//#define NDEBUG

#pragma GCC optimize("-O3,inline,omit-frame-pointer,unroll-loops")

#include <iostream>

#include <array>
#include <vector>
#include <list>
#include <stack>
#include <queue>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <algorithm>
#include <numeric>
#include <iomanip>

#include <string>
#include <sstream>

#include <memory>
#include <cassert>

#include <functional>
#include <chrono>

using namespace std;

namespace ValLib {

	const int MOD = 1000000007;

	typedef unsigned long long ull;

	template<typename Key, typename Value>
	using umap = std::unordered_map<Key, Value>;
	template<typename T>
	using uset = std::unordered_set<T>;
	template <typename T>
	using sptr = typename std::shared_ptr<T>;
	class vnode;
	class vegde;
	class vgraph;

	class NullStream {
	public:
		template<typename T> NullStream& operator<<(T const&) { return *this; }
	};

#ifdef NDEBUG
#define vcerr if(false) NullStream()
#define vendl nullptr
#else
#define vcerr std::cerr
#define vendl std::endl
#endif

	template<typename T>
	void fill(vector<T>& vec, const T& value) {
		std::fill(vec.begin(), vec.end(), value);
	}
	template<typename T>
	void fill(vector<vector<T>>& vec, const T& value) {
		for (vector<T>& t : vec) std::fill(t.begin(), t.end(), value);
	}
	template<typename T>
	void resize(vector<vector<T>>& vec, int size1, int size2) {
		vec.resize(size1);
		for (vector<T>& t : vec) t.resize(size2);
	}
	template <typename T>
	const typename vector<T>::const_iterator max_element(const vector<T>& vec) {
		assert(!vec.empty());
		return std::max_element(vec.begin(), vec.end());
	}
	template <typename T, typename _Pr>
	const typename vector<T>::const_iterator max_element(const vector<T>& vec, _Pr lessThan) {
		assert(!vec.empty());
		return std::max_element(vec.begin(), vec.end(), lessThan);
	}
	template <typename T>
	const typename vector<T>::const_iterator min_element(const vector<T>& vec) {
		assert(!vec.empty());
		return std::min_element(vec.begin(), vec.end());
	}
	template <typename T, typename _Pr>
	const typename vector<T>::const_iterator min_element(const vector<T>& vec, _Pr lessThan) {
		assert(!vec.empty());
		return std::min_element(vec.begin(), vec.end(), lessThan);
	}
	int accumulate(const vector<int>& vec) {
		return std::accumulate(vec.begin(), vec.end(), 0);
	}
	template <typename _Pr>
	int accumulate(const vector<int>& vec, _Pr func) {
		return std::accumulate(vec.begin(), vec.end(), 0, func);
	}
	double accumulate(const vector<double>& vec) {
		return std::accumulate(vec.begin(), vec.end(), 0.0);
	}
	template <typename _Pr>
	double accumulate(const vector<double>& vec, _Pr func) {
		return std::accumulate(vec.begin(), vec.end(), 0.0, func);
	}
	float accumulate(const vector<float>& vec) {
		return std::accumulate(vec.begin(), vec.end(), 0.f);
	}
	template <typename _Pr>
	float accumulate(const vector<float>& vec, _Pr func) {
		return std::accumulate(vec.begin(), vec.end(), 0.f, func);
	}
	template <typename T, typename _Pr>
	bool all_of(const vector<T>& vec, _Pr pred) {
		return std::all_of(vec.begin(), vec.end(), pred);
	}
	template <typename T, typename _Pr>
	bool any_of(const vector<T>& vec, _Pr pred) {
		return std::any_of(vec.begin(), vec.end(), pred);
	}
	template <typename T, typename _Pr>
	bool none_of(const vector<T>& vec, _Pr pred) {
		return std::none_of(vec.begin(), vec.end(), pred);
	}
	template <typename T>
	const typename vector<T>::const_iterator find(const vector<T>& vec, const T& val) {
		return std::find(vec.begin(), vec.end(), val);
	}
	template <typename T, typename _Pr>
	const typename vector<T>::const_iterator find_if(const vector<T>& vec, _Pr pred) {
		return std::find_if(vec.begin(), vec.end(), pred);
	}
	template <typename T>
	bool contains(const vector<T>& vec, const T& val) {
		return std::find(vec.begin(), vec.end(), val) != vec.end();
	}
	template <typename T, typename _Pr>
	bool contains_if(const vector<T>& vec, _Pr pred) {
		return std::find_if(vec.begin(), vec.end(), pred) != vec.end();
	}
	template <typename T, typename _Pr>
	typename iterator_traits<const typename vector<T>::const_iterator>::difference_type count_if(const vector<T>& vec, _Pr pred) {
		return std::count_if(vec.begin(), vec.end(), pred);
	}

