//ACL
#ifndef ATCODER_INTERNAL_BITOP_HPP
#define ATCODER_INTERNAL_BITOP_HPP 1
#ifdef _MSC_VER
#include <intrin.h>
#include<cassert>
#endif
namespace atcoder {
	namespace internal {
		int ceil_pow2(int n) {
			int x = 0;
			while ((1U << x) < (unsigned int)(n)) x++;
			return x;
		}
		int bsf(unsigned int n) {
#ifdef _MSC_VER
			unsigned long index;
			_BitScanForward(&index, n);
			return index;
#else
			return __builtin_ctz(n);
#endif
		}
	}
}
#endif
#ifndef ATCODER_INTERNAL_MATH_HPP
#define ATCODER_INTERNAL_MATH_HPP 1
#include <utility>
namespace atcoder {
	namespace internal {
		constexpr long long safe_mod(long long x, long long m) {
			x %= m;
			if (x < 0) x += m;
			return x;
		}
		struct barrett {
			unsigned int _m;
			unsigned long long im;
			barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
			unsigned int umod() const { return _m; }
			unsigned int mul(unsigned int a, unsigned int b) const {
				unsigned long long z = a;
				z *= b;
#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;
			}
		};
		constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
			if (m == 1) return 0;
			unsigned int _m = (unsigned int)(m);
			unsigned long long r = 1;
			unsigned long long y = safe_mod(x, m);
			while (n) {
				if (n & 1) r = (r * y) % _m;
				y = (y * y) % _m;
				n >>= 1;
			}
			return r;
		}
		constexpr bool is_prime_constexpr(int n) {
			if (n <= 1) return false;
			if (n == 2 || n == 7 || n == 61) return true;
			if (n % 2 == 0) return false;
			long long d = n - 1;
			while (d % 2 == 0) d /= 2;
			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);
		constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
			a = safe_mod(a, b);
			if (a == 0) return { b, 0 };
			long long s = b, t = a;
			long long m0 = 0, m1 = 1;
			while (t) {
				long long u = s / t;
				s -= t * u;
				m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b
				auto tmp = s;
				s = t;
				t = tmp;
				tmp = m0;
				m0 = m1;
				m1 = tmp;
			}
			if (m0 < 0) m0 += b / s;
			return { s, m0 };
		}
		constexpr int primitive_root_constexpr(int m) {
			if (m == 2) return 1;
			if (m == 167772161) return 3;
			if (m == 469762049) return 3;
			if (m == 754974721) return 11;
			if (m == 998244353) return 3;
			int divs[20] = {};
			divs[0] = 2;
			int cnt = 1;
			int x = (m - 1) / 2;
			while (x % 2 == 0) x /= 2;
			for (int i = 3; (long long)(i)*i <= x; i += 2) {
				if (x % i == 0) {
					divs[cnt++] = i;
					while (x % i == 0) {
						x /= i;
					}
				}
			}
			if (x > 1) {
				divs[cnt++] = x;
			}
			for (int g = 2;; g++) {
				bool ok = true;
				for (int i = 0; i < cnt; i++) {
					if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
						ok = false;
						break;
					}
				}
				if (ok) return g;
			}
		}
		template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
	}
}
#endif
#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++; }
		};
	}
}
#endif
#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;
				}
			}
		};
		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} }); }
			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;
		};
	}
}
#endif
#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;
	}
}
#endif
#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>;
	}
	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>;
	}
}
#endif
#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];
			if (first) {
				first = false;
				mint es[30], ies[30];
				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];
			if (first) {
				first = false;
				mint es[30], ies[30];
				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))];
				}
			}
		}
	}
	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;
		static constexpr unsigned long long MOD2 = 167772161;
		static constexpr unsigned long long MOD3 = 469762049;
		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;
			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;
	}
}
#endif
#ifndef ATCODER_DSU_HPP
#define ATCODER_DSU_HPP 1
#include <algorithm>
#include <cassert>
#include <vector>
namespace atcoder {
	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;
		std::vector<int> parent_or_size;
	};
}
#endif
#ifndef ATCODER_FENWICKTREE_HPP
#define ATCODER_FENWICKTREE_HPP 1
#include <cassert>
#include <vector>
namespace atcoder {
	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;
		}
	};
}
#endif
#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();
		}
	};
}
#endif
#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;
	}
	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());
		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;
			}
			long long g, im;
			std::tie(g, im) = internal::inv_gcd(m0, m1);
			long long u1 = (m1 / g);
			if ((r1 - r0) % g) return { 0, 0 };
			long long x = (r1 - r0) / g % u1 * im % u1;
			r0 += x * m0;
			m0 *= u1;
			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;
	}
}
#endif
#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;
	};
}
#endif
#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);
			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;
					for (int i = 0; i < int(g[v].size()); i++) {
						auto e = g[v][i];
						if (vis[e.to] || !e.cap) continue;
						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] -= 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;
	};
}
#endif
#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;
	};
}
#endif
#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]); }
	};
}
#endif
#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;
		}
		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;
		}
	}
	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);
	}
	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);
	}
	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);
	}
}
#endif
#ifndef ATCODER_TWOSAT_HPP
#define ATCODER_TWOSAT_HPP 1
#include <cassert>
#include <vector>
namespace atcoder {
	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;
	};
}
#endif
//ACL end
#pragma warning(disable:4996)
#include <bits/stdc++.h> 
using namespace std;
using namespace atcoder;
using mint = static_modint<1000000007>;
using vmint = vector<mint>;
using vvmint = vector<vmint>;
using vvvmint = vector<vvmint>;
using vvvvmint = vector<vvvmint>;
istream& operator>>(istream& is, mint& a) {
	long long v;
	is >> v;
	a = v;
	return is;
}
ostream& operator<<(ostream& os, mint a) {
	return os << a.val();
}
#ifdef _MSC_VER
#include <intrin.h>
#define __builtin_popcount __popcnt
#define __builtin_popcountll __popcnt64
#endif
template<typename T>
using vec = vector<T>;
using ll = long long;
using ull = unsigned long long;
using stirng = string;
using itn = int;
using vs = vec<string>;
using vc = vec<char>;
using vvc = vec<vc>;
using vvvc = vec<vvc>;
using vi = vec<int>;
using vvi = vec<vi>;
using vvvi = vec<vvi>;
using vvvvi = vec<vvvi>;
using vll = vec<ll>;
using vvll = vec<vll>;
using vvvll = vec<vvll>;
using vvvvll = vec<vvvll>;
using si = set<int>;
using vsi = vec<si>;
using vvsi = vec<vsi>;
using vvvsi = vec<vvsi>;
using sll = set<ll>;
using vsll = vec<sll>;
using vvsll = vec<vsll>;
using vvvsll = vec<vvsll>;
using vd = vec<double>;
using vvd = vec<vd>;
using vvvd = vec<vvd>;
using vvvvd = vec<vvvd>;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using pd = pair<double, double>;
using mii = map<int, int>;
using mll = map<ll, ll>;
using msi = multiset<int>;
using msll = multiset<ll>;
using vpii = vec<pii>;
using vvpii = vec<vpii>;
using vvvpii = vec<vvpii>;
using vpll = vec<pll>;
using vvpll = vec<vpll>;
using vvvpll = vec<vvpll>;
using vpd = vec<pd>;
using spii = set<pii>;
using spll = set<pll>;
using spd = set<pd>;
using vspll = vec<spll>;
using vvspll = vec<vspll>;
using vb = vec<bool>;
using vvb = vec<vb>;
using vvvb = vec<vvb>;
#define rep(i,j,n) for(ll i=j;i<(n);i++)
#define rrep(i,j,n) for (ll i = (n)-1; i >=j; i--)
#define Call_4th(_1, _2, _3, _4, ...) _4
#define rp(...) Call_4th(__VA_ARGS__, _1, rp2)(__VA_ARGS__)
#define rp2(i, n) rep(i, 0, n)
#define rrp(i,n) rrep(i,0,(n))
#define all(v) v.begin(), v.end()
#define pb push_back
#define pp pop_back()
#define bk back()
#define ins insert
#define yes cout<<"Yes"<<endl;
#define no cout<<"No"<<endl; 
#define NL {cout<<endl;}
#define n0 {cout<<"-1"<<endl;}
#define lwb lower_bound
#define upb upper_bound
#define reutrn return
#define Endl endl
template<typename T>
int len(T& x) { return int(x.size()); }
#define sz size()
#define fi first
#define se second
#define nxp next_permutation
ll mod0 = 998244353, lmax = 4e18, inf = 1050000000;
double pi = 3.141592653589793238462643;

