#define _USE_MATH_DEFINES
#define _CRT_SECURE_NO_WARNINGS
#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 = (int)1e9;
constexpr ll inf_ll = (ll)1e18, MOD = 1000000007;
constexpr LD 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 get_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;
	}
#define input(T) Input::in<T>()
#define INT input(int)
#define LL input(ll)
#define STR input(string)
#define inputs(T, ...) T __VA_ARGS__; in(__VA_ARGS__)
#define ini(...) inputs(int, __VA_ARGS__)
#define inl(...) inputs(ll, __VA_ARGS__)
#define ins(...) inputs(string, __VA_ARGS__)
}in;

// --- 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 {
		if (v < 0)putchar_unlocked('-'), v = -v;
		char b[10]; int i = 0;
		while (v)b[i++] = '0' + v % 10, v /= 10;
		if (!i)b[i++] = '0';
		while (i--)putchar_unlocked(b[i]);
	}
	void p(ll v)const {
		if (v < 0)putchar_unlocked('-'), v = -v;
		char b[20]; int i = 0;
		while (v)b[i++] = '0' + v % 10, v /= 10;
		if (!i)b[i++] = '0';
		while (i--)putchar_unlocked(b[i]);
	}
	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]); }
	}
	template<size_t N = 0, class T>void output_tuple(T& v)const {
		if constexpr (N < tuple_size_v<T>) {
			if constexpr (N != 0) {
				p(D.d);
			}
			p(get<N>(v));
			output_tuple<N + 1>(v);
		}
	}
	template<class...T>void p(const tuple<T...>& v)const {
		output_tuple(v);
	}
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 count()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 count()const {
		return be.count();
	}
	constexpr T step()const {
		return be.step();
	}
	constexpr T sum()const {
		return start() * count() + step() * (count() * (count() - 1) / 2);
	}
	operator vector<T>()const {
		return as_vector();
	}
	template<class F>void each(const F& f)const {
		for (T i : *this)f(i);
	}
	auto as_vector()const {
		vector<T> res; res.reserve(count());
		each([&](T i) {res.push_back(i); });
		return res;
	}
	template<class F, class U = invoke_result_t<F, T>>auto map(const F& f)const {
		vector<U> res; res.reserve(count());
		each([&](T i) {res.push_back(f(i)); });
		return res;
	}
	template<class F>auto select(const F& f)const {
		vector<T> res;
		each([&](T i) {if (f(i))res.push_back(i); });
		return res;
	}
	template<class F>int count_if(const F& f)const {
		int res = 0;
		each([&](T i) {if (f(i))++res; });
		return res;
	}
	template<class F>optional<T> find_if(const F& f)const {
		for (T i : *this)if (f(i))return i;
		return nullopt;
	}
	template<class F>auto max_by(const F& f)const {
		auto v = map(f); return *max_element(v.begin(), v.end());
	}
	template<class F>auto min_by(const F& f)const {
		auto v = map(f); return *min_element(v.begin(), v.end());
	}
	template<class F>bool all_of(const F& f)const {
		for (T i : *this)if (!f(i))return false;
		return true;
	}
	template<class F>bool any_of(const F& f)const {
		for (T i : *this)if (f(i))return true;
		return false;
	}
	template<class F, class U = invoke_result_t<F, T>>auto sum(const F& f)const {
		U res = 0; each([&](T i) {res += static_cast<U>(f(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);
}