	template<typename T, size_t N>
	void fill(array<T, N>& ary, const T& value) {
		std::fill(ary.begin(), ary.end(), value);
	}
	template<typename T, size_t N, size_t M>
	void fill(array<array<T, M>, N>& ary, const T& value) {
		for (array<T, M>& t : ary) std::fill(t.begin(), t.end(), value);
	}
	template <typename T, size_t N, typename _Pr>
	const typename array<T, N>::const_iterator max_element(const array<T, N>& ary) {
		assert(!ary.empty());
		return std::max_element(ary.begin(), ary.end());
	}
	template <typename T, size_t N, typename _Pr>
	const typename vector<T>::const_iterator max_element(const array<T, N>& ary) {
		assert(!ary.empty());
		return std::max_element(ary.begin(), ary.end());
	}
	template <typename T, size_t N, typename _Pr>
	const typename array<T, N>::const_iterator max_element(const array<T, N>& ary, _Pr lessThan) {
		assert(!ary.empty());
		return std::max_element(ary.begin(), ary.end(), lessThan);
	}
	template <typename T, size_t N, typename _Pr>
	const typename array<T, N>::const_iterator min_element(const array<T, N>& ary) {
		assert(!ary.empty());
		return std::min_element(ary.begin(), ary.end());
	}
	template <typename T, size_t N, typename _Pr>
	const typename array<T, N>::const_iterator min_element(const array<T, N>& ary, _Pr lessThan) {
		assert(!ary.empty());
		return std::min_element(ary.begin(), ary.end(), lessThan);
	}
	template <size_t N>
	int accumulate(const array<int, N>& ary) {
		return std::accumulate(ary.begin(), ary.end(), 0);
	}
	template <size_t N, typename _Pr>
	int accumulate(const array<int, N>& ary, _Pr func) {
		return std::accumulate(ary.begin(), ary.end(), 0, func);
	}
	template <size_t N>
	double accumulate(const array<double, N>& ary) {
		return std::accumulate(ary.begin(), ary.end(), 0.0);
	}
	template <size_t N, typename _Pr>
	double accumulate(const array<double, N>& ary, _Pr func) {
		return std::accumulate(ary.begin(), ary.end(), 0.0, func);
	}
	template <size_t N>
	float accumulate(const array<float, N>& ary) {
		return std::accumulate(ary.begin(), ary.end(), 0.f);
	}
	template <size_t N, typename _Pr>
	float accumulate(const array<float, N>& ary, _Pr func) {
		return std::accumulate(ary.begin(), ary.end(), 0.f, func);
	}
	template <typename T, size_t N, typename _Pr>
	bool all_of(const array<T, N>& ary, _Pr pred) {
		return std::all_of(ary.begin(), ary.end(), pred);
	}
	template <typename T, size_t N, typename _Pr>
	bool any_of(const array<T, N>& ary, _Pr pred) {
		return std::any_of(ary.begin(), ary.end(), pred);
	}
	template <typename T, size_t N, typename _Pr>
	bool none_of(const array<T, N>& ary, _Pr pred) {
		return std::none_of(ary.begin(), ary.end(), pred);
	}
	template <typename T, size_t N>
	const typename array<T, N>::const_iterator find(const array<T, N>& ary, const T& val) {
		return std::find(ary.begin(), ary.end(), val);
	}
	template <typename T, size_t N, typename _Pr>
	const typename array<T, N>::const_iterator find_if(const array<T, N>& ary, _Pr pred) {
		return std::find_if(ary.begin(), ary.end(), pred);
	}
	template <typename T, size_t N>
	bool contains(const array<T, N>& ary, const T& val) {
		return std::find(ary.begin(), ary.end(), val) != ary.end();
	}
	template <typename T, size_t N, typename _Pr>
	bool contains_if(const array<T, N>& ary, _Pr pred) {
		return std::find_if(ary.begin(), ary.end(), pred) != ary.end();
	}
	template <typename T, size_t N, typename _Pr>
	typename iterator_traits<const typename array<T, N>::const_iterator>::difference_type count_if(const array<T, N>& ary, _Pr pred) {
		return std::count_if(ary.begin(), ary.end(), pred);
	}

	template<typename Key, typename Value>
	bool containsKey(const umap<Key, Value>& m, const Key& key) {
		return m.find(key) != m.end();
	}
	template<typename Key, typename Value>
	bool containsValue(const umap<Key, Value>& m, const Value& val) {
		for (auto it = m.begin(); it != m.end(); ++it)
			if (it->second == val)
				return true;
		return false;
	}
	template<typename T>
	const typename uset<T>::const_iterator find(const uset<T>& s, const T& key) {
		return s.find(key);
	}
	template<typename T>
	bool contains(const uset<T>& s, const T& key) {
		return s.find(key) != s.end();
	}

	void test() {
		vector<int> v = {};
		bool c = contains(v, 2);
		int x = count_if(v, [](int y) { return y <= 0; });
		cout << x << endl;

		uset<int> s;
		s.insert(1);
		s.insert(4);
		s.insert(1);
		s.insert(2);
		bool bb = contains(s, 1);
		cout << bb << endl;
		bb = contains(s, 3);
		cout << bb << endl;

		umap<int, string> m;
		m[0] = "a";
		m[4] = "b";
		m[2] = "c";
		auto au = m.find(4);
		if (au != m.end()) {
			string val = (*au).second;
		}
		bool b = containsKey(m, 1);
		cout << b << endl;
		b = containsKey(m, 3);
		cout << b << endl;
		b = containsValue(m, string("a"));
		cout << b << endl;
		b = containsValue(m, string("d"));
		cout << b << endl;

		getchar();
		getchar();
	}

	class Point {
	public:

		Point() = default;

		//inline Point() :
		//x(-1),
		//y(-1)
		//{}

		inline Point(int _x, int _y) :
			x(_x),
			y(_y)
		{}

		static Point getDistance(const Point& p1, const Point& p2) {
			return move(Point(abs(p1.x - p2.x), abs(p1.y - p2.y))); 
		}

		void setPos(const Point& pos) { x = pos.x; y = pos.y; }
		void setPos(int x, int y) { this->x = x; this->y = y; }

		Point operator+ (const Point &p) const { return move(Point(x + p.x, y + p.y)); }
		Point operator- (const Point &p) const { return move(Point(x - p.x, y - p.y)); }
		void operator+= (const Point &p) { x += p.x; y += p.y; }
		void operator-= (const Point &p) { x -= p.x; y -= p.y; }
		bool operator== (const Point &p) const { return x == p.x && y == p.y; }
		bool operator!= (const Point &p) const { return x != p.x || y != p.y; }
		//bool operator<(const Point &p) const {return x * x + y * y < p.x * p.x + p.y * p.y;}
		const Point& getPos() const { return *this; }

		string to_string() const { return "(" + std::to_string(x) + ", " + std::to_string(y) + ")"; }

		int x;
		int y;
	};

	/// <summary>
	/// 素集合データ構造
	/// </summary>
	class UnionFind {
	public:

		/// <summary>
		/// コンストラクタ
		/// </summary>
		/// <param name="N">要素数</param>
		UnionFind(int N) {
			parent.resize(N);
			for (int i = 0; i < N; ++i) {
				parent[i] = i;
			}
		}

		/// <summary>
		/// 要素xの根を探索
		/// </summary>
		/// <param name="x">探索する要素番号(0～N-1)</param>
		/// <returns>要素xの根(0～N-1)</returns>
		int find(int x) {
			if (parent[x] == x) {
				return x;
			} else {
				parent[x] = find(parent[x]); // 経路圧縮(間接的な要素をrootに直接つなぐ)
				return parent[x];
			}
		}

		/// <summary>
		/// ２つの要素の根が同じか
		/// </summary>
		/// <param name="x1">探索する要素番号１(0～N-1)</param>
		/// <param name="x2">探索する要素番号２(0～N-1)</param>
		/// <returns>true: 同じ根, false: 異なる根</returns>
		bool same(int x1, int x2) {
			return find(x1) == find(x2);
		}

		/// <summary>
		/// ２つの要素の根同士を繋ぐ
		/// </summary>
		/// <param name="x1">要素番号１(0～N-1)</param>
		/// <param name="x2">要素番号１(0～N-1)</param>
		void union_elements(int x1, int x2) {
			int rootX1 = find(x1);
			int rootX2 = find(x2);
			parent[rootX2] = rootX1;
		}

	private:
		vector<int> parent;

	};

	class vmath {
	public:

		// 約数を全て求める O(√n)
		static ull calcDivisors(list<ull>* divisors, ull n) {
			divisors->clear();
			if (n <= 0ull) {
				return 0ull;
			}
			divisors->push_back(1ull);
			if (n != 1ull) {
				divisors->push_back(n);
			}
			for (ull i = 2ull; i * i <= n; ++i) {
				if (n % i == 0ull) {
					divisors->push_back(i);
					if (i != n / i) {
						divisors->push_back(n / i);
					}
				}
			}
			return divisors->size();
		}

		// 約数の個数を返す O(√n)
		static ull calcDivisorNum(ull n) {
			if (n <= 0ull) {
				return 0ull;
			}
			ull count = 1ull; // for 1
			if (n != 1ull) {
				++count; // for n
			}
			// for 2～n-1
			for (ull i = 2ull; i * i <= n; ++i) {
				if (n % i == 0ull) {
					count += 1ull;
					if (i != n / i) {
						count += 1ull;
					}
				}
			}
			return count;
		}

		// 素因数分解 O(√n)
		static int calcDecompositPrime(list<ull>* primes, ull n) {
			primes->clear();
			if (n == 0) {
				return 0ull;
			}
			if (n == 1) {
				primes->push_back(1);
				return 1ull;
			}
			// 割る数の初期値
			ull a = 2ull;
			// √n ≧ a ( n ≧ a * a ) の間ループ処理
			while (n >= a * a) {
				if (n % a == 0ull) {
					// a で割り切れたら、a は素因数
					primes->push_back(a);
					// そして、割られる数を a で割る
					n /= a;
				} else {
					// a で割り切れなかったら、 a を 1 増加させる
					a++;
				}
			}
			primes->push_back(n);
			return primes->size();
		}

		// 素因数の数を返す O(√n)
		static ull calcDecompositPrimeNum(ull n) {
			if (n <= 1) {
				return n;
			}
			ull count = 0ull;
			// 割る数の初期値
			ull a = 2ull;
			// √n ≧ a ( n ≧ a * a ) の間ループ処理
			while (n >= a * a) {
				if (n % a == 0ull) {
					// a で割り切れたら、a は素因数
					++count;
					// そして、割られる数を a で割る
					n /= a;
				} else {
					// a で割り切れなかったら、 a を 1 増加させる
					a++;
				}
			}
			++count; // for n
			return count;
		}

