#define _USE_MATH_DEFINES
#include <bits/stdc++.h>
#define rep(i, n) for (int i = 0; i < (n); ++i)
#define FOR(i, m, n) for (int i = (m); i < (n); ++i)
#define rrep(i, n) for (int i = (n)-1; i >= 0; --i)
#define rfor(i, m, n) for (int i = (m); i >= (n); --i)
#define unless(c) if (!(c))
#define sz(x) ((int)(x).size())
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define range_it(a, l, r) (a).begin() + (l), (a).begin() + (r)

using namespace std;
using ll = long long;
using LD = long double;
using VB = vector<bool>;
using VVB = vector<VB>;
using VI = vector<int>;
using VVI = vector<VI>;
using VL = vector<ll>;
using VVL = vector<VL>;
using VS = vector<string>;
using VD = vector<LD>;
using PII = pair<int, int>;
using VP = vector<PII>;
using PLL = pair<ll, ll>;
using VPL = vector<PLL>;
template <class T> using PQ = priority_queue<T>;
template <class T> using PQS = priority_queue<T, vector<T>, greater<T>>;
constexpr int inf = 1e9;
constexpr long long inf_ll = 1e18, MOD = 1000000007;
constexpr long double PI = M_PI, EPS = 1e-12;

// --- input --- //
#ifdef _WIN32
#define getchar_unlocked _getchar_nolock
#define putchar_unlocked _putchar_nolock
#define fwrite_unlocked fwrite
#define fflush_unlocked fflush
#endif
class Input {
	static int gc() {
		return getchar_unlocked();
	}
	template <class T> static void i(T& v) {
		cin >> v;
	}
	static void i(char& v) {
		while (isspace(v = gc()))
			;
	}
	static void i(bool& v) {
		v = in<char>() != '0';
	}
	static void i(string& v) {
		v.clear();
		char c;
		for (i(c); !isspace(c); c = gc()) v += c;
	}
	static void i(int& v) {
		bool neg = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void i(long long& v) {
		bool neg = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c); c = gc()) v = v * 10 + (c - '0');
		if (neg) v = -v;
	}
	static void i(double& v) {
		double dp = 1;
		bool neg = false, adp = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c) || c == '.'; c = gc()) {
			if (c == '.')
				adp = true;
			else if (adp)
				v += (c - '0') * (dp *= 0.1);
			else
				v = v * 10 + (c - '0');
		}
		if (neg) v = -v;
	}
	static void i(long double& v) {
		long double dp = 1;
		bool neg = false, adp = false;
		v = 0;
		char c;
		i(c);
		if (c == '-') {
			neg = true;
			c = gc();
		}
		for (; isdigit(c) || c == '.'; c = gc()) {
			if (c == '.')
				adp = true;
			else if (adp)
				v += (c - '0') * (dp *= 0.1);
			else
				v = v * 10 + (c - '0');
		}
		if (neg) v = -v;
	}
	template <class T, class U> static void i(pair<T, U>& v) {
		i(v.first);
		i(v.second);
	}
	template <class T> static void i(vector<T>& v) {
		for (auto& e : v) i(e);
	}
	template <size_t N = 0, class T> static void input_tuple(T& v) {
		if constexpr (N < tuple_size_v<T>) {
			i(get<N>(v));
			input_tuple<N + 1>(v);
		}
	}
	template <class... T> static void i(tuple<T...>& v) {
		input_tuple(v);
	}
	struct InputV {
		int n, m;
		InputV(int _n) : n(_n), m(0) {}
		InputV(const pair<int, int>& nm) : n(nm.first), m(nm.second) {}
		template <class T> operator vector<T>() {
			vector<T> v(n);
			i(v);
			return v;
		}
		template <class T> operator vector<vector<T>>() {
			vector<vector<T>> v(n, vector<T>(m));
			i(v);
			return v;
		}
	};