// --- functions --- //
inline namespace {
	template<class T>inline void Sort(T& a) {
		sort(all(a));
	}
	template<class T>inline void RSort(T& a) {
		sort(rall(a));
	}
	template<class T, class F>inline void Sort(T& a, const F& f) {
		sort(all(a), f);
	}
	template<class T, class F>inline void RSort(T& a, const F& f) {
		sort(rall(a), f);
	}
	template<class T>inline T Sorted(T a) {
		Sort(a); return a;
	}
	template<class T>inline T RSorted(T a) {
		RSort(a); return a;
	}
	template<class T, class F>inline T Sorted(T& a, const F& f) {
		Sort(a, f); return a;
	}
	template<class T, class F>inline T RSorted(T& a, const F& f) {
		RSort(a, f); return a;
	}
	template<class T, class F>inline void SortBy(T& a, const F& f) {
		sort(all(a), [&](const auto& x, const auto& y) {return f(x) < f(y); });
	}
	template<class T, class F>inline void RSortBy(T& a, const F& f) {
		sort(rall(a), [&](const auto& x, const auto& y) {return f(x) < f(y); });
	}
	template<class T>inline void Reverse(T& a) {
		reverse(all(a));
	}
	template<class T>inline void Unique(T& a) {
		a.erase(unique(all(a)), a.end());
	}
	template<class T>inline void Uniq(T& a) {
		Sort(a); Unique(a);
	}
	template<class T>inline void Rotate(T& a, int left) {
		rotate(a.begin(), a.begin() + left, a.end());
	}
	template<class T>inline T Reversed(T a) {
		Reverse(a); return a;
	}
	template<class T>inline T Uniqued(T a) {
		Unique(a); return a;
	}
	template<class T>inline T Uniqed(T a) {
		Uniq(a); return a;
	}
	template<class T>inline T Rotated(T a, int left) {
		Rotate(a, left); return a;
	}
	template<class T>inline auto Max(const T& a) {
		return *max_element(all(a));
	}
	template<class T>inline auto Min(const T& a) {
		return *min_element(all(a));
	}
	template<class T>inline int MaxPos(const T& a) {
		return max_element(all(a)) - a.begin();
	}
	template<class T>inline int MinPos(const T& a) {
		return min_element(all(a)) - a.begin();
	}
	template<class T, class F>inline auto MaxBy(const T& a, const F& f) {
		return *max_element(all(a), [&](const auto& x, const auto& y) {return f(x) < f(y); });
	}
	template<class T, class F>inline auto MinBy(const T& a, const F& f) {
		return *min_element(all(a), [&](const auto& x, const auto& y) {return f(x) < f(y); });
	}
	template<class T, class U>inline int Count(const T& a, const U& v) {
		return count(all(a), v);
	}
	template<class T, class F>inline int CountIf(const T& a, const F& f) {
		return count_if(all(a), f);
	}
	template<class T, class U>inline int Find(const T& a, const U& v) {
		return find(all(a), v) - a.begin();
	}
	template<class T, class F>inline int FindIf(const T& a, const F& f) {
		return find_if(all(a), f) - a.begin();
	}
	template<class T, class U = typename T::value_type>inline U Sum(const T& a) {
		return accumulate(all(a), U());
	}
	template<class T, class U>inline bool Includes(const T& a, const U& v) {
		return find(all(a), v) != a.end();
	}
	template<class T, class F>inline auto Sum(const T& v, const F& f) {
		return accumulate(next(v.begin()), v.end(), f(*v.begin()), [&](auto a, auto b) {return a + f(b); });
	}
	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, class F>inline void RemoveIf(T& a, const F& f) {
		a.erase(remove_if(all(a), f), a.end());
	}
	template<class F>inline auto Vector(size_t size, const F& f) {
		vector<invoke_result_t<F, size_t>> res(size);
		for (size_t i = 0; i < size; ++i)res[i] = f(i);
		return res;
	}
	template<class T>inline auto Grid(size_t h, size_t w, const T& v = T()) {
		return vector<vector<T>>(h, vector<T>(w, v));
	}
	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();
	}
	template<class T, class F>inline auto Each(T& v, F&& f) {
		for (auto& i : v)f(i);
	}
	template<class T, class F>inline auto Select(const T& v, const F& f) {
		T res;
		for (const auto& e : v)if (f(e))res.push_back(e);
		return res;
	}
	template<class T, class F>inline auto Map(const T& v, F&& f) {
		vector<invoke_result_t<F, typename T::value_type>> res(v.size());
		size_t i = 0; for (const auto& e : v)res[i++] = f(e);
		return res;
	}
	template<class T, class F>inline auto MapIndex(const T& v, const F& f) {
		vector<invoke_result_t<F, size_t, typename T::value_type>> res(v.size());
		size_t i = 0; for (auto it = v.begin(); it != v.end(); ++it, ++i)res[i] = f(i, *it);
		return res;
	}
	template<class T, class F>inline auto TrueIndex(const T& v, const F& f) {
		vector<size_t> res;
		for (size_t i = 0; i < v.size(); ++i)if (f(v[i]))res.push_back(i);
		return res;
	}
	template<class T, class U = typename T::value_type>inline auto Indexed(const T& v) {
		vector<pair<U, int>> res(v.size());
		for (int i = 0; i < (int)v.size(); ++i)res[i] = make_pair(static_cast<U>(v[i]), i);
		return res;
	}
	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;
	}
	inline int pop_count(int n) {
		return bitset<32>(n).count();
	}
	inline int pop_count(ll n) {
		return bitset<64>(n).count();
	}
	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 = ll>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;
	}
}