//cin系統
template<typename... T>
void in(T&... X) { (cin >> ... >> X); }
template<typename T>
void ain(vec<T>& Y) { rp(K, Y.size()) cin >> Y[K]; }
template<typename T>
void ain(set<T>& Y, auto n) {
	rp(i, n) {
		T YY; cin >> YY;
		Y.insert(YY);
	}
}
template<typename T>
void ain(multiset<T>& Y, auto n) {
	rp(i, n) {
		T YY; cin >> YY;
		Y.insert(YY);
	}
}
template<typename T>
void ain(pair<T, T>& a) { cin >> a.fi >> a.se; }
template<typename T>
void bin(vec<T>& Z) { rp(L, Z.size()) ain(Z[L]); }

//cout系統
// vector
template <typename T> struct is_vector : false_type {};
template <typename T, typename Alloc>
struct is_vector<vector<T, Alloc>> : true_type {};
// queue
template <typename T> struct is_queue : false_type {};
template <typename T, typename Container>
struct is_queue<queue<T, Container>> : true_type {};
// stack
template <typename T> struct is_stack : false_type {};
template <typename T, typename Container>
struct is_stack<stack<T, Container>> : true_type {};
// pair
template <typename T> struct is_pair : false_type {};
template <typename T1, typename T2>
struct is_pair<pair<T1, T2>> : true_type {};
// set / multiset
template <typename T> struct is_set_like : false_type {};
template <typename Key, typename Compare, typename Alloc>
struct is_set_like<set<Key, Compare, Alloc>> : true_type {};
template <typename Key, typename Compare, typename Alloc>
struct is_set_like<multiset<Key, Compare, Alloc>> : true_type {};
// map / multimap
template <typename T> struct is_map_like : false_type {};
template <typename Key, typename Val, typename Compare, typename Alloc>
struct is_map_like<map<Key, Val, Compare, Alloc>> : true_type {};
template <typename Key, typename Val, typename Compare, typename Alloc>
struct is_map_like<multimap<Key, Val, Compare, Alloc>> : true_type {};

template <typename T>
void aout(const T& x) {
	if constexpr (is_vector<T>::value) {
		for (const auto& e : x) aout(e);
		cout << endl;
	}
	else if constexpr (is_queue<T>::value) {
		auto copy = x;
		while (!copy.empty()) {
			aout(copy.front());
			copy.pop();
		}
		cout << endl;
	}
	else if constexpr (is_stack<T>::value) {
		auto copy = x;
		while (!copy.empty()) {
			aout(copy.top());
			copy.pop();
		}
		cout << endl;
	}
	else if constexpr (is_pair<T>::value) {
		aout(x.first);
		aout(x.second);
		cout << endl;
	}
	else if constexpr (is_set_like<T>::value) {
		for (const auto& e : x) aout(e);
		cout << endl;
	}
	else if constexpr (is_map_like<T>::value) {
		for (const auto& e : x) aout(e); // pair<const Key, Val>
		cout << endl;
	}
	else {
		cout << x << " ";
	}
}