public:
	static string read_line() {
		string v;
		char c;
		for (i(c); c != '\n' && c != '\0'; c = gc()) v += c;
		return v;
	}
	template <class T> static T in() {
		T v;
		i(v);
		return v;
	}
	template <class T> operator T() const {
		return in<T>();
	}
	int operator--(int) const {
		return in<int>() - 1;
	}
	InputV operator[](int n) const {
		return InputV(n);
	}
	InputV operator[](const pair<int, int>& n) const {
		return InputV(n);
	}
	void operator()() const {}
	template <class H, class... T> void operator()(H&& h, T&&... t) const {
		i(h);
		operator()(forward<T>(t)...);
	}

private:
	template <template <class...> class, class...> struct Multiple;
	template <template <class...> class V, class Head, class... Tail> struct Multiple<V, Head, Tail...> {
		template <class... Args> using vec = V<vector<Head>, Args...>;
		using type = typename Multiple<vec, Tail...>::type;
	};
	template <template <class...> class V> struct Multiple<V> { using type = V<>; };
	template <class... T> using multiple_t = typename Multiple<tuple, T...>::type;
	template <size_t N = 0, class T> void in_multiple(T& t) const {
		if constexpr (N < tuple_size_v<T>) {
			auto& vec = get<N>(t);
			using V = typename remove_reference_t<decltype(vec)>::value_type;
			vec.push_back(in<V>());
			in_multiple<N + 1>(t);
		}
	}

public:
	template <class... T> auto multiple(int H) const {
		multiple_t<T...> res;
		while (H--) in_multiple(res);
		return res;
	}
} in;
#define input(T) Input::in<T>()
#define INT input(int)
#define LL input(long long)
#define STR input(string)
#define inputs(T, ...) \
	T __VA_ARGS__;     \
	in(__VA_ARGS__)
#define ini(...) inputs(int, __VA_ARGS__)
#define inl(...) inputs(long long, __VA_ARGS__)
#define ins(...) inputs(string, __VA_ARGS__)

// --- output --- //
struct BoolStr {
	const char *t, *f;
	BoolStr(const char* _t, const char* _f) : t(_t), f(_f) {}
} Yes("Yes", "No"), yes("yes", "no"), YES("YES", "NO"), Int("1", "0");
struct DivStr {
	const char *d, *l;
	DivStr(const char* _d, const char* _l) : d(_d), l(_l) {}
} spc(" ", "\n"), no_spc("", "\n"), end_line("\n", "\n"), comma(",", "\n"), no_endl(" ", "");
class Output {
	BoolStr B{Yes};
	DivStr D{spc};
	void p(int v) const {
		char buf[12]{};
		if (auto [ptr, e] = to_chars(begin(buf), end(buf), v); e == errc{}) {
			fwrite(buf, sizeof(char), ptr - buf, stdout);
		} else {
			assert(false);
		}
	}
	void p(long long v) const {
		char buf[21]{};
		if (auto [ptr, e] = to_chars(begin(buf), end(buf), v); e == errc{}) {
			fwrite(buf, sizeof(char), ptr - buf, stdout);
		} else {
			assert(false);
		}
	}
	void p(bool v) const {
		p(v ? B.t : B.f);
	}
	void p(char v) const {
		putchar_unlocked(v);
	}
	void p(const char* v) const {
		fwrite_unlocked(v, 1, strlen(v), stdout);
	}
	void p(double v) const {
		printf("%.20f", v);
	}
	void p(long double v) const {
		printf("%.20Lf", v);
	}
	template <class T> void p(const T& v) const {
		cout << v;
	}
	template <class T, class U> void p(const pair<T, U>& v) const {
		p(v.first);
		p(D.d);
		p(v.second);
	}
	template <class T> void p(const vector<T>& v) const {
		rep(i, sz(v)) {
			if (i) p(D.d);
			p(v[i]);
		}
	}
	template <class T> void p(const vector<vector<T>>& v) const {
		rep(i, sz(v)) {
			if (i) p(D.l);
			p(v[i]);
		}
	}