		// 階乗
		static ull fact(ull x) {
			if (x == 0ull) {
				return 1ull;
			} else {
				return x * fact(x - 1ull);
			}
		}

		// 順列 nPr
		static ull permutation(int n, int r) {
			assert(n >= r);
			//return fact(n) / fact(n - r);
			ull result = 1;
			for (ull i = n; i > n - r; --i) {
				result *= i;
			}
			return result;
		}

		// 組み合わせ nCr
		// 先にmakePascalTriangleでパスカルの三角形を作成しておく必要がある
		static ull combination(int n, int r) {
			assert(n >= r);
			assert(n <= mPascalTriangleDepth);
			return mPascalTriangle[n][r];
		}

		// 重複組合せ nHr = n+r-1Cr
		// 使いどころ：n人にr個を配るとき、同じ人に何個配っても良い場合とか
		// 4人に5個の◯を配るときa=2,b=0,c=2,d=1のとき、◯◯||◯◯|◯みたいになる。
		// これは◯と|を混ぜた組み合わせで、◯の数がn,棒の数はr-1だから全体でn+r-1
		// (n-r)で割ったものが順列n+r-1Prで、それを更にrで割っているからnHr = n+r-1Cr
		static ull repeatedCombination(int n, int r) {
			return combination(n + r - 1, r);
		}

		// パスカルの三角形。組み合わせの計算に使用するのでこれを先に作成しておく必要がある。
		static void makePascalTriangle(int depth) {
			resize(mPascalTriangle, depth + 1, depth + 1);
			for (int i = 0; i <= depth; ++i) {
				mPascalTriangle[i][0] = 1;
				for (int j = 1; j <= i; ++j) {
					mPascalTriangle[i][j] = mPascalTriangle[i - 1][j] + mPascalTriangle[i - 1][j - 1];
				}
			}
			mPascalTriangleDepth = depth;
		}

		// xのN桁目の数値を得る
		static ull getNDigitNumber(ull x, ull n) {
			return (x / pow(10ull, n - 1ull)) % 10;
		}

		// xのN桁目の数値を得る
		static int getNDigitNumber(int x, int n) {
			assert(n <= 10);
			return (x / pow(10, n - 1)) % 10;
		}

		// xのN乗を求める(バイナリ法) O(logN)
		static ull pow(ull x, ull n) {
			assert(x >= 0ull);
			assert(n >= 0ull);
			if (x == 0ull) {
				if (n == 0ull) {
					return 1ull;
				} else {
					return 0ull;
				}
			}
			ull result = 1ull;
			ull z = x;
			while (n > 0ull) {
				if (n & 1ull) {
					result *= z;
				}
				z *= z;
				n >>= 1;
			}
			return result;
		}

		// xのN乗を求める(バイナリ法) O(logN)
		static int pow(int x, int n) {
			assert(x >= 0);
			assert(n >= 0);
			if (x == 0) {
				if (n == 0) {
					return 1;
				} else {
					return 0;
				}
			}
			int result = 1;
			int z = x;
			while (n > 0) {
				if (n & 1) {
					result *= z;
				}
				z *= z;
				n >>= 1;
			}
			return result;
		}

	private:
		static int mPascalTriangleDepth;
		static vector<vector<ull>> mPascalTriangle;
	};
	int vmath::mPascalTriangleDepth = 0;
	vector<vector<ull>> vmath::mPascalTriangle;

	class vmath_mod {
	public:

		// 階乗
		static ull fact(ull x) {
			ull result = 1ull;
			for (ull i = 1ull; i <= x; ++i) {
				result *= i;
				result %= MOD;
			}
			return result;
		}

		// 順列 nPr
		static ull permutation(int n, int r) {
			assert(n >= r);
			//return fact(n) / fact(n - r);
			ull result = 1;
			for (ull i = n; i > n - r; --i) {
				result *= i;
				result %= MOD;
			}
			return result;
		}

		// 組み合わせ nCr
		// 先にmakePascalTriangleでパスカルの三角形を作成しておく必要がある
		static ull combination(int n, int r) {
			assert(n >= r);
			assert(n <= mPascalTriangleDepth);
			return mPascalTriangle[n][r];
		}

		// 重複組合せ nHr = n+r-1Cr
		// 使いどころ：n人にr個を配るとき、同じ人に何個配っても良い場合とか
		// 4人に5個の◯を配るときa=2,b=0,c=2,d=1のとき、◯◯||◯◯|◯みたいになる。
		// これは◯と|を混ぜた組み合わせで、◯の数がn,棒の数はr-1だから全体でn+r-1
		// (n-r)で割ったものが順列n+r-1Prで、それを更にrで割っているからnHr = n+r-1Cr
		static ull repeatedCombination(int n, int r) {
			return combination(n + r - 1, r);
		}