//数学系
ll modcnt(ll l, ll r, ll m, ll a) {//l以上r以下の整数iでi≡a(mod m)となるものの個数(l>r,a<0等も可)
	assert(m != 0);
	m = abs(m);
	ll x = l + ((a - l) % m + m) % m;//l以上で最小のi≡a(mod m)となるi
	if (x > r) return 0;
	return (r - x) / m + 1;
}
ll dsum(ll a) {//桁和
	ll r = 0;
	while (a != 0) {
		r += a % 10;
		a /= 10;
	}
	return r;
}
ll dnum(ll a) {//桁数(0は0桁)
	ll r = 0;
	while (a != 0) {
		r++;
		a /= 10;
	}
	return r;
}
sll yk(ll n) {//約数全列挙O(sqrt(N)),set<int>
	sll yaku;
	for (ll i = 1; i * i <= n; i++) if (n % i == 0) {
		yaku.insert(i);
		yaku.insert(n / i);
	}
	return yaku;
}
vi ssa(int n) {//n:必要な素数の最大値 素数全列挙 線形篩
	vector<int> prime;
	vector<int>lpf(n + 1, 0);//最小素因数
	rep(i, 2, n + 1) {
		if (lpf[i] == 0) {
			lpf[i] = i;
			prime.push_back(i);
		}
		for (int k : prime) {
			if (k * i > n || k > lpf[i]) break;//k最小素因数よりk<=i
			lpf[k * i] = k;//kを最小素因数に持つ数k*iを拾う
		}
	}
	return prime;
}
vector<pll> sib(ll x) {//素因数分解
	if (x == 1) return vector<pll>(1, { 1,1 });
	vector<pll> ans;
	for (ll i = 2; i * i <= x; i++) {//rootx回iを回しxで割る
		ll kaisu = 0;
		while (x % i == 0) {
			x /= i; kaisu++;
		}
		if (kaisu > 0) ans.pb({ i,kaisu });
	}
	if (x != 1) ans.pb({ x,1 });
	return ans;
}
bool isp(ll x) {//素数判定 O(sqrt(x))
	auto y = sib(x);
	return (x != 1 && y.sz == 1 && y[0] == pll{ x, 1 });
}
vec<vec<pll>> sib(ll x, ll y) {//閉区間[x,y]全素因数分解 x=1は何も返さない O(max(sqrt(y),y-x)loglogy) ABC227G ABC412E
	vi b = ssa(sqrt(y) + 100);
	vec<vec<pll>> r(y - x + 1);
	vll num(y - x + 1);
	rp(i, y - x + 1) num[i] = x + i;
	for (ll k : b) for (ll a = (1 + (x - 1) / k) * k; a <= y; a += k) {
		ll kaisu = 0;
		while (num[a - x] % k == 0) {
			num[a - x] /= k; kaisu++;
		}
		if (kaisu) r[a - x].pb({ k,kaisu });
	}
	rp(i, y - x + 1) if (num[i] != 1) r[i].pb({ num[i],1 });
	return r;
}
template<typename T>
ll sqrtll(T a) {//整数sqrt a<0orr非整数の場合-1を返す
	if (a < 0) return -1;
	ll r = sqrt(a);
	while (r * r > a) r--;
	while ((r + 1) * (r + 1) <= a) r++;
	if (r * r == a)return r;
	else return -1;
}
#define llsqrt sqrtll
ll powll(ll a, ll b) {//llでpow
	ll c = 1;
	assert(pow(a, b) <= 9e18);
	rp(i, b) {
		c *= a;
	}
	return c;
}
#define llpow powll
template<typename T>
T sq(T a) {//2乗
	return a * a;
}
template<typename T>
T cb(T a) {//3乗
	return a * a * a;
}
string basea(ll x, ll a) {//xをa進法に変換
	string r;
	while (x != 0) {
		r.pb(x % a + '0');
		x /= a;
	}
	reverse(all(r));
	return r;
}
ll base10(string x, ll a) {//a進数のxを10進法に戻す
	ll r = 0;
	rp(i, x.sz) {
		r *= a;
		r += x[i] - '0';
	}
	return r;
}
vll vbasea(ll x, ll a) {//xをa進法に変換(vector)
	vll r;
	while (x != 0) {
		r.pb(x % a);
		x /= a;
	}
	reverse(all(r));
	return r;
}
template<typename T>
ll bc(T x) {//bitcount
	assert(x >= 0);
	return __builtin_popcount(x);
}
double ds(double x1, double x2, double y1, double y2) {//dist
	return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
}
ll ds2(ll x1, ll x2, ll y1, ll y2) {//distの2乗(整数値)
	return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
}
template<typename T>
ll ds2(vec<T> a) {
	return (a[0] - a[2]) * (a[0] - a[2]) + (a[1] - a[3]) * (a[1] - a[3]);
}
ll gpsum(ll a, ll r, ll n, ll m) {//初項a,公比r,項数nの等比数列の総和mod m r=10でレピュニットも可能
	if (n == 0) return 0;
	if (n == 1) return a % m;
	modint::set_mod(m);
	if (n % 2 == 1) return (a + gpsum(a * r % m, r, n - 1, m)) % m;
	else return ((modint(r).pow(n / 2) + 1) * gpsum(a, r, n / 2, m)).val();
}