public:
	Output& operator()() {
		p(D.l);
		return *this;
	}
	template <class H> Output& operator()(H&& h) {
		p(h);
		p(D.l);
		return *this;
	}
	template <class H, class... T> Output& operator()(H&& h, T&&... t) {
		p(h);
		p(D.d);
		return operator()(forward<T>(t)...);
	}
	template <class It> Output& range(const It& l, const It& r) {
		for (It i = l; i != r; i++) {
			if (i != l) p(D.d);
			p(*i);
		}
		p(D.l);
		return *this;
	}
	template <class T> Output& range(const T& a) {
		range(a.begin(), a.end());
		return *this;
	}
	template <class... T> void exit(T&&... t) {
		operator()(forward<T>(t)...);
		std::exit(EXIT_SUCCESS);
	}
	Output& flush() {
		fflush_unlocked(stdout);
		return *this;
	}
	Output& set(const BoolStr& b) {
		B = b;
		return *this;
	}
	Output& set(const DivStr& d) {
		D = d;
		return *this;
	}
	Output& set(const char* t, const char* f) {
		B = BoolStr(t, f);
		return *this;
	}
} out;

// --- step --- //
template <class T> struct Step {
	class It {
		T a, b, c;

	public:
		constexpr It() : a(T()), b(T()), c(T()) {}
		constexpr It(T _b, T _c, T _s) : a(_b), b(_c), c(_s) {}
		constexpr It& operator++() {
			--b;
			a += c;
			return *this;
		}
		constexpr It operator++(int) {
			It tmp = *this;
			--b;
			a += c;
			return tmp;
		}
		constexpr const T& operator*() const {
			return a;
		}
		constexpr const T* operator->() const {
			return &a;
		}
		constexpr bool operator==(const It& i) const {
			return b == i.b;
		}
		constexpr bool operator!=(const It& i) const {
			return !(b == i.b);
		}
		constexpr T start() const {
			return a;
		}
		constexpr T size() const {
			return b;
		}
		constexpr T step() const {
			return c;
		}
	};
	constexpr Step(T b, T c, T s) : be(b, c, s) {}
	constexpr It begin() const {
		return be;
	}
	constexpr It end() const {
		return en;
	}
	constexpr T start() const {
		return be.start();
	}
	constexpr T size() const {
		return be.size();
	}
	constexpr T step() const {
		return be.step();
	}
	constexpr T sum() const {
		return start() * size() + step() * (size() * (size() - 1) / 2);
	}
	operator vector<T>() const {
		return to_a();
	}
	auto to_a() const {
		vector<T> res;
		res.reserve(size());
		for (auto i : *this) {
			res.push_back(i);
		}
		return res;
	}
	using value_type = T;
	using iterator = It;

private:
	It be, en;
};
template <class T> inline constexpr auto step(T a) {
	return Step<T>(0, a, 1);
}
template <class T> inline constexpr auto step(T a, T b) {
	return Step<T>(a, b - a, 1);
}
template <class T> inline constexpr auto step(T a, T b, T c) {
	return Step<T>(a, a < b ? (b - a - 1) / c + 1 : 0, c);
}

namespace Ruby {
	template <class F> struct Callable {
		F func;
		Callable(const F& f) : func(f) {}
	};
	template <class T, class F> auto operator|(const T& v, const Callable<F>& c) {
		return c.func(v);
	}

