#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx")
#pragma GCC optimize("unroll-loops")
//#pragma warning(disable : 4996)

#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;
			int v[] = { 2, 7, 61 };
			for (long long a : v) {
				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

#ifdef _MSC_VER
#include <intrin.h>

#define __builtin_popcount __popcnt
#define __builtin_popcountll __popcnt64
#endif

#include <limits.h>
#include <math.h>
#include <time.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstring>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <map>
#include <numeric>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <vector>

//#include<atcoder/all>

#include <vector>




using namespace std;
//using namespace atcoder;


#define REP(i, n) for (int i = 0; i < (n); ++i)
#define REPR(i, n) for (int i = n - 1; i >= 0; --i)
#define FOR(i, m, n) for (int i = m; i < n; ++i)
#define FORR(i, m, n) for (int i = m - 1; i >= n; --i)
#define SORT(v, n) sort(v, v + n);
#define VSORT(v) sort(v.begin(), v.end());
#define REVERSE(v, n) reverse(v, v + n);
#define VREVERSE(v) reverse(v.begin(), v.end())
using ll = long long;
#define print(x) cout << (x) << endl
#define pe(x) cout << (x) << " "
#define DEBUG(x) cout << #x << ": " << x << endl
#define lb(v, n) lower_bound(v.begin(), v.end(), (n))
#define ub(v, n) upper_bound(v.begin(), v.end(), (n))
//#define int long long
//#define double long double
#define all(x) (x).begin(), (x).end()
#define print_space(v) REP(i, v.size()) cout << v[i] << " \n"[i==(int)v.size()-1]
template <typename T1, typename T2> inline void chmin(T1& a, T2 b) { if (a > b) a = b; }
template <typename T1, typename T2> inline void chmax(T1& a, T2 b) { if (a < b) a = b; }
typedef pair<int, int> pii;
typedef pair<long long, long long> pll;
std::random_device rd;
std::mt19937 mt(rd());
constexpr ll MOD = 1e9+7;
constexpr int MAX = 300010;
const double pi = acos(-1);
constexpr double EPS = 1e-8;
constexpr ll LINF = 1e18 + 1;
constexpr int INF = 1e9 + 1;
void Yes(bool cond) { cout << (cond ? "Yes" : "No") << '\n'; }
void YES(bool cond) { cout << (cond ? "YES" : "NO") << '\n'; }


#include<algorithm>
using namespace std;


#define N 200020

class SegmentTree {
	const ll inf = 1e18;
	int n, n0;
	ll max_v[4 * N], smax_v[4 * N];
	ll sum[4 * N], max_c[4 * N];

	void update_node_max(int k, ll x) {
		sum[k] += (x - max_v[k]) * max_c[k];
		max_v[k] = x;
	}

	void push(int k) {
		if (max_v[k] < max_v[2 * k + 1]) {
			update_node_max(2 * k + 1, max_v[k]);
		}
		if (max_v[k] < max_v[2 * k + 2]) {
			update_node_max(2 * k + 2, max_v[k]);
		}
	}

	void update(int k) {
		sum[k] = sum[2 * k + 1] + sum[2 * k + 2];

		if (max_v[2 * k + 1] < max_v[2 * k + 2]) {
			max_v[k] = max_v[2 * k + 2];
			max_c[k] = max_c[2 * k + 2];
			smax_v[k] = max(max_v[2 * k + 1], smax_v[2 * k + 2]);
		}
		else if (max_v[2 * k + 1] > max_v[2 * k + 2]) {
			max_v[k] = max_v[2 * k + 1];
			max_c[k] = max_c[2 * k + 1];
			smax_v[k] = max(smax_v[2 * k + 1], max_v[2 * k + 2]);
		}
		else {
			max_v[k] = max_v[2 * k + 1];
			max_c[k] = max_c[2 * k + 1] + max_c[2 * k + 2];
			smax_v[k] = max(smax_v[2 * k + 1], smax_v[2 * k + 2]);
		}
	}

	void _update_min(ll x, int a, int b, int k, int l, int r) {
		if (b <= l || r <= a || max_v[k] <= x) {
			return;
		}
		if (a <= l && r <= b && smax_v[k] < x) {
			update_node_max(k, x);
			return;
		}

		push(k);
		_update_min(x, a, b, 2 * k + 1, l, (l + r) / 2);
		_update_min(x, a, b, 2 * k + 2, (l + r) / 2, r);
		update(k);
	}

	ll _query_max(int a, int b, int k, int l, int r) {
		if (b <= l || r <= a) {
			return 0;
		}
		if (a <= l && r <= b) {
			return max_v[k];
		}
		push(k);
		ll lv = _query_max(a, b, 2 * k + 1, l, (l + r) / 2);
		ll rv = _query_max(a, b, 2 * k + 2, (l + r) / 2, r);
		return max(lv, rv);
	}

	ll _query_sum(int a, int b, int k, int l, int r) {
		if (b <= l || r <= a) {
			return 0;
		}
		if (a <= l && r <= b) {
			return sum[k];
		}
		push(k);
		ll lv = _query_sum(a, b, 2 * k + 1, l, (l + r) / 2);
		ll rv = _query_sum(a, b, 2 * k + 2, (l + r) / 2, r);
		return lv + rv;
	}

public:
	SegmentTree(int n) {
		SegmentTree(n, nullptr);
	}

	SegmentTree(int n, ll *a) : n(n) {
		n0 = 1;
		while (n0 < n) n0 <<= 1;

		for (int i = 0; i<n; ++i) {
			max_v[n0 - 1 + i] = sum[n0 - 1 + i] = (a != nullptr ? a[i] : 0);
			smax_v[n0 - 1 + i] = -inf;
			max_c[n0 - 1 + i] = 1;
		}

		for (int i = n; i<n0; ++i) {
			max_v[n0 - 1 + i] = smax_v[n0 - 1 + i] = -inf;
			sum[n0 - 1 + i] = max_c[n0 - 1 + i] = 0;
		}
		for (int i = n0 - 2; i >= 0; i--) update(i);
	}

	// range minimize query
	void update_min(int a, int b, ll x) {
		return _update_min(x, a, b, 0, 0, n0);
	}

	// range maximum query
	ll query_max(int a, int b) {
		return _query_max(a, b, 0, 0, n0);
	}

	// range sum query
	ll query_sum(int a, int b) {
		return _query_sum(a, b, 0, 0, n0);
	}
};

ll a[200020], b[200020];
void solve() {
	ll n, Q; cin >> n >> Q;
	FOR(i, 0, n-1) {
		a[i] = n - 1;
		b[i] = n - 1;
	}
	//print("a");
	SegmentTree segX(n, a);
	SegmentTree segY(n, b);
	//print("b");
	ll sum = (n - 2)*(n - 2);
	ll res = 0;
	REP(_, Q) {
		int sign, x; cin >> sign>> x;
		x--;
		if (sign == 1) {
			int mn = segY.query_max(x, x + 1);
			res += mn - 1;
			segX.update_min(0, mn, x);
		}
		else {
			int mn = segX.query_max(x, x + 1);
			res += mn - 1;
			segY.update_min(0, mn, x);
		}
		
	}
	print(sum - res);
}



signed main() {
	cin.tie(0);
	ios::sync_with_stdio(false);

	//int q;
	//cin >> q;
	//while (q--)
	solve();

}


Measured memory_bytes: 39608