		// パスカルの三角形。組み合わせの計算に使用するのでこれを先に作成しておく必要がある。
		static void makePascalTriangle(int depth) {
			resize(mPascalTriangle, depth + 1, depth + 1);
			for (int i = 0; i <= depth; ++i) {
				mPascalTriangle[i][0] = 1;
				for (int j = 1; j <= i; ++j) {
					mPascalTriangle[i][j] = (mPascalTriangle[i - 1][j] + mPascalTriangle[i - 1][j - 1]) % MOD;
				}
			}
			mPascalTriangleDepth = depth;
		}

		// xのN桁目の数値を得る
		static ull getNDigitNumber(ull x, ull n) {
			return (x / pow(10ull, n - 1ull)) % 10;
		}

		// xのN桁目の数値を得る
		static int getNDigitNumber(int x, int n) {
			assert(n <= 10);
			return (x / pow(10, n - 1)) % 10;
		}

		// xのN乗を求める O(n)
		static ull pow(ull x, ull n) {
			if (x == 0ull) {
				if (n == 0ull) {
					return 1ull;
				} else {
					return 0ull;
				}
			}
			ull result = 1ull;
			for (ull i = 0ull; i < n; ++i) {
				result *= x;
				result %= MOD;
			}
			return result;
		}

		// xのN乗を求める O(n)
		static int pow(int x, int n) {
			assert(x >= 0);
			assert(n >= 0);
			if (x == 0) {
				if (n == 0) {
					return 1;
				} else {
					return 0;
				}
			}
			int result = 1;
			for (int i = 0; i < n; ++i) {
				result *= x;
				result %= MOD;
			}
			return result;
		}

	private:
		static int mPascalTriangleDepth;
		static vector<vector<ull>> mPascalTriangle;
	};
	int vmath_mod::mPascalTriangleDepth = 0;
	vector<vector<ull>> vmath_mod::mPascalTriangle;

	class vegde {
	public:
		vegde() : vegde(-1) {

		}

		vegde(int cost) :
			mCost(cost) {

		}

		int getCost() const { return mCost; }

	private:
		int mCost;
	};

	class vnode {
	public:
		vnode() : vnode(-1) {

		}

		vnode(int id) :
			mID(id) {

		}

		void addEgde(const vegde& egde, const vnode* node) {
			mEgdeList.emplace_back(egde, node);
		}
		void removeEgde(int nodeID) {
			auto itrNewEnd = std::remove_if(mEgdeList.begin(), mEgdeList.end(), [=](const pair<vegde, const vnode*>& p)->bool {
				return (p.second->getID() == nodeID);
			});
			mEgdeList.erase(itrNewEnd, mEgdeList.end());
		}

		int getID() const { return mID; }
		const list<pair<vegde, const vnode*>>& getEgde() const { return mEgdeList; }
	private:
		list<pair<vegde, const vnode*>> mEgdeList;
		int mID;
	};

	class AdjacencyMatrix {
	public:

		AdjacencyMatrix() {
		}

		AdjacencyMatrix(int nodeNum) {
			resize(mConnectionMap, nodeNum, nodeNum);
			resize(mCostMap, nodeNum, nodeNum);
			resize(mMinimumDistMap, nodeNum, nodeNum);
			resize(mPrevNodeMap, nodeNum, nodeNum);
		}

		void addEgde(int nodeID1, int nodeID2, int cost) {
			mConnectionMap[nodeID1][nodeID2] = true;
			mConnectionMap[nodeID2][nodeID1] = true;
			mCostMap[nodeID1][nodeID2] = cost;
			mCostMap[nodeID2][nodeID1] = cost;
		}

		void removeEgde(int nodeID1, int nodeID2) {
			mConnectionMap[nodeID1][nodeID2] = false;
			mConnectionMap[nodeID2][nodeID1] = false;
			mCostMap[nodeID1][nodeID2] = 0;
			mCostMap[nodeID2][nodeID1] = 0;
		}

		void warshallFloyd(int nodeNum) {
			for (int k = 0; k < nodeNum; ++k) {
				for (int i = 0; i < nodeNum; ++i) {
					for (int j = 0; j < nodeNum; ++j) {
						if (mConnectionMap[i][j]) {
							mMinimumDistMap[i][j] = mCostMap[i][j];
						} else {
							mMinimumDistMap[i][j] = 99999999;
						}
					}
				}
			}
			for (int k = 0; k < nodeNum; ++k) {
				for (int i = 0; i < nodeNum; ++i) {
					for (int j = 0; j < nodeNum; ++j) {
						mMinimumDistMap[i][j] = min(mMinimumDistMap[i][j], mMinimumDistMap[i][k] + mMinimumDistMap[k][j]);
					}
				}
			}
			//for (int i = 0; i < mNodeNum; ++i) {
			//	for (int j = 0; j < mNodeNum; ++j) {
			//		cerr << mMinimumDistMap[i][j] << ", ";
			//	}
			//	cerr << endl;
			//}
		}

		const vector<vector<bool>>& getConnectionMap() const { return mConnectionMap; }
		const vector<vector<int>>& getCostMap() const { return mCostMap; }

		const vector<vector<int>>& getMinimumDistMap() const { return mMinimumDistMap; }
		const vector<vector<int>>& getPrevNodeMap() const { return mPrevNodeMap; }
	private:
		vector<vector<bool>> mConnectionMap;
		vector<vector<int>> mCostMap;

		vector<vector<int>> mMinimumDistMap;
		vector<vector<int>> mPrevNodeMap;