	struct Sort_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				sort(begin(v), end(v), f);
				return v;
			});
		}
		template <class T> friend auto operator|(T v, const Sort_impl& c) {
			sort(begin(v), end(v));
			return v;
		}
	} Sort;
	struct RSort_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				sort(rbegin(v), rend(v), f);
				return v;
			});
		}
		template <class T> friend auto operator|(T v, const RSort_impl& c) {
			sort(rbegin(v), rend(v));
			return v;
		}
	} RSort;
	struct Reverse_impl {
		template <class T> friend auto operator|(T v, const Reverse_impl& c) {
			reverse(begin(v), end(v));
			return v;
		}
	} Reverse;
	struct Unique_impl {
		template <class T> friend auto operator|(T v, const Unique_impl& c) {
			v.erase(unique(begin(v), end(v), end(v)));
			return v;
		}
	} Unique;
	struct Uniq_impl {
		template <class T> friend auto operator|(T v, const Uniq_impl& c) {
			sort(begin(v), end(v));
			v.erase(unique(begin(v), end(v), end(v)));
			return v;
		}
	} Uniq;
	struct Rotate_impl {
		template <class F> auto operator()(int left) {
			return Callable([&](auto v) {
				rotate(begin(v), begin(v) + left, end(v));
				return v;
			});
		}
	} Rotate;
	struct Max_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				return *max_element(begin(v), end(v), f);
			});
		}
		template <class T> friend auto operator|(T v, const Max_impl& c) {
			return *max_element(begin(v), end(v));
		}
	} Max;
	struct Min_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				return *min_element(begin(v), end(v), f);
			});
		}
		template <class T> friend auto operator|(T v, const Min_impl& c) {
			return *min_element(begin(v), end(v));
		}
	} Min;
	struct MaxPos_impl {
		template <class T> friend auto operator|(T v, const MaxPos_impl& c) {
			return max_element(begin(v), end(v)) - begin(v);
		}
	} MaxPos;
	struct MinPos_impl {
		template <class T> friend auto operator|(T v, const MinPos_impl& c) {
			return min_element(begin(v), end(v)) - begin(v);
		}
	} MinPos;
	struct MaxBy_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				auto max_it = begin(v);
				auto max_val = f(*max_it);
				for (auto it = next(begin(v)); it != end(v); ++it) {
					if (auto val = f(*it); max_val < val) {
						max_it = it;
						max_val = val;
					}
				}
				return *max_it;
			});
		}
	} MaxBy;
	struct MinBy_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				auto min_it = begin(v);
				auto min_val = f(*min_it);
				for (auto it = next(begin(v)); it != end(v); ++it) {
					if (auto val = f(*it); min_val > val) {
						min_it = it;
						min_val = val;
					}
				}
				return *min_it;
			});
		}
	} MinBy;
	struct MaxOf_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				auto max_val = f(*begin(v));
				for (auto it = next(begin(v)); it != end(v); ++it) {
					if (auto val = f(*it); max_val < val) {
						max_val = val;
					}
				}
				return max_val;
			});
		}
	} MaxOf;
	struct MinOf_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				auto min_val = f(*begin(v));
				for (auto it = next(begin(v)); it != end(v); ++it) {
					if (auto val = f(*it); min_val > val) {
						min_val = val;
					}
				}
				return min_val;
			});
		}
	} MinOf;
	struct Count_impl {
		template <class V> auto operator()(const V& val) {
			return Callable([&](auto v) {
				return count(begin(v), end(v), val);
			});
		}
	} Count;
	struct CountIf_impl {
		template <class F> auto operator()(const F& f) {
			return Callable([&](auto v) {
				return count_if(begin(v), end(v), f);
			});
		}
	} CountIf;
	struct Index_impl {
		template <class V> auto operator()(const V& val) {
			return Callable([&](auto v) -> optional<int> {
				auto res = find(begin(v), end(v), val);
				return res != end(v) ? optional(res - begin(v)) : nullopt;
			});
		}
	} Index;
	struct IndexIf_impl {
		template <class F> auto operator()(const F& f) {
			return Callable([&](auto v) -> optional<int> {
				auto res = find_if(begin(v), end(v), f);
				return res != end(v) ? optional(res - begin(v)) : nullopt;
			});
		}
	} IndexIf;
	struct FindIf_impl {
		template <class F> auto operator()(const F& f) {
			return Callable([&](auto v) -> optional<typename decltype(v)::value_type> {
				auto res = find_if(begin(v), end(v), f);
				return res != end(v) ? optional(*res) : nullopt;
			});
		}
	} FindIf;
	struct Sum_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				return accumulate(next(begin(v)), end(v), f(*begin(v)), [&](const auto& a, const auto& b) {
					return a + f(b);
				});
			});
		}
		template <class T> friend auto operator|(T v, const Sum_impl& c) {
			return accumulate(begin(v), end(v), typename T::value_type());
		}
	} Sum;
	struct Includes {
		template <class V> auto operator()(const V& val) {
			return Callable([&](auto v) {
				return find(begin(v), end(v), val) != end(v);
			});
		}
	} Includes;
	struct IncludesIf_impl {
		template <class F> auto operator()(const F& f) {
			return Callable([&](auto v) {
				return find_if(begin(v), end(v), f) != end(v);
			});
		}
	} IncludesIf;
	struct RemoveIf_impl {
		template <class F> auto operator()(const F& f) {
			return Callable([&](auto v) {
				v.erase(remove_if(begin(v), end(v), f), end(v));
				return v;
			});
		}
	} RemoveIf;
	struct Each_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				for (const auto& i : v) {
					f(i);
				}
			});
		}
	} Each;
	struct Select_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				using value_type = typename decltype(v)::value_type;
				vector<value_type> res;
				for (const auto& i : v) {
					if (f(i)) res.push_back(i);
				}
				return res;
			});
		}
	} Select;
	struct Map_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				using result_type = invoke_result_t<F, typename decltype(v)::value_type>;
				vector<result_type> res;
				res.reserve(size(v));
				for (const auto& i : v) {
					res.push_back(f(i));
				}
				return res;
			});
		}
	} Map;
	struct Indexed_impl {
		template <class T> friend auto operator|(const T& v, Indexed_impl& c) {
			using value_type = typename T::value_type;
			vector<pair<value_type, int>> res;
			res.reserve(size(v));
			int index = 0;
			for (const auto& i : v) {
				res.emplace_back(i, index++);
			}
			return res;
		}
	} Indexed;
	struct AllOf_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				for (const auto& i : v) {
					if (!f(i)) return false;
				}
				return true;
			});
		}
	} AllOf;
	struct AnyOf_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				for (const auto& i : v) {
					if (f(i)) return true;
				}
				return false;
			});
		}
	} AnyOf;
	struct NoneOf_impl {
		template <class F> auto operator()(F&& f) {
			return Callable([&](auto v) {
				for (const auto& i : v) {
					if (f(i)) return false;
				}
				return true;
			});
		}
	} NoneOf;
};  // namespace Ruby
using namespace Ruby;