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

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

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);
	}
	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;
	}
	vector<Point> adjacent4()const {
		vector<Point> v(4); for (int i = 0; i < 4; ++i)v[i] = *this + d[i]; return v;
	}
	vector<Point> adjacent8()const {
		vector<Point> v(8); for (int i = 0; i < 8; ++i)v[i] = *this + d[i]; return v;
	}
	vector<Point> adj4_in_range()const {
		vector<Point> v; for (const auto& P : adjacent4())if (P.in_range())v.push_back(P); return v;
	}
	vector<Point> adj8_in_range()const {
		vector<Point> v; for (const auto& P : adjacent8())if (P.in_range())v.push_back(P); return v;
	}
	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} };

template<class T>class SegmentTree {
	using Func = function<T(T, T)>;
	int N; T init;
	vector<T> A;
	Func F;
	static int ceil2(int n) { int m = 1; while (m < n)m *= 2; return m; }

	template<class C>int ff(int a, int b, const C& c, int k, int l, int r)const {
		if (!c(A[k]) || r <= a || b <= l) {
			return -1;
		} else if (k >= N) {
			return k - N;
		} else {
			int lv = ff(a, b, c, 2 * k, l, (l + r) / 2);
			if (lv != -1)return lv;
			return ff(a, b, c, 2 * k + 1, (l + r) / 2, r);
		}
	}
	template<class C>int fl(int a, int b, const C& c, int k, int l, int r)const {
		if (!c(A[k]) || r <= a || b <= l) {
			return -1;
		} else if (k >= N) {
			return k - N;
		} else {
			int rv = fl(a, b, c, 2 * k + 1, (l + r) / 2, r);
			if (rv != -1)return rv;
			return fl(a, b, c, 2 * k, l, (l + r) / 2);
		}
	}

public:
	SegmentTree(int n, const T& init_, const Func& f)
		: N(ceil2(n)), init(init_), A(N * 2, init), F(f) {}
	SegmentTree(const vector<T>& vec, const T& init_, const Func& f)
		: init(init_), F(f) {
		build(vec);
	}
	void build(const vector<T>& v) {
		N = ceil2(v.size());
		A.assign(N * 2, init);
		for (int i = 0; i < v.size(); ++i) {
			A[i + N] = v[i];
		}
		for (int i = N - 1; i > 0; --i) {
			A[i] = F(A[i * 2], A[i * 2 + 1]);
		}
	}
	void update(int i, const T& x) {
		assert(0 <= i && i < N);
		A[i += N] = x;
		while (i >>= 1) {
			A[i] = F(A[i * 2], A[i * 2 + 1]);
		}
	}
	T operator[](int i)const {
		assert(0 <= i && i < N);
		return A[i + N];
	}
	T operator()(int l, int r)const { // [l, r)
		assert(0 <= l && l < r&& r <= N);
		T L = init, R = init;
		for (l += N, r += N; l < r; l >>= 1, r >>= 1) {
			if (l & 1)L = F(L, A[l++]);
			if (r & 1)R = F(A[--r], R);
		}
		return F(L, R);
	}
	template<class C>int find_first(int l, int r, const C& c)const {
		return ff(l, r, c, 1, 0, N);
	}
	template<class C>int find_last(int l, int r, const C& c)const {
		return fl(l, r, c, 1, 0, N);
	}
	vector<T> as_vector()const {
		vector<T> res(N);
		for (int i = 0; i < N; ++i)res[i] = operator[](i);
		return res;
	}
};

