MAXFLOW_HPP
#ifndef ATCODER_MINCOSTFLOW_HPP
#define ATCODER_MINCOSTFLOW_HPP 1
#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <vector>
namespace atcoder {
template <class Cap, class Cost> struct mcf_graph {
  public:
	mcf_graph() {}
	mcf_graph(int n) : _n(n), g(n) {}
	int add_edge(int from, int to, Cap cap, Cost cost) {
		assert(0 <= from && from < _n);
		assert(0 <= to && to < _n);
		int m = int(pos.size());
		pos.push_back({from, int(g[from].size())});
		g[from].push_back(_edge{to, int(g[to].size()), cap, cost});
		g[to].push_back(_edge{from, int(g[from].size()) - 1, 0, -cost});
		return m;
	}
	struct edge {
		int from, to;
		Cap cap, flow;
		Cost cost;
	};
	edge get_edge(int i) {
		int m = int(pos.size());
		assert(0 <= i && i < m);
		auto _e = g[pos[i].first][pos[i].second];
		auto _re = g[_e.to][_e.rev];
		return edge{
			pos[i].first, _e.to, _e.cap + _re.cap, _re.cap, _e.cost,
		};
	}
	std::vector<edge> edges() {
		int m = int(pos.size());
		std::vector<edge> result(m);
		for (int i = 0; i < m; i++) {
			result[i] = get_edge(i);
		}
		return result;
	}
	std::pair<Cap, Cost> flow(int s, int t) {
		return flow(s, t, std::numeric_limits<Cap>::max());
	}
	std::pair<Cap, Cost> flow(int s, int t, Cap flow_limit) {
		return slope(s, t, flow_limit).back();
	}
	std::vector<std::pair<Cap, Cost>> slope(int s, int t) {
		return slope(s, t, std::numeric_limits<Cap>::max());
	}
	std::vector<std::pair<Cap, Cost>> slope(int s, int t, Cap flow_limit) {
		assert(0 <= s && s < _n);
		assert(0 <= t && t < _n);
		assert(s != t);
		// variants (C = maxcost):
		// -(n-1)C <= dual[s] <= dual[i] <= dual[t] = 0
		// reduced cost (= e.cost + dual[e.from] - dual[e.to]) >= 0 for all edge
		std::vector<Cost> dual(_n, 0), dist(_n);
		std::vector<int> pv(_n), pe(_n);
		std::vector<bool> vis(_n);
		auto dual_ref = [&]() {
			std::fill(dist.begin(), dist.end(),
					  std::numeric_limits<Cost>::max());
			std::fill(pv.begin(), pv.end(), -1);
			std::fill(pe.begin(), pe.end(), -1);
			std::fill(vis.begin(), vis.end(), false);
			struct Q {
				Cost key;
				int to;
				bool operator<(Q r) const { return key > r.key; }
			};
			std::priority_queue<Q> que;
			dist[s] = 0;
			que.push(Q{0, s});
			while (!que.empty()) {
				int v = que.top().to;
				que.pop();
				if (vis[v]) continue;
				vis[v] = true;
				if (v == t) break;
				// dist[v] = shortest(s, v) + dual[s] - dual[v]
				// dist[v] >= 0 (all reduced cost are positive)
				// dist[v] <= (n-1)C
				for (int i = 0; i < int(g[v].size()); i++) {
					auto e = g[v][i];
					if (vis[e.to] || !e.cap) continue;
					// |-dual[e.to] + dual[v]| <= (n-1)C
					// cost <= C - -(n-1)C + 0 = nC
					Cost cost = e.cost - dual[e.to] + dual[v];
					if (dist[e.to] - dist[v] > cost) {
						dist[e.to] = dist[v] + cost;
						pv[e.to] = v;
						pe[e.to] = i;
						que.push(Q{dist[e.to], e.to});
					}
				}
			}
			if (!vis[t]) {
				return false;
			}
			for (int v = 0; v < _n; v++) {
				if (!vis[v]) continue;
				// dual[v] = dual[v] - dist[t] + dist[v]
				//         = dual[v] - (shortest(s, t) + dual[s] - dual[t]) + (shortest(s, v) + dual[s] - dual[v])
				//         = - shortest(s, t) + dual[t] + shortest(s, v)
				//         = shortest(s, v) - shortest(s, t) >= 0 - (n-1)C
				dual[v] -= dist[t] - dist[v];
			}
			return true;
		};
		Cap flow = 0;
		Cost cost = 0, prev_cost = -1;
		std::vector<std::pair<Cap, Cost>> result;
		result.push_back({flow, cost});
		while (flow < flow_limit) {
			if (!dual_ref()) break;
			Cap c = flow_limit - flow;
			for (int v = t; v != s; v = pv[v]) {