// --- functions --- //
namespace Functions {
	template <class T, class U> inline int Lower(const T& a, const U& v) {
		return lower_bound(all(a), v) - a.begin();
	}
	template <class T, class U> inline int Upper(const T& a, const U& v) {
		return upper_bound(all(a), v) - a.begin();
	}
	template <class T> inline auto Slice(const T& v, size_t i, size_t len) {
		return i < v.size() ? T(v.begin() + i, v.begin() + min(i + len, v.size())) : T();
	}
	inline auto operator*(string s, size_t n) {
		string res;
		for (size_t i = 0; i < n; ++i) res += s;
		return res;
	}
	template <class T> inline auto& operator<<(vector<T>& v, const vector<T>& v2) {
		v.insert(v.end(), all(v2));
		return v;
	}
	template <class T> inline T Ceil(T n, T m) {
		return (n + m - 1) / m;
	}
	template <class T> inline T Ceil2(T n, T m) {
		return Ceil(n, m) * m;
	}
	template <class T> inline T Tri(T n) {
		return (n & 1) ? (n + 1) / 2 * n : n / 2 * (n + 1);
	}
	template <class T> inline T nC2(T n) {
		return (n & 1) ? (n - 1) / 2 * n : n / 2 * (n - 1);
	}
	template <class T> inline T Mid(const T& l, const T& r) {
		return l + (r - l) / 2;
	}
	template <class T> inline bool chmax(T& a, const T& b) {
		if (a < b) {
			a = b;
			return true;
		}
		return false;
	}
	template <class T> inline bool chmin(T& a, const T& b) {
		if (a > b) {
			a = b;
			return true;
		}
		return false;
	}
	template <class T> inline bool inRange(const T& v, const T& min, const T& max) {
		return min <= v && v < max;
	}
	template <class T> inline bool isSquere(T n) {
		T s = sqrt(n);
		return s * s == n || (s + 1) * (s + 1) == n;
	}
	template <class T = long long> inline T BIT(int b) {
		return T(1) << b;
	}
	template <class T, class U = typename T::value_type> inline U Gcdv(const T& v) {
		return accumulate(next(v.begin()), v.end(), U(*v.begin()), gcd<U, U>);
	}
	template <class T, class U = typename T::value_type> inline U Lcmv(const T& v) {
		return accumulate(next(v.begin()), v.end(), U(*v.begin()), lcm<U, U>);
	}
	template <class T> inline T Pow(T a, T n) {
		T r = 1;
		while (n > 0) {
			if (n & 1) r *= a;
			a *= a;
			n /= 2;
		}
		return r;
	}
	template <class T> inline T Powmod(T a, T n, T m = MOD) {
		T r = 1;
		while (n > 0) {
			if (n & 1)
				r = r * a % m, n--;
			else
				a = a * a % m, n /= 2;
		}
		return r;
	}
}  // namespace Functions
using namespace Functions;

