#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using grid = vector<vector<char>>;
constexpr ll MOD = 1000000007;
//constexpr ll MOD = 998244353;
constexpr ll INF = 1050000000;
constexpr ll LONGINF = 1050000000000000000;
struct all_init {
	all_init() {
		cout.tie(nullptr);
		cin.tie(nullptr);
		ios::sync_with_stdio(false);
		cout << fixed << setprecision(11);
	};
} ALL_INIT;
struct edge {
 
	int from, to;
	ll cost;
	ll capa;
 
	edge(int s, int d) : from(s), to(d) {
		cost = 0;
		capa = 0;
	}
	edge(int s, int d, ll w) : from(s), to(d), cost(w) { capa = 0; }
	edge(int s, int d, ll x, ll y) : from(s), to(d), cost(x), capa(y) {}
 
	bool operator<(const edge& x) const { return cost < x.cost; }
};
using graph = vector<vector<edge>>;
 
#define CIN(vector_array_etc, n)         \
  for (int loop = 0; loop < n; loop++) { \
    cin >> vector_array_etc[loop];       \
  }
#define COUT(vector_array_etc, n)                                   \
  for (int LOOP = 0; LOOP < n; LOOP++) {                            \
    cout << vector_array_etc[LOOP] << (LOOP == n - 1 ? '\n' : ' '); \
  }
#define VC(Type_name) vector<Type_name>
#define SORT(vector_etc) sort(vector_etc.begin(), vector_etc.end())
#define ALL(vec_etc) vec_etc.begin(), vec_etc.end()
#define VCVC(Type_name) vector<vector<Type_name>>  
#define WARSHALL vector<vector<ll>> g(n, vector<ll>(n, LONGINF))
#define endl '\n'

 
template <class T>
bool chmax(T& a, const T& b) {
	if (a < b) {
		a = b;
		return true;
	}
	return false;
}
template <class T>
bool chmin(T& a, const T& b) {
	if (b < a) {
		a = b;
		return true;
	}
	return false;
}
template <typename T>
istream& operator>>(istream& is, vector<T>& Vec) {
	for (T& x : Vec) {
		is >> x;
	}
	return is;
}
template <typename V, typename H>
void resize(vector<V>& vec, const H head) {
	vec.resize(head);
}
template <typename V, typename H, typename... T>
void resize(vector<V>& vec, const H& head, const T... tail) {
	vec.resize(head);
	for (auto& v : vec) {
		resize(v, tail...);
	}
}
template <ll mod>
struct ModInt {
	long long val;
	constexpr ModInt(long long v = 0) noexcept : val(v% mod) {
		if (val < 0) v += mod;
	}
	constexpr int getmod() { return mod; }
	constexpr ModInt operator-() const noexcept { return val ? mod - val : 0; }
	constexpr ModInt operator+(const ModInt& r) const noexcept {
		return ModInt(*this) += r;
	}
	constexpr ModInt operator-(const ModInt& r) const noexcept {
		return ModInt(*this) -= r;
	}
	constexpr ModInt operator*(const ModInt& r) const noexcept {
		return ModInt(*this) *= r;
	}
	constexpr ModInt operator/(const ModInt& r) const noexcept {
		return ModInt(*this) /= r;
	}
	constexpr ModInt& operator+=(const ModInt& r) noexcept {
		val += r.val;
		if (val >= mod) val -= mod;
		return *this;
	}
	constexpr ModInt& operator-=(const ModInt& r) noexcept {
		val -= r.val;
		if (val < 0) val += mod;
		return *this;
	}
	constexpr ModInt& operator*=(const ModInt& r) noexcept {
		val = val * r.val % mod;
		return *this;
	}
	constexpr ModInt& operator/=(const ModInt& r) noexcept {
		long long a = r.val, b = mod, u = 1, v = 0;
		while (b) {
			long long t = a / b;
			a -= t * b;
			swap(a, b);
			u -= t * v;
			swap(u, v);
		}
		val = val * u % mod;
		if (val < 0) val += mod;
		return *this;
	}
	constexpr bool operator==(const ModInt& r) const noexcept {
		return this->val == r.val;
	}
	constexpr bool operator!=(const ModInt& r) const noexcept {
		return this->val != r.val;
	}
	friend ostream& operator<<(ostream& os, const ModInt<mod>& x) noexcept {
		return os << x.val;
	}
	friend istream& operator>>(istream& is, ModInt<mod>& x) noexcept {
		return is >> x.val;
	}
	friend constexpr ModInt<mod> modpow(const ModInt<mod>& a,
		long long n) noexcept {
		if (n == 0) return 1;
		auto t = modpow(a, n / 2);
		t = t * t;
		if (n & 1) t = t * a;
		return t;
	}
};
template <class T>
struct nCk {
	vector<T> fact_, inv_, finv_;
	constexpr nCk() {}
	constexpr nCk(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
		init(n);
	}
	constexpr void init(int n) noexcept {
		fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
		ll MOD = 1000000007;
		for (ll i = 2; i < n; i++) {
			fact_[i] = fact_[i - 1] * i;
			inv_[i] = -inv_[MOD % i] * (MOD / i);
			finv_[i] = finv_[i - 1] * inv_[i];
		}
	}
	constexpr T com(ll n, ll k) const noexcept {
		if (n < k || n < 0 || k < 0) return 0;
		return fact_[n] * finv_[k] * finv_[n - k];
	}
	constexpr T fact(int n) const noexcept {
		if (n < 0) return 0;
		return fact_[n];
	}
	constexpr T inv(int n) const noexcept {
		if (n < 0) return 0;
		return inv_[n];
	}
	constexpr T finv(int n) const noexcept {
		if (n < 0) return 0;
		return finv_[n];
	}
};
 