//二分探索
template<typename T>
auto lwa(typename std::vector<T>::iterator b, typename std::vector<T>::iterator e, T c) {//[b,e)でcより小さい最大の値のイテレータ(複数あれば右端)
	auto it = std::lower_bound(b, e, c);
	if (it == b) return e;
	--it;
	return it;
}
template<typename T>
auto lwa(vector<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	return lwa(all(a), b);
}
template<typename T>
auto lwa(set<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	auto i = a.lwb(b);
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}
template<typename T>
auto lwa(multiset<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	auto i = a.lwb(b);
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}

template<typename T>
auto lwc(typename std::vector<T>::iterator b, typename std::vector<T>::iterator e, T c) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = lwb(b, e, c);
	if (i != e && *i == c) return i;
	if (i == b) return e;
	else i--;
	return i;
}
template<typename T>
auto lwc(vector<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	return lwc(all(a), b);
}
template<typename T>
auto lwc(set<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = a.lwb(b);
	if (i != a.end() && *i == b) return i;
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}
template<typename T>
auto lwc(multiset<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = a.lwb(b);
	if (i != a.end() && *i == b) return i;
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}

template <typename T>
std::pair<typename vector<T>::iterator, typename vector<T>::iterator> its(vec<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = lwb(all(v), a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}
template <typename T>
std::pair<typename set<T>::iterator, typename set<T>::iterator> its(set<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = v.lwb(a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}
template <typename T>
std::pair<typename multiset<T>::iterator, typename multiset<T>::iterator> its(multiset<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = v.lwb(a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}

template <typename T>
typename vector<T>::iterator nits(vec<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}
template <typename T>
typename set<T>::iterator nits(set<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}
template <typename T>
typename multiset<T>::iterator nits(multiset<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}

template <typename T>
int rcount(vec<T>& a, T l, T r) {//ソート済配列aから[l,r]の範囲(l以上r以下)の要素カウント
	//右端を超える最小のi-左端に引っかかる最小のi
	//l,rのどちらに対しても境界を含む/含まないを入れ替えるときupb/lwbを変えれば良い
	return upb(all(a), r) - lwb(all(a), l);
}
template <typename T>
int rcount(set<T>& a, T l, T r) {//ソート済配列aから[l,r]の範囲(l以上r以下)の要素カウント
	return a.upb(r) - a.lwb(l);
}

template <class Container, class Key>
bool has(const Container& c, const Key& x) {//sort済vector/set/map等に対しその値を持つかどうか返す
	if constexpr (is_set_like<Container>::value || is_map_like<Container>::value) {
		// set / multiset / map
		return c.find(x) != c.end();
	}
	else {
		// vector 等（ソート済前提）
		return std::binary_search(c.begin(), c.end(), x);
	}
}

//配列便利関数系
template<typename T>
bool chmin(T& M1, T M2) {
	if (M1 > M2) { M1 = M2; return 1; }
	else return 0;
}
template<typename T>
bool chmax(T& M1, T M2) {
	if (M1 < M2) { M1 = M2; return 1; }
	else return 0;
}
template<typename T>
auto vmax(T a) {
	return *max_element(all(a));
}
template<typename T>
auto vmin(T a) {
	return *min_element(all(a));
}
template<typename T>
auto vmaxi(T a) {
	return max_element(all(a)) - a.begin();
}
template<typename T>
auto vmini(T a) {
	return min_element(all(a)) - a.begin();
}
template<typename T>
auto vsum(vector<T> a) {
	T x = 0;
	for (auto i : a)x += i;
	return x;
}
template<typename T>
vec<T> inv(vec<T> a) {//0-indexed 逆関数
	vec<T> ans(a.sz);
	rp(i, a.sz) {
		ans[a[i]] = i;
	}
	return ans;
}
template<typename T>
vec<T> inv1(vec<T> a) {//1-indexed
	vec<T> ans(a.sz);
	rp(i, a.sz) {
		ans[a[i] - 1] = i + 1;
	}
	return ans;
}
template<typename T>
vec<T> zat(vec<T> a) {//座標圧縮 T=ll,int
	auto b = a;
	sort(all(b));
	b.erase(unique(all(b)), b.end());
	vec<T> c(a.size());
	rp(i, a.size()) {
		c[i] = lwb(all(b), a[i]) - b.begin();
	}
	return c;
}
template<typename T>
vec<T> rsw(vec<T> a, bool is_put0 = 1) {//累積和 T=ll,int
	if (is_put0 == 0) {
		vec<T> b = {};
		rp(i, a.sz) b.pb(a[i] + (b.sz == 0 ? 0 : b[b.sz - 1]));
		return b;
	}
	vec<T> b = { 0 };
	rp(i, a.sz) b.pb(a[i] + b[b.sz - 1]);
	return b;
}
template<typename T>
T rswsum(vec<T>& a, int l, int r, bool is_put0 = 1) {//rsw配列から元の配列の[l,r)番目(0indexed)の和を取得
	if (is_put0 == 0) {
		assert(0 <= l && l <= r);
		assert(r <= a.sz);
		return (r == 0 ? 0 : a[r - 1]) - (l == 0 ? 0 : a[l - 1]);
	}
	assert(0 <= l && l <= r);
	assert(r < a.sz);
	return a[r] - a[l];
}
template<typename T>
T rswsum2(vec<T>& a, int l, int r, bool is_put0 = 1) {//rsw配列から元の配列の[l,r]番目(0indexed)の和を取得
	return rswsum(a, l, r + 1, is_put0);
}