// --- dump --- //
#if __has_include("/home/yuruhiya/contest/library/dump.hpp")
#include "/home/yuruhiya/contest/library/dump.hpp"
#else
#define dump(...) ((void)0)
#endif

// ---------------------------------------------------------------- //

#include <atcoder/all>

struct Point {
	static int H, W;
	static const vector<Point> d;
	static void set_range(int _H, int _W) {
		H = _H;
		W = _W;
	}
	static constexpr Point zero() {
		return {0, 0};
	}
	static constexpr Point one() {
		return {1, 1};
	}
	int x, y;
	constexpr Point() : x(0), y(0) {}
	constexpr Point(int _x, int _y) : x(_x), y(_y) {}
	constexpr Point(const pair<int, int>& xy) : x(xy.first), y(xy.second) {}
	Point(int n) : x(n % W), y(n / W) {}
	constexpr Point operator+() const {
		return *this;
	}
	constexpr Point operator-() const {
		return {-x, -y};
	}
	constexpr Point operator+(const Point& p) const {
		return Point(*this) += p;
	}
	constexpr Point operator-(const Point& p) const {
		return Point(*this) -= p;
	}
	constexpr Point operator*(const Point& p) const {
		return Point(*this) *= p;
	}
	constexpr Point operator/(const Point& p) const {
		return Point(*this) /= p;
	}
	constexpr Point operator%(const Point& p) const {
		return Point(*this) %= p;
	}
	constexpr Point operator+(int n) const {
		return Point(*this) += n;
	}
	constexpr Point operator-(int n) const {
		return Point(*this) -= n;
	}
	constexpr Point operator*(int n) const {
		return Point(*this) *= n;
	}
	constexpr Point operator/(int n) const {
		return Point(*this) /= n;
	}
	constexpr Point operator%(int n) const {
		return Point(*this) %= n;
	}
	constexpr Point& operator+=(const Point& p) {
		x += p.x;
		y += p.y;
		return *this;
	}
	constexpr Point& operator-=(const Point& p) {
		x -= p.x;
		y -= p.y;
		return *this;
	}
	constexpr Point& operator*=(const Point& p) {
		x *= p.x;
		y *= p.y;
		return *this;
	}
	constexpr Point& operator/=(const Point& p) {
		x /= p.x;
		y /= p.y;
		return *this;
	}
	constexpr Point& operator%=(const Point& p) {
		x %= p.x;
		y %= p.y;
		return *this;
	}
	constexpr Point& operator+=(int n) {
		x += n;
		y += n;
		return *this;
	}
	constexpr Point& operator-=(int n) {
		x -= n;
		y -= n;
		return *this;
	}
	constexpr Point& operator*=(int n) {
		x *= n;
		y *= n;
		return *this;
	}
	constexpr Point& operator/=(int n) {
		x /= n;
		y /= n;
		return *this;
	}
	constexpr Point& operator%=(int n) {
		x %= n;
		y %= n;
		return *this;
	}
	constexpr bool operator==(const Point& p) const {
		return x == p.x && y == p.y;
	}
	constexpr bool operator!=(const Point& p) const {
		return x != p.x || y != p.y;
	}
	bool operator<(const Point& p) const {
		return to_i() < p.to_i();
	}
	bool operator<=(const Point& p) const {
		return to_i() <= p.to_i();
	}
	bool operator>(const Point& p) const {
		return to_i() > p.to_i();
	}
	bool operator>=(const Point& p) const {
		return to_i() >= p.to_i();
	}
	constexpr int operator[](int i) const {
		return i == 0 ? x : i == 1 ? y : 0;
	}
	bool in_range() const {
		return 0 <= x && x < W && 0 <= y && y < H;
	}
	int to_i() const {
		return x + y * W;
	}
	constexpr pair<int, int> to_pair() const {
		return {x, y};
	}
	int dist(const Point& p) const {
		return std::abs(x - p.x) + std::abs(y - p.y);
	}
	int dist_square(const Point& p) const {
		return (x - p.x) * (x - p.x) + (y - p.y) * (y - p.y);
	}
	Point abs(const Point& p) const {
		return {std::abs(x - p.x), std::abs(y - p.y)};
	}
	Point abs() const {
		return {std::abs(x), std::abs(y)};
	}
	Point& swap() {
		std::swap(x, y);
		return *this;
	}