int dx[] = { 0, 1, -1, 0, 1, -1, 1, -1 };  // i<4:4way i<8:8way
int dy[] = { 1, 0, 0, -1, 1, -1, -1, 1 };
 
ll PowMod(ll n, ll k, ll mod) {
	ll r = 1;
 
	for (; k > 0; k >>= 1) {
		if (k & 1) {
			r = (r * n) % mod;
		}
		n = (n * n) % mod;
	}
	return r;
}
ll Gcd(ll a, ll b) {
	return b != 0 ? Gcd(b, a % b) : a;
}
ll Lcm(ll a, ll b) {
	return a / Gcd(a, b) * b;
}
vector<string> Split(string s, string t) {
	vector<string> v;
	int p = s.find(t);
	while (p != s.npos) {
		v.emplace_back(s.substr(0, p));
		s = s.substr(p + (int)t.size());
		p = s.find(t);
	}
	v.emplace_back(s);
	return v;
}
vector<int> Lis(const vector<int>& a) {
	#define Index_of(as, x) distance(as.begin(), lower_bound(as.begin(), as.end(),x))
//#define Index_of(as, x) \
  distance(as.begin(), upper_bound(as.begin(), as.end(), x))
	const int n = a.size();
	vector<int> A(n, INF);
	vector<int> id(n);
	for (int i = 0; i < n; ++i) {
		id[i] = Index_of(A, a[i]);
		A[id[i]] = a[i];
	}
	int m = *max_element(id.begin(), id.end());
	vector<int> b(m + 1);
	for (int i = n - 1; i >= 0; --i)
		if (id[i] == m) b[m--] = a[i];
	return b;
}
string ReplaceString(string s, string target, string replacestring) {
	string::size_type Pos(s.find(target));
 
	while (Pos != string::npos) {
		s.replace(Pos, target.length(), replacestring);
		Pos = s.find(target, Pos + replacestring.length());
	}
 
	return s;
}
string LcsAlphabeticalMinOrder(string a, string b) {
	if (a.size() < b.size()) {
		swap(a, b);
	}
 
	int n = a.size(), m = b.size();
 
	vector<string> dp(m + 1);
 
	for (int i = 0; i < n; i++) {
		vector<string> to(m + 1);
		for (int j = 0; j < m; j++) {
			if (a[i] == b[j]) {
				to[j + 1] = dp[j] + a[i];
			}
			else {
				if (to[j].size() > dp[j + 1].size()) {
					to[j + 1] = to[j];
				}
				else if (to[j].size() < dp[j + 1].size()) {
					to[j + 1] = dp[j + 1];
				}
				else if (to[j] < dp[j + 1]) {
					to[j + 1] = to[j];
				}
				else {
					to[j + 1] = dp[j + 1];
				}
			}
		}
		dp.swap(to);
	}
	return dp[m];
}
string Lcs(const string& s, const string& t) {
	int dp[3001][3001];
	int n = s.size();
	int m = t.size();
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			if (s[i - 1] == t[j - 1]) {
				dp[i][j] = dp[i - 1][j - 1] + 1;
			}
			else {
				dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
			}
		}
	}
	string ans = "";
	int i = s.size(), j = t.size();
	while (i > 0 && j > 0) {
		if (s[i - 1] == t[j - 1]) {
			ans += s[i - 1];
			i--;
			j--;
		}
		else if (dp[i - 1][j] >= dp[i][j - 1])
			i--;
		else
			j--;
	}
	reverse(ans.begin(), ans.end());
	return ans;
}
vector<int> LcsInteger(const vector<int>& a, const vector<int>& b) {
#define index_of(as, x) \
  distance(as.begin(), lower_bound(as.begin(), as.end(), x))
	struct node {
		int value;
		node* next;
		node(int value, node* next) : value(value), next(next) {}
	};
	const int n = a.size(), m = b.size();
	map<int, vector<int>> M;
	for (int j = m - 1; j >= 0; --j) M[b[j]].push_back(j);
	vector<int> xs(n + 1, INF);
	xs[0] = -INF;
	vector<node*> link(n + 1);
	for (int i = 0; i < n; ++i) {
		if (M.count(a[i])) {
			vector<int> ys = M[a[i]];
			for (int j = 0; j < (int)ys.size(); ++j) {
				int k = index_of(xs, ys[j]);
				xs[k] = ys[j];
				link[k] = new node(b[ys[j]], link[k - 1]);
			}
		}
	}
	vector<int> c;
	int l = index_of(xs, INF - 1) - 1;
	for (node* p = link[l]; p; p = p->next) c.push_back(p->value);
	reverse(c.begin(), c.end());
	return c;
}
bool IsPrime(ll n) {
	if (n < 2) return false;
	for (ll i = 2; i * i <= n; i++)
		if (!(n % i)) return false;
	return true;
}
vector<bool> Eratosthenes(int n) {
	vector<int> res;
	vector<bool> Prime(n + 1, true);
	Prime[0] = Prime[1] = false;
	for (int i = 2; i * i <= n; i++) {
		if (Prime[i]) {
			for (int j = 2; i * j <= n; j++) {
				Prime[i * j] = false;
			}
		}
	}
	for (int i = 2; i <= n; i++) {
		if (Prime[i]) {
			res.emplace_back(i);
		}
	}
	return Prime;
}
ll maxSubArraySum(vector<ll> a,int left,int right) {
	//[left,right)
	int size=a.size(); 
    ll max_so_far = -LONGINF, max_ending_here = 0; 
  