//文字列系
vector<pair<char, ll>>  rle(string rr) {
	vector<pair<char, ll>> ret;
	char lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
vector<pair<int, ll>>  rle(vi rr) {
	vector<pair<int, ll>> ret;
	int lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
vector<pair<ll, ll>>  rle(vll rr) {
	vector<pair<ll, ll>> ret;
	ll lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
bool kai(string S) {//回文か判定
	bool ans = 1;
	rp(i, S.size()) if (S[i] != S[S.size() - 1 - i]) ans = 0;
	return ans;
}
template <typename... Args>
string concat(const Args&... args) {//任意個のchat・string・int等を引数に取り文字列として繋げたstringを返す(整数はasciiでcharに変換)
	string res;
	(res += ... += args);
	return res;
}

//その他
ll max(ll a, int b) { return max(a, ll(b)); }
ll max(int a, ll b) { return max(ll(a), b); }
ll min(ll a, int b) { return min(a, ll(b)); }
ll min(int a, ll b) { return min(ll(a), b); }
bool yn(bool jg) {
	if (jg) {
		cout << "Yes" << endl; return true;
	}
	else {
		cout << "No" << endl; return false;
	}
}
bool ynab(bool jg) {
	if (jg) {
		cout << "Alice" << endl; return true;
	}
	else {
		cout << "Bob" << endl; return false;
	}
}
template<typename T>
void cp(map < T, ll >& a, T b, ll c) {//countplus
	if (a.find(b) == a.end()) a[b] = c;
	else a[b] += c;
	if (a[b] == 0) a.erase(b);
}
template<typename T>
bool cross(T a, T b, T c, T d) {//区間(a,b)と区間(c,d)が重なるか判定
	return !(b <= c || d <= a);
}
template<typename T>
bool crosse(T a, T b, T c, T d) {//区間[a,b]と区間[c,d]が重なるか判定
	return !(b < c || d < a);
}
template<typename T>
vec<vll> absort(vec<T> a, vec<T> b) {//l,rのl昇順r逆順sort
	vec<vec<T>> ab(a.sz);
	rp(i, a.sz) ab[i] = { a[i],-b[i],i };
	sort(all(ab));
	rp(i, a.sz) ab[i][1] = -ab[i][1];
	return ab;
}
template<typename T>
bool rabc(T a, T b, T c) {//円環状でa,b,cがこの順か判定 同じ値はfalse
	return ((a - b) / abs(a - b)) * ((b - c) / abs(b - c)) * ((c - a) / abs(c - a)) > 0;
}
bool labc(ll a, ll b, ll c) {//a<=b<cか判定
	return a <= b && b < c;
}
#define inr labc
bool labc2(ll a, ll b, ll c) {//a<=b<=cか判定
	return a <= b && b <= c;
}
#define inr2 labc2

vector<vector<char>> rot(vs a) {//時計回り90度回転
	vector<vector<char>> r(a[0].sz, vector<char>(a.sz));
	rp(i, a.sz)rp(j, a[0].sz) r[j][a[0].sz - 1 - i] = a[i][j];
	return r;
}
template<typename T>
vector<vector<T>> rot(vector<vector<T>> a) {
	vector<vector<T>> r(a[0].sz, vector<T>(a.sz));
	rp(i, a.sz)rp(j, a[0].sz) r[j][a.sz - 1 - i] = a[i][j];
	return r;
}


//実行時間調整付きassert
std::chrono::steady_clock::time_point START_TIME;
void start() {
	START_TIME = std::chrono::steady_clock::now();
}

void assertt(bool b, int t) {
	if (b) return;
	if (START_TIME.time_since_epoch().count() == 0) {
		start();
	}
	auto target_time = START_TIME + std::chrono::milliseconds(t - 100);
	while (std::chrono::steady_clock::now() < target_time);
	assert(0);
}
bool testt(int t) {//tミリ秒経っていないならtrue (while(testt(1950))のように使う)
	assert(START_TIME.time_since_epoch().count() != 0);
	auto target_time = START_TIME + std::chrono::milliseconds(t - 100);
	return (std::chrono::steady_clock::now() < target_time);
}
//lazy_segtree<S, op, e, F, mapping, composition, id> seg(n);
//S:中身型 op:左右合成 e:初期値
//F:遅延型 mapping:遅延適用後の要素 composition:遅延合成( 関数f(g()) ) id:遅延初期値(何もさせない)

using S = ll;
using F = ll;
S op(S a, S b) {
	return max(a, b);
}

S e() {
	return -lmax;
}
using segt = segtree<S, op, e>;
/*

S mapping(F x, S a) {//xをaに作用(x初期値に注意) S[l]~S[r]のopであるSにF適用するとF(S[l])~F(S[r])のopになる必要がある 、つまりL~Rの代表にFかけてもL~Rの代表であり続けられるようS作る必要がある
	if(x==0) return a;
	else return {a.se,a.se};
}
F composition(F f, F g) {//fの方が後(f初期値に注意) Fを何個合成しても1つのFで表せる
	if(f==0 && g==0 )return 0;
	return 1;
}
F id() { return 0; }
using lsegt = lazy_segtree<S, op, e, F, mapping, composition, id>;
*/
//cout<<setprecision(15)<<fixed<<
//function<返り値の型(引数の型1, 引数の型2, ...)> 関数名 = [&](引数の型1 引数名1, 引数の型2, 引数名2, ...)
//function<ll(int,int)> f=[&](int a,int b)
//再帰しない場合 auto f=[&](int a,int b) ->ll{} 