int main() {
	ini(h, w, k);
	Point start = in, goal = in;
	start -= 1; goal -= 1;
	start.swap(); goal.swap();
	VS s = in[h];
	Point::set_range(h, w);

	vector<SegmentTree<int>> seg_y(h, SegmentTree<int>(w, 0, [](int x, int y) {return min(x, y); }));
	vector<SegmentTree<int>> seg_x(w, SegmentTree<int>(h, 0, [](int x, int y) {return min(x, y); }));
	const Point nil(-1, -1);

	auto print_seg = [&] {
		VVI dp(h, VI(w));
		rep(i, h)rep(j, w) {
			int Y = seg_y[i][j];
			int X = seg_x[j][i];
			assert(Y == X);
			dp[i][j] = Y;
		}
		return dp;
	};

	auto set_1 = [&](int i, int j) {
		seg_y[i].update(j, 1);
		seg_x[j].update(i, 1);
	};

	queue<Point> q;
	q.push(start);
	VVI vis(h, VI(w, inf));
	vis[start.y][start.x] = 0;
	set_1(start.y, start.x);
	while (q.size()) {
		Point f = q.front(); q.pop();
		int dist = vis[f.y][f.x];
		if (f == goal) {
			break;
		}
		{
			Point d(1, 0), now = f;
			auto calc_next = [&](const Point& p) {
				int res = seg_y[p.y].find_first(p.x + 1, w, [](int x) {return x <= 0; });
				dump(res);
				if (res == -1) {
					return nil;
				} else {
					return Point(res, p.y);
				}
			};
			dump(d, now);
			for (now = calc_next(now); now != nil && s[now.y][now.x] != '@' && f.dist(now) <= k; now = calc_next(now)) {
				dump(now);
				assert(vis[now.y][now.x] == inf);
				vis[now.y][now.x] = dist + 1;
				set_1(now.y, now.x);
				q.push(now);
			}
		}
		{
			Point d(-1, 0), now = f;
			auto calc_next = [&](const Point& p) {
				int res = seg_y[p.y].find_last(0, p.x, [](int x) {return x <= 0; });
				dump(res);
				if (res == -1) {
					return nil;
				} else {
					return Point(res, p.y);
				}
			};
			dump(d, now);
			for (now = calc_next(now); now != nil && s[now.y][now.x] != '@' && f.dist(now) <= k; now = calc_next(now)) {
				dump(now);
				assert(vis[now.y][now.x] == inf);
				vis[now.y][now.x] = dist + 1;
				set_1(now.y, now.x);
				q.push(now);
			}
		}
		{
			Point d(0, 1), now = f;
			auto calc_next = [&](const Point& p) {
				int res = seg_x[p.x].find_first(p.y + 1, h, [](int x) {return x <= 0; });
				dump(res);
				if (res == -1) {
					return nil;
				} else {
					return Point(p.x, res);
				}
			};
			dump(d, now);
			for (now = calc_next(now); now != nil && s[now.y][now.x] != '@' && f.dist(now) <= k; now = calc_next(now)) {
				dump(now);
				assert(vis[now.y][now.x] == inf);
				vis[now.y][now.x] = dist + 1;
				set_1(now.y, now.x);
				q.push(now);
			}
		}
		{
			Point d(0, -1), now = f;
			auto calc_next = [&](const Point& p) {
				int res = seg_x[p.x].find_last(0, p.y, [](int x) {return x <= 0; });
				dump(res);
				if (res == -1) {
					return nil;
				} else {
					return Point(p.x, res);
				}
			};
			dump(d, now);
			for (now = calc_next(now); now != nil && s[now.y][now.x] != '@' && f.dist(now) <= k; now = calc_next(now)) {
				dump(now);
				assert(vis[now.y][now.x] == inf);
				vis[now.y][now.x] = dist + 1;
				set_1(now.y, now.x);
				q.push(now);
			}
		}
		dump(vis, print_seg());
	}

	int ans = vis[goal.y][goal.x];
	out(ans != inf ? ans : -1);
}

Measured memory_bytes: 113988