    for (int i = left; i < right; i++) { 
        max_ending_here = max_ending_here + a[i]; 
        if (max_so_far < max_ending_here) {
            max_so_far = max_ending_here; 
		}
  
        if (max_ending_here < 0) {
            max_ending_here = 0; 
		}
    } 
    return max_so_far; 
} 
ll MergeCount(vector<int>& a) {
	ll count = 0;
	int n = a.size();
	if (n > 1) {
		vector<int> b(a.begin(), a.begin() + n / 2);
		vector<int> c(a.begin() + n / 2, a.end());
		count += MergeCount(b);
		count += MergeCount(c);
		for (int i = 0, j = 0, k = 0; i < n; ++i)
			if (k == (int)c.size())
				a[i] = b[j++];
			else if (j == (int)b.size())
				a[i] = c[k++];
			else if (b[j] <= c[k])
				a[i] = b[j++];
			else {
				a[i] = c[k++];
				count += n / 2 - j;
			}
	}
	return count;
}
bool WarshallFloyd(vector<vector<ll>>& c) {
	int V = c.size();
	for (int i = 0; i < V; i++) {
		c[i][i] = 0;
	}
 
	for (int i = 0; i < V; i++) {
		for (int j = 0; j < V; j++) {
			for (int k = 0; k < V; k++) {
				if (c[j][k] > c[j][i] + c[i][k]) {
					if(c[j][i] != INF && c[i][k] != INF){
						c[j][k] = c[j][i] + c[i][k];
					}
				}
			}
		}
	}
 
	for (int i = 0; i < V; i++) {
		if (c[i][i] < 0) {
			return false;
		}
	}
 
	return true;
}
vector<ll> Dijkstra(int i, vector<vector<edge>> graph) {
	int n = graph.size();
	vector<ll> d(n, LONGINF);
	d[i] = 0;
	priority_queue<pair<ll, int>, vector<pair<ll, int>>, greater<pair<ll, int>>>
		q;
	q.push(make_pair(0, i));
	while (!q.empty()) {
		pair<ll, int> p = q.top();
		q.pop();
		int v = p.second;
		if (d[v] < p.first) {
			continue;
		}
		for (auto x : graph[v]) {
			if (d[x.to] > d[v] + x.cost) {
				d[x.to] = d[v] + x.cost;
				q.push(make_pair(d[x.to], x.to));
			}
		}
	}
	return d;
}
bool BellmanFord(int start, int V, int E, vector<edge> Edge, vector<ll>& d) {
	resize(d, V);
	fill(d.begin(), d.end(), LONGINF);
	d[start] = 0;
	vector<bool> t(V, false);
	for (int i = 0; i < V - 1; i++) {
		for (int j = 0; j < E; j++) {
			edge e = Edge[j];
			if (d[e.from] == LONGINF) {
				continue;
			}
			if (d[e.to] > d[e.from] + e.cost) {
				d[e.to] = d[e.from] + e.cost;
			}
		}
	}
	for (int i = 0; i < V; i++) {
		for (int j = 0; j < E; j++) {
			edge e = Edge[j];
			if (d[e.from] == LONGINF) {
				continue;
			}
			if (d[e.to] > d[e.from] + e.cost) {
				d[e.to] = d[e.from] + e.cost;
				t[e.to] = true;
				/*
				if (i == V - 1) {
						return false;
				}
				*/
			}
			if (t[e.from]) {
				t[e.to] = true;
			}
		}
	}
 