				c = std::min(c, g[pv[v]][pe[v]].cap);
			}
			for (int v = t; v != s; v = pv[v]) {
				auto& e = g[pv[v]][pe[v]];
				e.cap -= c;
				g[v][e.rev].cap += c;
			}
			Cost d = -dual[s];
			flow += c;
			cost += c * d;
			if (prev_cost == d) {
				result.pop_back();
			}
			result.push_back({flow, cost});
			prev_cost = cost;
		}
		return result;
	}
  private:
	int _n;
	struct _edge {
		int to, rev;
		Cap cap;
		Cost cost;
	};
	std::vector<std::pair<int, int>> pos;
	std::vector<std::vector<_edge>> g;
};
}  // namespace atcoder
#endif  // ATCODER_MINCOSTFLOW_HPP
#ifndef ATCODER_SCC_HPP
#define ATCODER_SCC_HPP 1
#include <algorithm>
#include <cassert>
#include <vector>
namespace atcoder {
struct scc_graph {
  public:
	scc_graph() : internal(0) {}
	scc_graph(int n) : internal(n) {}
	void add_edge(int from, int to) {
		int n = internal.num_vertices();
		assert(0 <= from && from < n);
		assert(0 <= to && to < n);
		internal.add_edge(from, to);
	}
	std::vector<std::vector<int>> scc() { return internal.scc(); }
  private:
	internal::scc_graph internal;
};
}  // namespace atcoder
#endif  // ATCODER_SCC_HPP
#ifndef ATCODER_SEGTREE_HPP
#define ATCODER_SEGTREE_HPP 1
#include <algorithm>
#include <cassert>
#include <vector>
namespace atcoder {
template <class S, S (*op)(S, S), S (*e)()> struct segtree {
  public:
	segtree() : segtree(0) {}
	segtree(int n) : segtree(std::vector<S>(n, e())) {}
	segtree(const std::vector<S>& v) : _n(int(v.size())) {
		log = internal::ceil_pow2(_n);
		size = 1 << log;
		d = std::vector<S>(2 * size, e());
		for (int i = 0; i < _n; i++) d[size + i] = v[i];
		for (int i = size - 1; i >= 1; i--) {
			update(i);
		}
	}
	void set(int p, S x) {
		assert(0 <= p && p < _n);
		p += size;
		d[p] = x;
		for (int i = 1; i <= log; i++) update(p >> i);
	}
	S get(int p) {
		assert(0 <= p && p < _n);
		return d[p + size];
	}
	S prod(int l, int r) {
		assert(0 <= l && l <= r && r <= _n);
		S sml = e(), smr = e();
		l += size;
		r += size;
		while (l < r) {
			if (l & 1) sml = op(sml, d[l++]);
			if (r & 1) smr = op(d[--r], smr);
			l >>= 1;
			r >>= 1;
		}
		return op(sml, smr);
	}
	S all_prod() { return d[1]; }
	template <bool (*f)(S)> int max_right(int l) {
		return max_right(l, [](S x) { return f(x); });
	}
	template <class F> int max_right(int l, F f) {
		assert(0 <= l && l <= _n);
		assert(f(e()));
		if (l == _n) return _n;
		l += size;
		S sm = e();
		do {
			while (l % 2 == 0) l >>= 1;
			if (!f(op(sm, d[l]))) {
				while (l < size) {
					l = (2 * l);
					if (f(op(sm, d[l]))) {
						sm = op(sm, d[l]);
						l++;
					}
				}
				return l - size;
			}
			sm = op(sm, d[l]);
			l++;
		} while ((l & -l) != l);
		return _n;
	}
	template <bool (*f)(S)> int min_left(int r) {
		return min_left(r, [](S x) { return f(x); });
	}
	template <class F> int min_left(int r, F f) {
		assert(0 <= r && r <= _n);
		assert(f(e()));
		if (r == 0) return 0;
		r += size;
		S sm = e();
		do {
			r--;
			while (r > 1 && (r % 2)) r >>= 1;
			if (!f(op(d[r], sm))) {
				while (r < size) {
					r = (2 * r + 1);
					if (f(op(d[r], sm))) {
						sm = op(d[r], sm);
						r--;
					}
				}
				return r + 1 - size;
			}
			sm = op(d[r], sm);
		} while ((r & -r) != r);
		return 0;
	}
  private:
	int _n, size, log;
	std::vector<S> d;
	void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
};
}  // namespace atcoder
#endif  // ATCODER_SEGTREE_HPP
#ifndef ATCODER_STRING_HPP
#define ATCODER_STRING_HPP 1
#include <algorithm>
#include <cassert>
#include <numeric>
#include <string>
#include <vector>
namespace atcoder {
namespace internal {