	};

	// グラフ
	class vgraph {
	public:

		const int INF = 1000000;

		vgraph(int nodeNum) {
			mNodeNum = nodeNum;
			mNodes.resize(nodeNum);
			for (int i = 0; i < nodeNum; ++i) {
				mNodes[i] = move(vnode(i));
			}
			mMinimumDists.resize(mNodeNum);
			mPrevNodes.resize(mNodeNum);
		}

		void addEgde(int nodeID1, int nodeID2) {
			addEgde(nodeID1, nodeID2, 1);
		}

		virtual void addEgde(int nodeID1, int nodeID2, int cost) = 0;
		virtual void removeEgde(int nodeID1, int nodeID2) = 0;

		int dfs(int start) {
			vector<bool> check(mNodes.size());
			fill(check, false);
			int MAX_DEPTH = INF; // 探索の深さ制限があれば定義する
			return dfsSub(start, check, MAX_DEPTH, 0);
		}

		void bfs(int start) {
			vector<bool> check(mNodes.size());
			fill(check, false);
			int MAX_DEPTH = INF; // 探索の深さ制限があれば定義する
			return bfsSub(start, check, MAX_DEPTH);
		}

		// ベルマンフォード法を使ってある頂点から全ての頂点への最短距離を求める
		// startからたどり着ける場所に負のループが存在する場合はfalseを返す
		// ダイクストラ法と違い、負のコストの辺があっても最短距離を計算できる
		// O(V*E)
		bool bellmanFord(int start) {

			vector<int>& dist = mMinimumDists;

			fill(dist, INF);
			dist[start] = 0;

			for (int i = 0; i < mNodeNum; ++i) {
				bool update = false;
				for (vnode node : mNodes) {
					for (const pair<vegde, const vnode*> egde : node.getEgde()) {
						int from = node.getID();
						int to = egde.second->getID();
						if (dist[from] == INF) {
							continue;
						}
						if (dist[from] + egde.first.getCost() < dist[to]) {
							dist[to] = dist[from] + egde.first.getCost();
							update = true;

							if (i == mNodeNum - 1) {
								//return false;
							}
						}
					}
				}
				if (!update) {
					break;
				}
			}
			return true;
		}

		// ダイクストラ法を使ってある頂点から全ての頂点への最短距離を求める
		// 負のコストの辺があると最短距離を計算できない点に注意する
		// O(E*logV)
		void dijkstraSearch(int start) {

			// Minimum distances for each nodes
			vector<int>& dpMinDists = mMinimumDists;
			fill(dpMinDists, INF);

			// Result of the previous visited node
			vector<int>& resultPrev = mPrevNodes;
			fill(resultPrev, -1);

			// Create a priority_queue for search.
			typedef pair<int, int> P; // key: その頂点までの最短距離, value: 頂点の番号
			priority_queue<P, vector<P>, greater<P>> pq_node;

			// Mark the current node as visited and enqueue it
			pq_node.push(P(0, start));

			dpMinDists[start] = 0;

			while (!pq_node.empty()) {
				P p = pq_node.top();
				pq_node.pop();
				int nowDist = p.first;
				int nowNodeID = p.second;

				if (dpMinDists[nowNodeID] < nowDist) {
					continue;
				}

				for (const pair<vegde, const vnode*>& egde : mNodes[nowNodeID].getEgde()) {
					const vnode* nextNode = egde.second;
					int nextNodeID = nextNode->getID();

					int nextNodeDist = nowDist + egde.first.getCost();
					if (nextNodeDist < dpMinDists[nextNodeID]) {
						dpMinDists[nextNodeID] = nextNodeDist;
						resultPrev[nextNodeID] = nowNodeID;

						pq_node.push(P(nextNodeDist, nextNodeID));
					}
				}
			}
		}

		int calcMaxDepth() const {
			pair<int, int> farestNodeData = getFarestNodeID(0);
			pair<int, int> farestNodeData2 = getFarestNodeID(farestNodeData.first);
			return farestNodeData2.second;
		}

		int getNodeNum() const { return mNodeNum; }
		const vector<vnode>& getNodes() const { return mNodes; }

		const vector<int>& getMinimumDists() const { return mMinimumDists; }
		const vector<int>& getPrevNodes() const { return mPrevNodes; }
	protected:

		int dfsSub(int nodeIndex, vector<bool>& check, int MAX_DEPTH, int depth) {
			check[nodeIndex] = true;
			int result = 0;

			// 隣接する辺を探索
			if (depth < MAX_DEPTH) {
				for (auto t : mNodes[nodeIndex].getEgde()) {
					if (check[t.second->getID()]) {
						// 巡回済みの辺は見ない
						continue;
					}
					result += dfsSub(t.second->getID(), check, MAX_DEPTH, depth + 1);
				}
			}

			// 末端での処理
			result += 1; // この巡回順で末端に辿り着いた時、目的の条件を満たすなら+1する


			check[nodeIndex] = false;

			return result;
		}

		void bfsSub(int start, vector<bool>& check, int MAX_DEPTH) {

			queue<pair<int, int>> target;
			target.push(pair<int, int>(start, 0));
			while (!target.empty()) {
				pair<int, int> now = target.front();
				target.pop();