	if (t[V - 1]) {
		return false;
	}
 
	return true;
}
bool TopologicalSort(const vector<vector<edge>>& g, vector<int>& ans) {
	int n = g.size(), k = 0;
	vector<int> ord(n), in(n);
	for (auto& es : g) {
		for (auto& e : es) {
			in[e.to]++;
		}
	}
	queue<int> q;
	for (int i = 0; i < n; ++i) {
		if (in[i] == 0) q.push(i);
	}
	while (!q.empty()) {
		int v = q.front();
		q.pop();
		ord[k++] = v;
		for (auto& e : g[v]) {
			if (--in[e.to] == 0) q.push(e.to);
		}
	}
	ans = ord;
	if (*max_element(in.begin(), in.end()) == 0) {
		return true;
	}
	return false;
}
vector<int> ArticulationNode(const vector<vector<edge>>& g) {
	int n = g.size(), idx;
	vector<int> low(n), ord(n), art;
	function<void(int)> DFS = [&](int v) {
		low[v] = ord[v] = ++idx;
		for (auto& e : g[v]) {
			int w = e.to;
			if (ord[w] == 0) {
				DFS(w);
				low[v] = min(low[v], low[w]);
				if ((ord[v] == 1 && ord[w] != 2) || (ord[v] != 1 && low[w] >= ord[v])) {
					art.push_back(v);
				}
			}
			else {
				low[v] = min(low[v], ord[w]);
			}
		}
	};
	for (int u = 0; u < n; u++) {
		if (ord[u] == 0) {
			idx = 0;
			DFS(u);
		}
	}
 
	sort(art.begin(), art.end());
	art.erase(unique(art.begin(), art.end()),
		art.end());
 
	return art;
}
vector<vector<edge>> ToRootTree(const vector<vector<edge>>& g, int r) {
	int n = g.size();
	vector<vector<edge>> G(n);
	vector<int> ord(n, -1);
 
	queue<int> q;
 
	q.push(r);
	int k = 0;
 
	while (q.size()) {
		int u = q.front();
		q.pop();
 
		for (auto& e : g[u]) {
			int v = e.to;
			if (ord[v] == -1) {
				ord[v] = k;
				k++;
				q.push(v);
				G[u].emplace_back(e);
			}
		}
	}
 
	return G;
}
edge TreeDiameter(const vector<vector<edge>>& g) {
 
	int start = 0;
 
	static const auto bfs = [](const vector<vector<edge>>& g, int s,
		queue<int>& q, vector<ll>& dist) {
			while (!q.empty()) {
				q.pop();
			}
			q.push(s);
			int n = g.size();
			dist.assign(n, LONGINF);
			dist[s] = 0;
			while (q.size()) {
				int u = q.front();
				q.pop();
				for (auto& e : g[u]) {
					int v = e.to;
					if (dist[v] == LONGINF) {
						dist[v] = dist[u] + e.cost;
						q.push(v);
					}
				}
			}
			return dist;
	};
	vector<ll> dist;
	queue<int> q;
	bfs(g, start, q, dist);
	int n = g.size(), u = -1, v = -1;
	for (int i = 0; i < n; i++)
		if (dist[i] != LONGINF && (u == -1 || dist[i] > dist[u])) u = i;
	bfs(g, u, q, dist);
	for (int i = 0; i < n; i++)
		if (dist[i] != LONGINF && (v == -1 || dist[i] > dist[v])) v = i;
	ll d = dist[v];
	if (u > v) swap(u, v);
	return edge(u, v, d);
}
void add_edge(vector<vector<edge>>& g, int a, int b, ll cost, ll cap) {
	g[a].emplace_back(a, b, cost, cap);
	g[b].emplace_back(b, a, cost, cap);
}
pair<vector<int>, vector<edge>> bridge(const vector<vector<edge>>& g) {
	const int n = g.size();
	int idx = 0, s = 0, t = 0, k = 0;
	vector<int> ord(n, -1), onS(n), stk(n), roots(n), cmp(n);
	vector<edge> brdg;
	function<void(int, int)> dfs = [&](int v, int u) {
		ord[v] = idx++;
		stk[s++] = v;
		onS[v] = true;
		roots[t++] = v;
		for (auto& e : g[v]) {
			int w = e.to;
			if (ord[w] == -1) {
				dfs(w, v);
			}
			else if (u != w && onS[w]) {
				while (ord[roots[t - 1]] > ord[w]) {
					--t;
				}
			}
		}
		if (v == roots[t - 1]) {
			brdg.emplace_back(u, v, 0);
			while (1) {
				int w = stk[--s];
				onS[w] = false;
				cmp[w] = k;
				if (v == w) break;
			}
			--t;
			++k;
		}
	};
	for (int u = 0; u < n; u++) {
		if (ord[u] == -1) {
			dfs(u, n);
			brdg.pop_back();
		}
	}
	return make_pair(cmp, brdg);
}
 
class UnionFind {
private:
	std::vector<int> parent;
	std::vector<int> height;
	std::vector<int> m_size;
	int forest_num;
 
public:
	UnionFind(int size_) : parent(size_), height(size_, 0), m_size(size_, 1) {
		forest_num = size_;
		for (int i = 0; i < size_; ++i) parent[i] = i;
	}
	void init(int size_) {
		parent.resize(size_);
		height.resize(size_, 0);
		m_size.resize(size_, 1);
		forest_num = size_;
		for (int i = 0; i < size_; ++i) parent[i] = i;
	}
	int find(int x) {
		if (parent[x] == x) return x;
		return parent[x] = find(parent[x]);
	}
	void unite(int x, int y) {
		x = find(x);
		y = find(y);
		if (x == y) return;
		int t = size(x) + size(y);
		m_size[x] = m_size[y] = t;
		if (height[x] < height[y])
			parent[x] = y;
		else
			parent[y] = x;
		if (height[x] == height[y]) ++height[x];
		forest_num--;
	}
	bool same(int x, int y) { return find(x) == find(y); }
	int size(int x) {
		if (parent[x] == x) return m_size[x];
		return size(parent[x] = find(parent[x]));
	}
	int forest() { return forest_num; }
};
class Dinic {
private:
	int n, s, t;
	vector<int> level, prog, que;
	vector<vector<ll>> cap, flow;
	vector<vector<int>> g;
	ll inf;
 