std::vector<int> sa_naive(const std::vector<int>& s) {
	int n = int(s.size());
	std::vector<int> sa(n);
	std::iota(sa.begin(), sa.end(), 0);
	std::sort(sa.begin(), sa.end(), [&](int l, int r) {
		if (l == r) return false;
		while (l < n && r < n) {
			if (s[l] != s[r]) return s[l] < s[r];
			l++;
			r++;
		}
		return l == n;
	});
	return sa;
}
std::vector<int> sa_doubling(const std::vector<int>& s) {
	int n = int(s.size());
	std::vector<int> sa(n), rnk = s, tmp(n);
	std::iota(sa.begin(), sa.end(), 0);
	for (int k = 1; k < n; k *= 2) {
		auto cmp = [&](int x, int y) {
			if (rnk[x] != rnk[y]) return rnk[x] < rnk[y];
			int rx = x + k < n ? rnk[x + k] : -1;
			int ry = y + k < n ? rnk[y + k] : -1;
			return rx < ry;
		};
		std::sort(sa.begin(), sa.end(), cmp);
		tmp[sa[0]] = 0;
		for (int i = 1; i < n; i++) {
			tmp[sa[i]] = tmp[sa[i - 1]] + (cmp(sa[i - 1], sa[i]) ? 1 : 0);
		}
		std::swap(tmp, rnk);
	}
	return sa;
}
// SA-IS, linear-time suffix array construction
// Reference:
// G. Nong, S. Zhang, and W. H. Chan,
// Two Efficient Algorithms for Linear Time Suffix Array Construction
template <int THRESHOLD_NAIVE = 10, int THRESHOLD_DOUBLING = 40>
std::vector<int> sa_is(const std::vector<int>& s, int upper) {
	int n = int(s.size());
	if (n == 0) return {};
	if (n == 1) return {0};
	if (n == 2) {
		if (s[0] < s[1]) {
			return {0, 1};
		} else {
			return {1, 0};
		}
	}
	if (n < THRESHOLD_NAIVE) {
		return sa_naive(s);
	}
	if (n < THRESHOLD_DOUBLING) {
		return sa_doubling(s);
	}
	std::vector<int> sa(n);
	std::vector<bool> ls(n);
	for (int i = n - 2; i >= 0; i--) {
		ls[i] = (s[i] == s[i + 1]) ? ls[i + 1] : (s[i] < s[i + 1]);
	}
	std::vector<int> sum_l(upper + 1), sum_s(upper + 1);
	for (int i = 0; i < n; i++) {
		if (!ls[i]) {
			sum_s[s[i]]++;
		} else {
			sum_l[s[i] + 1]++;
		}
	}
	for (int i = 0; i <= upper; i++) {
		sum_s[i] += sum_l[i];
		if (i < upper) sum_l[i + 1] += sum_s[i];
	}
	auto induce = [&](const std::vector<int>& lms) {
		std::fill(sa.begin(), sa.end(), -1);
		std::vector<int> buf(upper + 1);
		std::copy(sum_s.begin(), sum_s.end(), buf.begin());
		for (auto d : lms) {
			if (d == n) continue;
			sa[buf[s[d]]++] = d;
		}
		std::copy(sum_l.begin(), sum_l.end(), buf.begin());
		sa[buf[s[n - 1]]++] = n - 1;
		for (int i = 0; i < n; i++) {
			int v = sa[i];
			if (v >= 1 && !ls[v - 1]) {
				sa[buf[s[v - 1]]++] = v - 1;
			}
		}
		std::copy(sum_l.begin(), sum_l.end(), buf.begin());
		for (int i = n - 1; i >= 0; i--) {
			int v = sa[i];
			if (v >= 1 && ls[v - 1]) {
				sa[--buf[s[v - 1] + 1]] = v - 1;
			}
		}
	};
	std::vector<int> lms_map(n + 1, -1);
	int m = 0;
	for (int i = 1; i < n; i++) {
		if (!ls[i - 1] && ls[i]) {
			lms_map[i] = m++;
		}
	}
	std::vector<int> lms;
	lms.reserve(m);
	for (int i = 1; i < n; i++) {
		if (!ls[i - 1] && ls[i]) {
			lms.push_back(i);
		}
	}
	induce(lms);
	if (m) {
		std::vector<int> sorted_lms;
		sorted_lms.reserve(m);
		for (int v : sa) {
			if (lms_map[v] != -1) sorted_lms.push_back(v);
		}
		std::vector<int> rec_s(m);
		int rec_upper = 0;
		rec_s[lms_map[sorted_lms[0]]] = 0;
		for (int i = 1; i < m; i++) {
			int l = sorted_lms[i - 1], r = sorted_lms[i];
			int end_l = (lms_map[l] + 1 < m) ? lms[lms_map[l] + 1] : n;
			int end_r = (lms_map[r] + 1 < m) ? lms[lms_map[r] + 1] : n;
			bool same = true;
			if (end_l - l != end_r - r) {
				same = false;
			} else {
				while (l < end_l) {
					if (s[l] != s[r]) {
						break;
					}
					l++;
					r++;
				}
				if (l == n || s[l] != s[r]) same = false;
			}
			if (!same) rec_upper++;
			rec_s[lms_map[sorted_lms[i]]] = rec_upper;