				// ノードに対して何かやることがあればここでやる
				//mNodes[now.first].something = someData;

				check[now.first] = true;
				int egdeCount = 0;
				if (now.second < MAX_DEPTH) {
					for (const pair<vegde, const vnode*>& egde : mNodes[now.first].getEgde()) {
						if (check[egde.second->getID()]) {
							// このノードはもっと少ない手順で巡回可能なので行かない
							continue;
						}
						++egdeCount;
						target.push(pair<int, int>(egde.second->getID(), now.second + 1));
					}
				}
				if (now.second == MAX_DEPTH || egdeCount == 0) {
					// 探索深さ制限またはこのノードから繋がる辺が全て巡回済み

					// doSomething.

				}
			}
		}

		// 引数で受け取ったノードから最も遠いノードのidと距離を返す
		// グラフにループがあると無限ループになるので注意する(つまり木専用)
		pair<int, int> getFarestNodeID(int start) const {
			queue<pair<int, int>> q; // nodeID, このノードまでの距離
			q.push(make_pair(start, 0));
			pair<int, int> finalNodeData;
			vector<bool> opened(mNodeNum);
			fill(opened, false);
			while (!q.empty()) {
				pair<int, int> nodeData = q.front();
				int nodeID = nodeData.first;
				int dist = nodeData.second;
				if (dist > finalNodeData.second) {
					finalNodeData.second = dist;
					finalNodeData.first = nodeID;
				}
				q.pop();
				for (const pair<vegde, const vnode*> egde : mNodes[nodeID].getEgde()) {
					int id = egde.second->getID();
					if (opened[id]) {
						continue;
					}
					opened[id] = true;
					q.push(make_pair(id, dist + egde.first.getCost()));
				}
			}
			return finalNodeData;
		}

		int mNodeNum;
		bool mUseMaps;
		vector<vnode> mNodes;

		vector<int> mMinimumDists;
		vector<int> mPrevNodes;
	};

	// 無向グラフ UnDirected Val Graph.
	class udvgraph : public vgraph {
	public:
		udvgraph(int nodeNum) :
			vgraph(nodeNum) {
		}

		void addEgde(int nodeID1, int nodeID2, int cost) {
			mNodes[nodeID1].addEgde(move(vegde(cost)), &mNodes[nodeID2]);
			mNodes[nodeID2].addEgde(move(vegde(cost)), &mNodes[nodeID1]);
		}

		void removeEgde(int nodeID1, int nodeID2) {
			mNodes[nodeID1].removeEgde(nodeID2);
			mNodes[nodeID2].removeEgde(nodeID1);
		}
	};

	// 隣接行列付きの無向グラフ。ワーシャルフロイドが使える。 UnDirected Val Graph Matrix
	class udvgraph_m : public udvgraph {
	public:

		udvgraph_m(int nodeNum) :
			udvgraph(nodeNum) {
			mAdjacencyMatrix = AdjacencyMatrix(nodeNum);
		}
		void addEgde(int nodeID1, int nodeID2, int cost) {
			udvgraph::addEgde(nodeID1, nodeID2, cost);
			mAdjacencyMatrix.addEgde(nodeID1, nodeID2, cost);
		}

		void removeEgde(int nodeID1, int nodeID2) {
			udvgraph::removeEgde(nodeID1, nodeID2);
			mAdjacencyMatrix.removeEgde(nodeID1, nodeID2);
		}

		void warshallFloyd() {
			mAdjacencyMatrix.warshallFloyd(mNodeNum);
		}

		const vector<vector<bool>>& getConnectionMap() const { return mAdjacencyMatrix.getConnectionMap(); }
		const vector<vector<int>>& getCostMap() const { return mAdjacencyMatrix.getCostMap(); }

		const vector<vector<int>>& getMinimumDistMap() const { return mAdjacencyMatrix.getMinimumDistMap(); }
		const vector<vector<int>>& getPrevNodeMap() const { return mAdjacencyMatrix.getPrevNodeMap(); }
	private:
		AdjacencyMatrix mAdjacencyMatrix;
	};


	// 有向グラフ Directed Val Graph.
	class dvgraph : public vgraph {
	public:
		dvgraph(int nodeNum) :
			vgraph(nodeNum) {
		}

		void addEgde(int nodeID1, int nodeID2, int cost) {
			mNodes[nodeID1].addEgde(move(vegde(cost)),&mNodes[nodeID2]);
		}

		void removeEgde(int nodeID1, int nodeID2) {
			mNodes[nodeID1].removeEgde(nodeID2);
		}

		// 入力のないノードのリストを返す
		list<int> searchStartNodes() const {
			list<int> startNodes;
			unordered_map<int, int> nodesInputs; // key:ノード番号, value:インプット数
			for (auto node : mNodes) {
				for (auto edge : node.getEgde()) {
					++nodesInputs[edge.second->getID()];
				}
			}
			for (int i = 0; i < mNodeNum; ++i) {
				if (nodesInputs.find(i) == nodesInputs.end()) {
					startNodes.push_back(i);
				}
			}
			return move(startNodes);
		}
	};