int mxkj;//階乗に入れる値の最大値
vector<mint> kj, gkj;
bool iskjset = 0;
void kjset(int maxkj) {
	iskjset = 1;
	mxkj = maxkj;
	kj.resize(mxkj + 1);
	gkj.resize(mxkj + 1);
	kj[0] = 1;//階乗
	rep(i, 1, mxkj + 1) {
		kj[i] = kj[i - 1] * i;
	}
	gkj[mxkj] = 1 / kj[mxkj];
	rrp(i, mxkj) gkj[i] = gkj[i + 1] * (i + 1);
}
auto cmb = [](int fst, int snd) {
	assert(iskjset);
	bool ck = 1;//範囲外値でエラーを返すなら1(ck=0でもpowの引数など他の値が範囲外にならないか要確認)
	if (0 <= fst && fst <= mxkj && 0 <= snd && snd <= fst);
	else {
		if (ck) assert(0);
		else return mint(0);
	}
	return  kj[fst] * gkj[snd] * gkj[fst - snd];
	};
auto pmt = [](int fst, int snd) {
	assert(iskjset);
	bool ck = 1;//範囲外値でエラーを返すなら1(ck=0でもpowの引数など他の値が範囲外にならないか要確認)
	if (0 <= fst && fst <= mxkj && 0 <= snd && snd <= fst);
	else {
		if (ck) assert(0);
		else return mint(0);
	}
	return  kj[fst] * gkj[fst - snd];
	};
vec<mint> conv_sp(vmint a, vmint sp, ll mxi = 200100) {//疎な場合のconv高速化 aの長さN,spの非0係数K個としてO(NK)
	vec<pair<int, mint>> sps;
	int spmxi = -1;
	rp(i, sp.size()) {
		if (sp[i] != 0) {
			sps.pb({ i,sp[i] });
			spmxi = i;
		}
	}
	if (spmxi == -1) return { 0 };
	vmint r(min(mxi, ll(a.sz) + spmxi), 0);
	rp(i, a.sz) {
		for (auto j : sps) {
			if (i + j.fi < mxi)r[i + j.fi] += j.se * a[i];
		}
	}
	return r;
}
vec<mint> div_sp(vmint f, vmint sp, ll mxi = 200100) {//f/spをmxiまで計算 ただしspのx^0項非0 spの非0係数K個としてO(mxiK)
	assert(sp.sz > 0 && sp[0] != 0);
	vec<pair<int, mint>> sps;
	int spmxi = -1;
	rp(i, sp.size()) {
		if (sp[i] != 0) {
			sps.pb({ i,sp[i] });
			spmxi = i;
		}
	}
	vec<mint> r(mxi, 0);
	f.resize(mxi);
	mint gsp0 = 1 / sp[0];
	rp(i, mxi) {
		r[i] = f[i] * gsp0;
		for (auto j : sps) {
			if (i + j.fi < mxi)f[i + j.fi] -= r[i] * j.se;
		}
	}
	return r;
}
vec<mint> conv(vmint a, vmint b, ll mxi = 200100) {//a.sz+b.sz=nとしてO(nlogn) a.sz+b.sz>2^23=8.3e6を超えると畳み込み不可能
	if (a.sz > mxi) a.resize(mxi);
	if (b.sz > mxi) b.resize(mxi);
	vmint r = convolution(a, b);
	if (r.size() > mxi) r.resize(mxi);
	return r;
}
mint convget(vmint a, vmint b, ll i) {//conv(a,b)[i]をO(a.sz+b.sz+i)で求める
	if (i < 0) return 0;
	mint r = 0;
	rp(ii, i + 1) {
		if (ii < a.sz && i - ii < b.sz)r += a[ii] * b[i - ii];
	}
	return r;
}
vec<mint> vinv(vmint f, ll mxi = 200100) {// O(mxi・logmxi)
	assert(f.sz > 0 && f[0] != 0);
	vec<mint> g = { mint(1) / f[0] };
	f.resize(mxi); // fを最初にリサイズ
	for (ll len = 1; len < mxi; len <<= 1) {//g←g(2-fg)をlog(mxi)回
		int need = min(mxi, len << 1);
		auto fg = conv(vec<mint>(f.begin(), f.begin() + need), g, need);    // f * g