		}
		auto rec_sa =
			sa_is<THRESHOLD_NAIVE, THRESHOLD_DOUBLING>(rec_s, rec_upper);
		for (int i = 0; i < m; i++) {
			sorted_lms[i] = lms[rec_sa[i]];
		}
		induce(sorted_lms);
	}
	return sa;
}
}  // namespace internal
std::vector<int> suffix_array(const std::vector<int>& s, int upper) {
	assert(0 <= upper);
	for (int d : s) {
		assert(0 <= d && d <= upper);
	}
	auto sa = internal::sa_is(s, upper);
	return sa;
}
template <class T> std::vector<int> suffix_array(const std::vector<T>& s) {
	int n = int(s.size());
	std::vector<int> idx(n);
	iota(idx.begin(), idx.end(), 0);
	sort(idx.begin(), idx.end(), [&](int l, int r) { return s[l] < s[r]; });
	std::vector<int> s2(n);
	int now = 0;
	for (int i = 0; i < n; i++) {
		if (i && s[idx[i - 1]] != s[idx[i]]) now++;
		s2[idx[i]] = now;
	}
	return internal::sa_is(s2, now);
}
std::vector<int> suffix_array(const std::string& s) {
	int n = int(s.size());
	std::vector<int> s2(n);
	for (int i = 0; i < n; i++) {
		s2[i] = s[i];
	}
	return internal::sa_is(s2, 255);
}
// Reference:
// T. Kasai, G. Lee, H. Arimura, S. Arikawa, and K. Park,
// Linear-Time Longest-Common-Prefix Computation in Suffix Arrays and Its
// Applications
template <class T>
std::vector<int> lcp_array(const std::vector<T>& s,
						   const std::vector<int>& sa) {
	int n = int(s.size());
	assert(n >= 1);
	std::vector<int> rnk(n);
	for (int i = 0; i < n; i++) {
		rnk[sa[i]] = i;
	}
	std::vector<int> lcp(n - 1);
	int h = 0;
	for (int i = 0; i < n; i++) {
		if (h > 0) h--;
		if (rnk[i] == 0) continue;
		int j = sa[rnk[i] - 1];
		for (; j + h < n && i + h < n; h++) {
			if (s[j + h] != s[i + h]) break;
		}
		lcp[rnk[i] - 1] = h;
	}
	return lcp;
}
std::vector<int> lcp_array(const std::string& s, const std::vector<int>& sa) {
	int n = int(s.size());
	std::vector<int> s2(n);
	for (int i = 0; i < n; i++) {
		s2[i] = s[i];
	}
	return lcp_array(s2, sa);
}
// Reference:
// D. Gusfield,
// Algorithms on Strings, Trees, and Sequences: Computer Science and
// Computational Biology
template <class T> std::vector<int> z_algorithm(const std::vector<T>& s) {
	int n = int(s.size());
	if (n == 0) return {};
	std::vector<int> z(n);
	z[0] = 0;
	for (int i = 1, j = 0; i < n; i++) {
		int& k = z[i];
		k = (j + z[j] <= i) ? 0 : std::min(j + z[j] - i, z[i - j]);
		while (i + k < n && s[k] == s[i + k]) k++;
		if (j + z[j] < i + z[i]) j = i;
	}
	z[0] = n;
	return z;
}
std::vector<int> z_algorithm(const std::string& s) {
	int n = int(s.size());
	std::vector<int> s2(n);
	for (int i = 0; i < n; i++) {
		s2[i] = s[i];
	}
	return z_algorithm(s2);
}
}  // namespace atcoder
#endif  // ATCODER_STRING_HPP
#ifndef ATCODER_TWOSAT_HPP
#define ATCODER_TWOSAT_HPP 1
#include <cassert>
#include <vector>
namespace atcoder {
// Reference:
// B. Aspvall, M. Plass, and R. Tarjan,
// A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean
// Formulas
struct two_sat {
  public:
	two_sat() : _n(0), scc(0) {}
	two_sat(int n) : _n(n), _answer(n), scc(2 * n) {}
	void add_clause(int i, bool f, int j, bool g) {
		assert(0 <= i && i < _n);
		assert(0 <= j && j < _n);
		scc.add_edge(2 * i + (f ? 0 : 1), 2 * j + (g ? 1 : 0));
		scc.add_edge(2 * j + (g ? 0 : 1), 2 * i + (f ? 1 : 0));
	}
	bool satisfiable() {
		auto id = scc.scc_ids().second;
		for (int i = 0; i < _n; i++) {
			if (id[2 * i] == id[2 * i + 1]) return false;
			_answer[i] = id[2 * i] < id[2 * i + 1];
		}
		return true;
	}
	std::vector<bool> answer() { return _answer; }
  private:
	int _n;
	std::vector<bool> _answer;
	internal::scc_graph scc;
};
}  // namespace atcoder
#endif  // ATCODER_TWOSAT_HPP