public:
	Dinic(const vector<vector<edge>>& graph)
		: n(graph.size()),
		cap(n, vector<ll>(n)),
		flow(n, vector<ll>(n)),
		g(n, vector<int>()),
		inf(LONGINF) {
		for (int i = 0; i < n; i++) {
			for (auto& e : graph[i]) {
				int u = e.from, v = e.to;
				ll c = e.capa;
				cap[u][v] += c;
				cap[v][u] += c;
				flow[v][u] += c;
				g[u].push_back(v);
				g[v].push_back(u);
			}
		}
	}
	inline ll residue(int u, int v) { return cap[u][v] - flow[u][v]; }
	ll solve(int s_, int t_) {
		this->t = t_, this->s = s_;
		que.resize(n + 1);
		ll res = 0;
		while (levelize()) {
			prog.assign(n, 0);
			res += augment(s, inf);
		}
		return res;
	}
	bool levelize() {
		int l = 0, r = 0;
		level.assign(n, -1);
		level[s] = 0;
		que[r++] = s;
		while (l != r) {
			int v = que[l++];
			if (v == t) break;
			for (const int& d : g[v])
				if (level[d] == -1 && residue(v, d) != 0) {
					level[d] = level[v] + 1;
					que[r++] = d;
				}
		}
		return level[t] != -1;
	}
	ll augment(int v, ll lim) {
		ll res = 0;
		if (v == t) return lim;
		for (int& i = prog[v]; i < (int)g[v].size(); i++) {
			const int& d = g[v][i];
			if (residue(v, d) == 0 || level[v] >= level[d]) continue;
			const ll aug = augment(d, min(lim, residue(v, d)));
			flow[v][d] += aug;
			flow[d][v] -= aug;
			res += aug;
			lim -= aug;
			if (lim == 0) break;
		}
		return res;
	}
};
class MinimumCostFlow {
private:
	using Flow = ll;
	using Cost = ll;
	struct Edge {
		int d;
		Flow c, f;
		Cost w;
		int r, is_r;
		Edge(int d_, Flow c_, Flow f_, Cost w_, int r_, bool is_r_)
			: d(d_), c(c_), f(f_), w(w_), r(r_), is_r(is_r_) {}
	};
	int n;
	vector<vector<Edge>> g;
 
public:
	MinimumCostFlow(int n_) : n(n_), g(vector<vector<Edge>>(n_)) {}
 
	void add_edge(int src, int dst, Cost cost,Flow cap) {
		int rsrc = g[dst].size();
		int rdst = g[src].size();
		g[src].emplace_back(dst, cap, 0, cost, rsrc, false);
		g[dst].emplace_back(src, cap, cap, -cost, rdst, true);
	}
 
	Cost solve(int s, int t, Flow f) {
		Cost res = 0;
 
		vector<Cost> h(n + 10), dist(n);
		vector<int> prevv(n + 10), preve(n + 10);
 
		using pcv = pair<Cost, int>;
		priority_queue<pcv, vector<pcv>, greater<pcv>> q;
		fill(h.begin(), h.end(), 0);
		while (f > 0) {
			fill(dist.begin(), dist.end(), LONGINF);
			dist[s] = 0;
			q.emplace(0, s);
			while (q.size()) {
				Cost cd;
				int v;
				tie(cd, v) = q.top();
				q.pop();
				if (dist[v] < cd) continue;
				for (int i = 0; i < (int)(g[v].size()); ++i) {
					Edge& e = g[v][i];
					if (residue(e) == 0) continue;
					if (dist[e.d] + h[e.d] > cd + h[v] + e.w) {
						dist[e.d] = dist[v] + e.w + h[v] - h[e.d];
						prevv[e.d] = v;
						preve[e.d] = i;
						q.emplace(dist[e.d], e.d);
					}
				}
			}
 
			if (dist[t] == LONGINF) return -1;
 
			for (int i = 0; i < n; ++i) h[i] += dist[i];
			Flow d = f;
			for (int v = t; v != s; v = prevv[v]) {
				chmin(d, residue(g[prevv[v]][preve[v]]));
			}
			f -= d;
			res += d * h[t];
			for (int v = t; v != s; v = prevv[v]) {
				Edge& e = g[prevv[v]][preve[v]];
				e.f += d;
				g[v][e.r].f -= d;
			}
		}
		return res;
	}
 