		for (int i = 0; i < need; i++) fg[i] = -fg[i];
		fg[0] += 2;  // 2 - f*g

		g = conv(g, fg, need);
	}
	return g;
}
vmint log(vmint f, ll mxi = 200100) {//log(f)を求める(ただしf[0]=1が必要) O(mxi・logmxi) ∫(f')/f
	if (f.sz > mxi) f.resize(mxi);
	assert(f[0] == 1);
	if (f.sz == 1) return { 0 };
	vmint bf(f.sz - 1);//fの微分
	rp(i, f.sz - 1) bf[i] = f[i + 1] * (i + 1);
	vmint bg = conv(bf, vinv(f, mxi), mxi);
	vmint g(bg.sz);
	g[0] = 0;
	rp(i, bg.sz - 1) g[i + 1] = bg[i] / (i + 1);
	return g;
}
vmint exp(vmint f, ll mxi = 200100) {//exp(f)をニュートン法で求める(ただしf[0]=0が必要) O(mxi・logmxi) 
	assert(f[0] == 0);
	vec<mint> g = { 1 };
	f.resize(mxi); // fを最初にリサイズ
	for (ll len = 1; len < mxi; len <<= 1) {//g←g(1-log(g)+f)をlog(mxi)回
		int need = min(mxi, len << 1);
		vmint x = log(g, need);
		x.resize(need);
		rp(i, need) x[i] = f[i] - x[i];
		x[0]++;
		g = conv(g, x, need);
	}
	return g;
}
vec<mint> vpow(vmint a, ll p, ll mxi = 200100) {//a[0]+a[1]x+a[2]x^2...のp乗をa^p=exp(log(a))を利用して計算 O(mxi・logmxi)
	int d = 0;//前から何個0が続くか
	rp(i, a.sz) {
		if (a[i] == 0) d++;
		else break;
	}
	assert(p >= 0 || d == 0);//p<0かつd>0は計算不能(FPSで定義されない)
	if (p == 0 || double(d) * p >= mxi || d == a.sz) {//0乗 or all zero
		vmint r(mxi, 0);
		if (p == 0) r[0] = 1;
		return r;
	}
	//a[0]=1になるように前d個の0を消し、a全体をa[0]で割る補正
	mint c = 1, ad = 1;
	if (a[d] != 1) {
		ad = a[d];
		c = 1 / a[d];
		rp(i, a.sz) a[i] *= c;
	}
	if (d > 0) {
		rp(i, a.sz - d)a[i] = a[i + d];
		a.resize(a.sz - d);
	}
	vmint ploga = log(a, mxi - d * p);//p*log(a)
	rp(i, ploga.sz) ploga[i] *= p;
	vmint r = exp(ploga, mxi - d * p);
	mint adp;//ad^p
	if (p < 0) adp = mint(c).pow(-p);
	else  adp = ad.pow(p);
	rp(i, r.sz) r[i] *= adp;
	r.resize(mxi);
	rrp(i, mxi) {
		if (i < d * p)r[i] = 0;
		else r[i] = r[i - d * p];
	}
	return r;
}
vec<mint> vap(vvmint a, ll mxi = 200100) {//多項式総積 次数の総和NとしてO(N(logN)^2)
	if (a.sz == 0) return { 1 };
	queue<vmint> q;
	for (auto& i : a) q.push(i);
	while (q.sz > 1) {
		vmint p1 = q.front();
		q.pop();
		vmint p2 = q.front();
		q.pop();
		vmint np = conv(p1, p2, mxi);
		q.push(np);
	}
	return q.front();
}
vec<mint> vquot(vmint g, vmint f) {//多項式除算 N次g,M次fに対してg=qf+r(m-1次以下)を満たすqをO(NlogN)で求める
	while (f.sz > 0 && f[f.sz - 1] == 0) f.pop_back();
	assert(f.sz != 0);
	while (g.sz > 0 && g[g.sz - 1] == 0) g.pop_back();
	if (g.sz < f.sz) return {};
	int n = int(g.sz) - 1, m = int(f.sz) - 1;
	vmint revg = g; reverse(all(revg));
	vmint revf = f; reverse(all(revf));
	vmint revq = conv(revg, vinv(revf, n - m + 1), n - m + 1);
	revq.resize(n - m + 1);
	reverse(all(revq));//ここでrevqがqを表すようになる
	return revq;
}
vec<mint> vmod(vmint g, vmint f) {//多項式の余り N次g,M次fに対してg=qf+r(m-1次以下)を満たすrをO(NlogN)で求める
	while (f.sz > 0 && f[f.sz - 1] == 0) f.pop_back();
	assert(f.sz != 0);
	vmint q = vquot(g, f);
	vmint qf = conv(q, f, f.sz - 1);
	vmint r(f.sz - 1);
	g.resize(f.sz - 1);
	qf.resize(f.sz - 1);
	rp(i, f.sz - 1) {
		r[i] = g[i] - qf[i];
	}
	while (r.sz > 0 && r[r.sz - 1] == 0) r.pop_back();
	return r;
}
vec<mint> vpowmod(vmint g, vmint f, ll p) {//多項式の余り g^p mod f f,gの次数NとしてO(NlogNlogp)
	vmint r = { 1 };
	assert(p < (ll(1) << 60));
	rrp(i, 60) {
		r = conv(r, r, 2 * r.sz);
		r = vmod(r, f);
		if (p & (ll(1) << i)) {
			r = conv(r, g, r.sz + g.sz);
			r = vmod(r, f);
		}
	}
	return r;
}
vec<mint> vplus(vmint f, vmint g) {//f+g
	if (f.sz > g.sz) swap(f, g);
	vmint r(g.sz, 0);
	rp(i, f.sz) r[i] = f[i] + g[i];
	rep(i, f.sz, g.sz) r[i] = g[i];
	return r;
}
mint vdn(vmint f, mint a) {//f(x)にaを代入(f(a)を計算)
	mint r = 0;
	rp(i, f.sz) r += f[i] * a.pow(i);
	return r;
}
vec<mint> v1pow(ll p, ll mxi = 200100) {//(1/(1-x))={1,1,1,...}^pをmxiまでO(mxi)で求める　p+mxiまでkjの前計算が必要 gkjが求まっていない場合はO(mxi・log998)
	assert(p + mxi - 1 <= mxkj);
	if (p == 0) {
		vmint r(mxi, 0);
		r[0] = 1;
		return r;
	}
	vmint r(mxi);
	rp(i, mxi) r[i] = cmb(p + i - 1, i);
	return r;
}
vec<mint> v1apow(mint a, ll p, ll mxi = 200100) {//(1+ax)^pをmxiまでa==1でO(mxi),a!=1でO(mxi・logp)で求める　mxiまでkjの前計算が必要 gkjが求まっていない場合はO(mxi・log998)
	assert(mxi <= mxkj);
	vmint r(mxi, 0);
	ll siz = min(mxi, p + 1);
	if (a == 1) rp(i, siz) {
		r[i] = cmb(p, i);
	}
	else rp(i, siz) {
		r[i] = cmb(p, i) * a.pow(i);
	}
	return r;
}
template<typename T>
mint parsum(vec<T> a, T s) {//部分和問題 配列aから0個以上選んで和をsにする通り数 aの総和xとしてO(x(logx)^2)
	if (s < 0) return 0;
	int n = a.sz;
	vvmint p(n);
	rp(i, n) {
		p[i] = vmint(a[i] + 1, 0);
		p[i][0] = 1;
		p[i][a[i]] += 1;
	}
	vmint ans = vap(p, s + 1);
	if (ans.sz <= s)return 0;
	return ans[s];
}