using namespace atcoder;

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef pair<ll,ll> P;
typedef pair<int,int> Pi;
#define rep(i,n) for(ll i=0;i<n;i++)
#define FOR(i,a,b) for(ll i=a;i<b;i++)
#define fi first
#define se second
#define endl "\n"

template<typename T> inline bool chmax(T &a, T b){if(a<b){a=b;return true;}return false;}
template<typename T> inline bool chmin(T &a, T b){if(a>b){a=b;return true;}return false;}
template<typename T> ostream& operator<<(ostream& s,const complex<T>& d) {return s<<"("<<d.real()<<", "<<d.imag()<< ")";}
template<typename T1, typename T2> ostream& operator<<(ostream& s,const pair<T1,T2>& d) {return s<<"("<<d.first<<", "<<d.second<<")";}
template<typename T> ostream& operator<<(ostream& s, const vector<T>& d){int len=d.size();rep(i,len){s<<d[i];if(i<len-1) s<<" ";}return s;}
template<typename T> ostream& operator<<(ostream& s,const vector<vector<T>>& d){int len=d.size();rep(i,len){s<<d[i]<<endl;}return s;}
template<typename T> ostream& operator<<(ostream& s,const set<T>& v){s<<"{ ";for(auto itr=v.begin();itr!=v.end();++itr) {if (itr!=v.begin()) {s<< ", ";}s<<(*itr);}s<<" }";return s;}
template<typename T> ostream& operator<<(ostream& s,const multiset<T>& v){s<<"{ ";for(auto itr=v.begin();itr!=v.end();++itr) {if (itr!=v.begin()) {s<< ", ";}s<<(*itr);}s<<" }";return s;}
template<typename T1, typename T2> ostream& operator<<(ostream& s,const map<T1,T2>& m){s<<"{"<<endl;for(auto itr=m.begin();itr!=m.end();++itr){s<<" "<<(*itr).first<<" : "<<(*itr).second<<endl;}s<<"}"<<endl;return s;}