	Flow residue(const Edge& e) { return e.c - e.f; }
 
	void show() {
		for (int i = 0; i < n; ++i) {
			for (int j = 0; j < (int)(g[i].size()); ++j) {
				Edge& e = g[i][j];
				if (e.is_r) continue;
				cout << i << "->" << e.d << "(flow:" << e.f << ")" << endl;
			}
		}
	}
};
class BipartiteMatching {
private:
	int V;
	vector<int> match;
	vector<bool> used;
	vector<vector<int>> g;
	vector<pair<int, int>> match_pair;
 
	bool dfs(int v) {
		used[v] = true;
		for (int i = 0; i < (int)g[v].size(); i++) {
			int u = g[v][i];
			int w = match[u];
			if (w < 0 || !used[w] && dfs(w)) {
				match[v] = u;
				match[u] = v;
				match_pair.emplace_back(make_pair(u, v));
				return true;
			}
		}
		return false;
	}
 
public:
	BipartiteMatching(int n) {
		V = n;
		resize(match, n);
		resize(used, n);
		resize(g, n);
	}
 
	void add_edge(int u, int v) {
		g[u].emplace_back(v);
		g[v].emplace_back(u);
	}
 
	int MatchingSolve() {
		int res = 0;
		fill(match.begin(), match.end(), -1);
 
		for (int v = 0; v < V; v++) {
			if (match[v] < 0) {
				fill(used.begin(), used.end(), false);
				if (dfs(v)) {
					res++;
				}
			}
		}
		return res;
	}
 
	vector<pair<int, int>> get_pair() {
		for (auto x : match_pair) {
			cout << x.first << "  " << x.second << endl;
		}
		return match_pair;
	}
};
class Lca {
private:
	int n;
	int log2_n;
	vector<vector<int>> parent;
	vector<int> depth;
	
	void dfs(const vector<vector<edge>>& g, int v, int p, int d) {
		parent[0][v] = p;
		depth[v] = d;
		for (auto& e : g[v]) {
			if (e.to != p) {
				dfs(g, e.to, v, d + 1);
			}
		}
	}
 
public:
	Lca(const vector<vector<edge>>& g, int root) {
		n = g.size();
		log2_n = (int)log2(n) + 1;
		resize(parent, log2_n, n);
		resize(depth, n);
 
		dfs(g, root, -1, 0);
 
		for (int k = 0; k + 1 < log2_n; k++) {
			for (int v = 0; v < (int)g.size(); v++) {
				if (parent[k][v] < 0) {
					parent[k + 1][v] = -1;
				}
				else {
					parent[k + 1][v] = parent[k][parent[k][v]];
				}
			}
		}
	}
 
	int get_lca(int u, int v) {
		if (depth[u] > depth[v]) {
			swap(u, v);
		}
 
		for (int k = 0; k < log2_n; k++) {
			if ((depth[v] - depth[u]) >> k & 1) {
				v = parent[k][v];
			}
		}
		if (u == v) {
			return u;
		}
 
		for (int k = log2_n - 1; k >= 0; k--) {
			if (parent[k][u] != parent[k][v]) {
				u = parent[k][u];
				v = parent[k][v];
			}
		}
		return parent[0][u];
	}
 
	int get_depth(int v) { return depth[v]; }
};
class DAG {
private:
	int n;
	vector<vector<edge>> g;
	vector<int> visited;
	vector<int> dp;
	vector<int> topological;
 
	int dfs(int s) {
		if ((int)g[s].size() == 0) {
			return 1;
		}
		if (dp[s] > 0) {
			return dp[s];
		}
 
		int mx = 1;
		for (auto j : g[s]) {
			if (visited[j.to] == 0) {
				visited[j.to] = 1;
				int l = 0;
				l = dfs(j.to);
				chmax(mx, l);
			}
			else {
				chmax(mx, dp[j.to]);
			}
		}
		return dp[s] = mx + 1;
	}
 
public:
	DAG(const vector<vector<edge>>& f) {
		g = f;
		n = f.size();
		resize(visited, n + 1);
		fill(visited.begin(), visited.end(), 0);
		resize(dp, n + 1);
		fill(dp.begin(), dp.end(), -1);
		resize(topological, n);
	}
	DAG(int x) {
		n = x;
		resize(g, n);
		resize(visited, n + 1);
		fill(visited.begin(), visited.end(), 0);
		resize(dp, n + 1);
		fill(dp.begin(), dp.end(), -1);
	}
 
	void add_edge(int a, int b) { g[a].emplace_back(a, b); }
	void add_edge(int a, int b, ll c) { g[a].emplace_back(a, b, c); }
	void add_edge(int a, int b, ll c, ll d) { g[a].emplace_back(a, b, c, d); }
 
	int longest_path() {
		int mx = -1;
		for (int i = 0; i < n; i++) {
			int h = 0;
			if (visited[i] == 0) {
				h = dfs(i);
				chmax(mx, h);
			}
		}
		return mx - 1;
	}
 
	bool TopologicalSort() {
		int k = 0;
		vector<int> ord(n), in(n);
		for (auto& es : g) {
			for (auto& e : es) {
				in[e.to]++;
			}
		}
		queue<int> q;
		for (int i = 0; i < n; ++i) {
			if (in[i] == 0) q.push(i);
		}
		while (!q.empty()) {
			int v = q.front();
			q.pop();
			ord[k++] = v;
			for (auto& e : g[v]) {
				if (--in[e.to] == 0) {
					q.push(e.to);
				}
			}
		}
		topological = ord;
		if (*max_element(in.begin(), in.end()) == 0) {
			return true;
		}
		return false;
	}