vmint apowsum(vmint a, ll m) {//長さNの配列aに対しΣ(a[i]^k)をk=0,1,2...Mに対してO(N(logN)^2+MlogM)で列挙
	int n = a.sz;
	vvmint x(n);
	rp(i, n) x[i] = { 1,-a[i] };
	vmint y = vap(x, n + 1);
	y = log(y, m + 1);
	rp(i, y.sz) y[i] = -y[i] * i;
	y[0] = n;
	return y;
}

vec<mint> ktms(vmint d, ll n) {//a.sz=d.sz=kとして aの漸化式ai=d_0*a_(i-1)+...+d_(k-1)*a_(i-k)のときa_n=x0a0+...+x_(k-1)a_(k-1)を満たす配列xをO(KlogKlogN)
	int k = d.sz;
	reverse(all(d));
	vmint q(k + 1);
	rp(i, k) q[i] = -d[i];
	q[k] = 1;
	vmint r = vpowmod({ 0,1 }, q, n);
	r.resize(k);
	return r;
}
mint bm(vmint p, vmint q, ll n) {// bostan-mori [x^n] P(x)/Q(x)をP,Qの次数KとしてO(KlogKlogN) 
	if (p.sz != 0) p.pb(0);
	if (n == 0) return p[0] / q[0];
	vmint mq = q;
	for (int i = 0; i < mq.sz; i += 2) mq[i] = -mq[i];
	vmint u = conv(p, mq, p.sz + mq.sz), v2 = conv(q, mq, q.sz + mq.sz);
	vmint v;
	for (int i = 0; i < v2.sz; i += 2) v.pb(v2[i]);
	if (n % 2 == 0) {
		vmint ue;
		for (int i = 0; i < u.sz; i += 2) ue.pb(u[i]);
		return bm(ue, v, n / 2);
	}
	else {
		vmint ue;
		for (int i = 1; i < u.sz; i += 2) ue.pb(u[i]);
		return bm(ue, v, n / 2);
	}
}
mint tobm(vmint a, vmint d, ll n) {//a.sz=d.sz=kとして aの漸化式ai=d_0*a_(i-1)+...+d_(k-1)*a_(i-k)のときa_n=x0a0+...+x_(k-1)a_(k-1)をO(KlogKlogN)
	assert(a.sz == d.sz);
	ll k = a.sz;
	vmint q = { 1 };
	for (auto i : d) q.pb(-i);
	vmint p = conv(a, q, k);
	return bm(p, q, n);
}
//Berlekamp–Massey aの漸化式ai=d_0*a_(i-1)+...+d_(k-1)*a_(i-k)のとき配列{1,-d_0,-d_1,...}をa.sz=NとしてO(N^2)
//L=q.sz(q:返り値)としてN<2*Lなら解は複数存在 2*L<=Nなら解は唯一
//qに対して末尾の0を削除してはならない
vmint bm2(vmint a) {
	vmint q = { 1 },b=q;
	int l = 0, pn = -1;
	mint pd = 1;
	rp(n, a.sz) {
		mint d = 0;
		rp(j, n + 1) if(j<q.sz && n-j<a.sz)d += q[j] * a[n - j];
		if (d == 0) continue;
		
		vmint qnew = q;
		mint k = d / pd;
		if (qnew.sz < b.sz + n - pn) qnew.resize(b.sz + n - pn);
		rp(j, b.sz) {
			qnew[j + n - pn] -= k * b[j];
		}

		if (2 * l <= n) {
			l = n + 1 - l;
			b = q;
			q = qnew;
			pn = n;
			pd = d;
		}
		else {
			q = qnew;
		}
	}
	return q;
}
//線形漸化式を持つ数列aのk番目の値を予測 
//返り値のfi=falseなら一意に定まらない(aが短すぎて複数解を持つ),trueなら一意に定まる
pair<bool,mint> bmbm(vmint a,ll k){
	pair<bool, mint> r;
	vmint q = bm2(a);
	if (q.sz * 2 <= a.sz) r.fi = true;
	else r.fi = false;
	r.se = bm(conv(a, q, q.sz), q, k);
	return r;
}
int main() {
	ll a, b, c, d, e, n;
	cin >> a >> b >> c >> d >> e >> n;
	vmint f(10);
	f[0] = a;
	f[1] = b;
	rep(i, 2, 10) f[i] = c * f[i - 1] + d * f[i - 2] + e;
	f = rsw(f, 0);
	auto ans = bmbm(f, n);
	cout << ans.se << endl;
}