const ll mod=998'244'353;
const ll inf=1'000'000'000'000'000'00;
const int INF=1'000'000'000;
const double EPS=1e-10;
const double PI=acos(-1);

using mint=modint998244353;

struct Mint{
	ll x;
	Mint(ll x=0):x((x%mod+mod)%mod){}
	Mint& operator+=(const Mint a){
		if((x+=a.x)>=mod) x-=mod;
		return *this;
	}
	Mint& operator-=(const Mint a){
		if((x+=mod-a.x)>=mod)x-=mod;
		return *this;
	}
	Mint& operator*=(const Mint a){(x*=a.x)%=mod;return *this;}
	Mint operator+(const Mint a) const{return Mint(*this)+=a;}
	Mint operator-(const Mint a) const{return Mint(*this)-=a;}
	Mint operator*(const Mint a) const{return Mint(*this)*=a;}
	Mint pow(ll t) const{
		if (!t) return 1;
		Mint a=pow(t>>1);
		a*=a;
		if(t&1) a*=*this;
		return a;
	}
	// for prime mod
	Mint inv() const{return pow(mod-2);}
	Mint& operator/=(const Mint a){return (*this)*=a.inv();}
	Mint operator/(const Mint a) const{return Mint(*this)/=a;}
};
istream& operator>>(istream& is, Mint& a) {return is>>a.x;}
ostream& operator<<(ostream& os, const Mint& a) {return os<<a.x;}

struct combination{
	vector<Mint> fact,ifact;
	combination(int n):fact(n+1),ifact(n+1){
		assert(n<mod);
		fact[0]=1;
		for(int i=1;i<=n;i++) fact[i]=fact[i-1]*i;
		ifact[n]=fact[n].inv();
		for(int i=n;i>=1;i--) ifact[i-1]=ifact[i]*i;
	}
	Mint operator()(int n,int k){
		if (k<0 || k>n) return 0;
		return fact[n]*ifact[k]*ifact[n-k];
	}
};

struct S{
    mint x;
	ll size;
};
using F=ll;
const F ID=inf;


vector<mint> poww(200'100),calc(200'100);

S op(S a,S b){return {a.x*mint(10).pow(b.size)+b.x,a.size+b.size};}
S e(){return {mint(0),0};}
S mapping(F f,S x){
    if(f!=ID){
		x.x=calc[x.size]*f;
	}
    return x;
}
F composition(F f,F g){
    if(f==ID) f=g;
    return f;
}
F id(){return ID;}

int main(){
	cin.tie(0);ios::sync_with_stdio(false);
	ll n,q;
	cin>>n>>q;
	poww[0]=1,calc[0]=1;
	FOR(i,1,n+10){
		poww[i]=poww[i-1]*10;
		calc[i]=(poww[i]-1)/9;
	}
	lazy_segtree<S, op, e, F, mapping, composition, id> seg(vector<S>(n+10,{mint(1),1}));
	while(q--){
		ll l,r,d;
		cin>>l>>r>>d;
		l--;
		seg.apply(l,r,d);
		cout<<seg.prod(0,n).x.val()<<endl;
	}
}

Measured memory_bytes: 18280