	vector<int> getTopologicalArray(){
		return topological;
	}
};
class RangeMinimumUpdateQuerySegmentTree {
private:
	int n;
	ll inf = (1LL << 31) - 1;  // 2^31-1
	vector<ll> dat, lazy;
 
	void eval(int len, int k) {
		if (lazy[k] == inf) return;
		if (k * 2 + 1 < n * 2 - 1) {
			lazy[2 * k + 1] = lazy[k];
			lazy[2 * k + 2] = lazy[k];
		}
		dat[k] = lazy[k];
		lazy[k] = inf;
	}
 
public:
	RangeMinimumUpdateQuerySegmentTree() {}
	RangeMinimumUpdateQuerySegmentTree(int n_) {
		n = 1;
		while (n < n_) n *= 2;
		dat.assign(n * 2, inf);
		lazy.assign(n * 2, inf);
	}
 
	// [a,b)
	ll update(int a, int b, ll x, int k, int l, int r) {
		eval(r - l, k);
		if (b <= l || r <= a) return dat[k];
		if (a <= l && r <= b) {
			lazy[k] = x;
			return lazy[k];
		}
		return dat[k] = min(update(a, b, x, 2 * k + 1, l, (l + r) / 2),
			update(a, b, x, 2 * k + 2, (l + r) / 2, r));
	}
	ll update(int a, int b, ll x) { return update(a, b, x, 0, 0, n); }
 
	// [a, b)
	ll query(int a, int b, int k, int l, int r) {
		eval(r - l, k);
		if (b <= l || r <= a) return inf;
		if (a <= l && r <= b) return dat[k];
		ll vl = query(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = query(a, b, 2 * k + 2, (l + r) / 2, r);
		return min(vl, vr);
	}
	ll query(int a, int b) { return query(a, b, 0, 0, n); }
};
class RangeSumQuerySegmentTree {
private:
	struct Node {
		Node* left, * right;
		ll v;
 
		Node() : left(nullptr), right(nullptr), v(0) {}
	};
	Node* root;
	ll n, n0;
	ll query(ll a, ll b, Node* n, ll l, ll r) {
		if (a <= l && r <= b) {
			return n->v;
		}
		if (r <= a || b <= l) {
			return 0;
		}
 
		ll lv = n->left ? query(a, b, n->left, l, (l + r) >> 1) : 0;
		ll rv = n->right ? query(a, b, n->right, (l + r) >> 1, r) : 0;
		return (lv + rv) % MOD;
	}
 
public:
	RangeSumQuerySegmentTree(ll n) : n(n) {
		n0 = 1;
		while (n0 < n) n0 <<= 1;
		root = new Node();
	}
	~RangeSumQuerySegmentTree() {
		delete root;
		root = nullptr;
	}
 
	void update(ll k, ll x) {
		Node* n = root;
		ll l = 0, r = n0;
		n->v = (n->v + x) % MOD;
		while (r - l > 1) {
			ll m = (l + r) >> 1;
			if (k < m) {
				if (!n->left) n->left = new Node();
				n = n->left;
 
				r = m;
			}
			else {
				if (!n->right) n->right = new Node();
				n = n->right;
 
				l = m;
			}
			n->v = (n->v + x) % MOD;
		}
	}
 
	ll query(ll a, ll b) { return query(a, b, root, 0, n0); }
 
	ll lquery(ll b) { return query(0, b, root, 0, n0); }
 
	ll rquery(ll a) { return query(a, n0, root, 0, n0); }
};
class KDimensionalTree {
public:
	struct Node {
		int location;
		int p, l, r;
		Node() {}
	};
	struct Point {
		int id, x, y;
		Point() {}
		Point(int i, int a, int b) {
			id = i;
			x = a;
			y = b;
		}
		bool operator<(const Point& p) const { return id < p.id; }
		void print() { cout << id << endl; }
	};
	static const ll NIL = -1;
	static bool lessX(const Point& p1, const Point& p2) { return p1.x < p2.x; }
	static bool lessY(const Point& p1, const Point& p2) { return p1.y < p2.y; }
 
	int N;
	vector<Point> P;
	vector<Node> T;
	int np;
 