	template <class It> vector<Point> enum_adjanect(It first, It last) const {
		vector<Point> res;
		for (; first != last; ++first) {
			res.push_back(operator+(*first));
		}
		return res;
	}
	template <class It> vector<Point> enum_adj_in_range(It first, It last) const {
		vector<Point> res;
		for (; first != last; ++first) {
			auto p = operator+(*first);
			if (p.in_range()) res.push_back(p);
		}
		return res;
	}
	vector<Point> adjacent4() const {
		return enum_adjanect(d.begin(), d.begin() + 4);
	}
	vector<Point> adjacent8() const {
		return enum_adjanect(d.begin(), d.end());
	}
	vector<Point> adj4_in_range() const {
		return enum_adj_in_range(d.begin(), d.begin() + 4);
	}
	vector<Point> adj8_in_range() const {
		return enum_adj_in_range(d.begin(), d.end());
	}
	constexpr Point left() const {
		return {x - 1, y};
	}
	constexpr Point right() const {
		return {x + 1, y};
	}
	constexpr Point up() const {
		return {x, y - 1};
	}
	constexpr Point down() const {
		return {x, y + 1};
	}
	constexpr Point moved(char c) const {
		return Point(*this).move(c);
	}
	constexpr Point& move(char c) {
		switch (c) {
			case 'L':
			case 'l':
			case 'W':
			case '>': x--; break;
			case 'R':
			case 'r':
			case 'E':
			case '<': x++; break;
			case 'U':
			case 'u':
			case 'N':
			case '^': y--; break;
			case 'D':
			case 'd':
			case 'S':
			case 'v': y++; break;
		}
		return *this;
	}
	constexpr Point rotate90() {
		return {y, -x};
	}
	constexpr Point rotate180() {
		return {-x, -y};
	}
	constexpr Point rotate270() {
		return {-y, x};
	}
	friend ostream& operator<<(ostream& os, const Point& p) {
		return os << '(' << p.x << ", " << p.y << ')';
	}
	friend istream& operator>>(istream& is, Point& p) {
		return is >> p.x >> p.y;
	}
};
int Point::H, Point::W;
const vector<Point> Point::d{{0, 1}, {1, 0}, {0, -1}, {-1, 0}, {1, 1}, {-1, -1}, {1, -1}, {-1, 1}};

int main() {
	ini(h, w);
	VS grid = in[h];
	Point::set_range(h, w);

	atcoder::mf_graph<int> g(h * w + 2);
	int s = h * w, t = h * w + 1;
	rep(i, h) rep(j, w) {
		if (grid[i][j] == '#') continue;
		Point p(j, i);
		if ((i + j) % 2 == 0) {
			g.add_edge(s, p.to_i(), 1);
			for (auto p2 : p.adj4_in_range()) {
				if (grid[p2.y][p2.x] == '.') {
					g.add_edge(p.to_i(), p2.to_i(), 1);
				}
			}
		} else {
			g.add_edge(p.to_i(), t, 1);
		}
	}

	out(g.flow(s, t));
	for (auto e : g.edges()) {
		if (e.from != s && e.to != t && e.flow != 0) {
			Point p1(e.from), p2(e.to);
			dump(p1, p2);
			if (p1.x == p2.x) {
				grid[min(p1.y, p2.y)][p1.x] = 'v';
				grid[max(p1.y, p2.y)][p1.x] = '^';
			} else {
				grid[p1.y][min(p1.x, p2.x)] = '>';
				grid[p1.y][max(p1.x, p2.x)] = '<';
			}
		}
	}
	out.set(end_line)(grid);
}

Measured memory_bytes: 5416