	KDimensionalTree() {}
	KDimensionalTree(int N) { init(N); }
 
	void init(int n) {
		N = n;
		P.clear();
		T.clear();
		resize(P, N);
		resize(T, N);
		np = 0;
	}
 
	int makeKDTree(int l, int r, int depth) {
		if (l >= r) {
			return NIL;
		}
		int mid = (l + r) / 2;
		int t = np++;
		if (depth & 1) {
			sort(P.begin() + l, P.begin() + r, lessY);
		}
		else {
			sort(P.begin() + l, P.begin() + r, lessX);
		}
		T[t].location = mid;
		T[t].l = makeKDTree(l, mid, depth + 1);
		T[t].r = makeKDTree(mid + 1, r, depth + 1);
		return t;
	}
	void find(int v, int sx, int tx, int sy, int ty, int depth,
		vector<Point>& ans) {
		int x = P[T[v].location].x;
		int y = P[T[v].location].y;
		if (sx <= x && x <= tx && sy <= y && y <= ty) {
			ans.push_back(P[T[v].location]);
		}
		if (depth % 2 == 0) {
			if (T[v].l != NIL) {
				if (sx <= x) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
			}
			if (T[v].r != NIL) {
				if (x <= tx) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
			}
		}
		else {
			if (T[v].l != NIL) {
				if (sy <= y) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
			}
			if (T[v].r != NIL) {
				if (y <= ty) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
			}
		}
	}
	void add_point(int i, int x, int y) {
		P[i].id = i;
		P[i].x = x;
		P[i].y = y;
		T[i].l = T[i].r = T[i].p = NIL;
	}
};
class RangeAddQuerySegmentTree {
private:
	int n;
	vector<ll> data;
 
public:
	RangeAddQuerySegmentTree() {}
	RangeAddQuerySegmentTree(int N) {
		n = N;
		resize(data, n + 1);
		fill(data.begin(), data.end(), 0);
	}
 
	void add(int i, ll x) {
		while (i) {
			data[i] += x;
			i -= (i & -i);
		}
	}
	void add(int i, int j, ll x) {
		add(j, x);
		add(i - 1, -x);
	}
 
	ll get(int i) {
		ll res = 0;
		while (i <= n) {
			res += data[i];
			i += (i & -i);
		}
		return res;
	}
};
class RangeSumAddQuerySegmentTree {
private:
	vector<ll> bit0, bit1;
	int n;
 
	ll sum(const vector<ll>& b, int i) {
		ll s = 0;
		while (i > 0) {
			s += b[i];
			i -= (i & -i);
		}
		return s;
	}
 
	void add(vector<ll>& b, int i, ll v) {
		while (i <= n) {
			b[i] += v;
			i += (i & -i);
		}
	}
 
public:
	RangeSumAddQuerySegmentTree() {}
	RangeSumAddQuerySegmentTree(int N) {
		n = N;
		resize(bit0, n + 1);
		resize(bit1, n + 1);
		fill(bit0.begin(), bit0.end(), 0);
		fill(bit1.begin(), bit1.end(), 0);
	}
 
	void update(int l, int r, ll x) {
		add(bit0, l, -x * (l - 1));
		add(bit1, l, x);
		add(bit0, r + 1, x * r);
		add(bit1, r + 1, -x);
	}
 
	ll query(int l, int r) {
		ll res = 0;
		res += sum(bit0, r) + sum(bit1, r) * r;
		res -= sum(bit0, l - 1) + sum(bit1, l - 1) * (l - 1);
		return res;
	}
};




using ld = double;
using P = complex<ld>;
using G = vector<P>;
const ld pi = acos(-1);
const ld eps = 1e-10;
const ld inf = 1e12;

#define rad first
#define pnt second

using circle = std::pair<P, long double>;
using C = pair<ld, P>;

ld cross(const P &a, const P &b) { return a.real() * b.imag() - a.imag() * b.real(); }
ld dot(const P &a, const P &b) { return a.real() * b.real() + a.imag() * b.imag(); }

C smallest_enclosing_disc(vector<P> ps){
    auto c3 = [](const P &a, const P &b, const P &c){
        ld A = norm(b - c);
        ld B = norm(c - a);
        ld C = norm(a - b);
        ld S = abs(cross(b - a, c - a));
        P p = (A*(B+C-A)*a + B*(C+A-B)*b + C*(A+B-C)*c) / (4*S*S);
        ld r = abs(p - a);
        return make_pair(r, p);
    };

    auto c2 = [](const P &a, const P &b){
        P c = (a + b) / (ld)2;
        ld r = abs(a - c);
        return make_pair(r, c);
    };

    auto in_circle = [](const P &a, const C &c){
        return norm(a - c.pnt) <= c.rad*c.rad + eps;
    };

    int n = ps.size();
    random_shuffle(ps.begin(), ps.end());
    C c = c2(ps[0], ps[1]);
    for(int i = 2; i < n; ++i){
        if(!in_circle(ps[i], c)){
            c = c2(ps[0], ps[i]);
            for(int j = 1; j < i; ++j){
                if(!in_circle(ps[j], c)){
                    c = c2(ps[j], ps[i]);
                    for(int k = 0; k < j; ++k){
                        if(!in_circle(ps[k], c)){
                            c = c3(ps[i], ps[j], ps[k]);
                        }
                    }
                }
            }
        }
    }
    return c;
}


int main() {
	int n,k;cin>>n>>k;
	vector<double> p(n);cin>>p;
	double sum=0;
	double ans=0;
	for(int i=0;i<k;i++){
		sum+=(p[i]+1)/2;
	}
	ans=max(ans,sum);
	for(int i=k;i<n;i++){
		sum-=(p[i-k]+1)/2;
		sum+=(p[i]+1)/2;
		ans=max(ans,sum);
	}

	cout<<ans<<endl;

    ;
    return 0;
}

Measured memory_bytes: 1792