	// 隣接行列付きの有向グラフ。ワーシャルフロイドが使える。 Directed Val Graph Matrix
	class dvgraph_m : public dvgraph {
	public:

		dvgraph_m(int nodeNum) :
			dvgraph(nodeNum) {
			mAdjacencyMatrix = AdjacencyMatrix(nodeNum);
		}
		void addEgde(int nodeID1, int nodeID2, int cost) {
			dvgraph::addEgde(nodeID1, nodeID2, cost);
			mAdjacencyMatrix.addEgde(nodeID1, nodeID2, cost);
		}

		void removeEgde(int nodeID1, int nodeID2) {
			dvgraph::removeEgde(nodeID1, nodeID2);
			mAdjacencyMatrix.removeEgde(nodeID1, nodeID2);
		}

		void warshallFloyd() {
			mAdjacencyMatrix.warshallFloyd(mNodeNum);
		}

		const vector<vector<bool>>& getConnectionMap() const { return mAdjacencyMatrix.getConnectionMap(); }
		const vector<vector<int>>& getCostMap() const { return mAdjacencyMatrix.getCostMap(); }

		const vector<vector<int>>& getMinimumDistMap() const { return mAdjacencyMatrix.getMinimumDistMap(); }
		const vector<vector<int>>& getPrevNodeMap() const { return mAdjacencyMatrix.getPrevNodeMap(); }
	private:
		AdjacencyMatrix mAdjacencyMatrix;
	};

	class Timer {
	public:
		Timer(const Timer&) = delete;
		Timer& operator=(const Timer&) = delete;
		Timer(Timer&&) = delete;
		Timer& operator=(Timer&&) = delete;
		static Timer& getInstance() {
			static Timer timer;
			return timer;
		}

		void setSize(int size) { mTimes.resize(size); mStopWatches.resize(size); }
		void startTimer(int index) { mStopWatches[index] = chrono::system_clock::now(); }
		void stopTimer(int index) { mTimes[index] += chrono::duration_cast<chrono::milliseconds>(chrono::system_clock::now() - mStopWatches[index]).count(); }
		void addTime(int index, long long millisec) { mTimes[index] += millisec; }
		void resetTimes() { fill(mTimes, 0LL); }
		void resetTime(int index) { mTimes[index] = 0LL; }
		long long getTime(int index) { return mTimes[index]; }
		void outputTimes() { outputTimes(0, mTimes.size()); }
		void outputTimes(int startIndex, int endIndex) {
			for (int i = startIndex; i <= endIndex; ++i) {
				cerr << "Time[" << i << "]: " << mTimes[i] << endl;
			}
		}
	private:
		Timer() {
			mTimes.resize(100);
			mStopWatches.resize(100);
			resetTimes();
		}
		~Timer() = default;

		vector<long long> mTimes;
		vector<chrono::system_clock::time_point> mStopWatches;
	};

	// 立っているビットの数を返す
	static int bitcount8(unsigned char b8) {
		// 8 bits 限定アルゴリズムを利用している
		//c_assert( 8 == (CHAR_BIT * sizeof( b8 )) );

		b8 = (unsigned char)( ((b8 & 0xAA) >> 1) + (b8 & 0x55) );
		b8 = (unsigned char)( ((b8 & 0xCC) >> 2) + (b8 & 0x33) );
		b8 = (unsigned char)( ((b8 & 0xF0) >> 4) + (b8 & 0x0F) );

		return b8;
	}

	// 立っているビットの数を返す
	static int bitcount16(unsigned short w16) {
		// 16 bits 限定アルゴリズムを利用している
		//c_assert( 16 == (CHAR_BIT * sizeof( w16 )) );

		w16 = (unsigned short)( ((w16 & 0xAAAA) >> 1) + (w16 & 0x5555) );
		w16 = (unsigned short)( ((w16 & 0xCCCC) >> 2) + (w16 & 0x3333) );
		w16 = (unsigned short)( ((w16 & 0xF0F0) >> 4) + (w16 & 0x0F0F) );
		w16 = (unsigned short)( ((w16 & 0xFF00) >> 8) + (w16 & 0x00FF) );

		return w16;
	}

	// 立っているビットの数を返す
	static int bitcount32(unsigned long dw32) {
		// 32 bits 限定アルゴリズムを利用している
		//c_assert( 32 == (CHAR_BIT * sizeof( dw32 )) );

		dw32 = ((dw32 & 0xAAAAAAAA) >>  1) + (dw32 & 0x55555555);
		dw32 = ((dw32 & 0xCCCCCCCC) >>  2) + (dw32 & 0x33333333);
		dw32 = ((dw32 & 0xF0F0F0F0) >>  4) + (dw32 & 0x0F0F0F0F);
		dw32 = ((dw32 & 0xFF00FF00) >>  8) + (dw32 & 0x00FF00FF);
		dw32 = ((dw32 & 0xFFFF0000) >> 16) + (dw32 & 0x0000FFFF);

		return dw32;
	}

	// 条件を満たす最小の要素のindexを返す
	// 存在しない場合は-1を返す
	// 使い方の例: int result = binarySearch<int>(A, 0, N - 1, [=](int x) { return x >= K; });
	template<typename T> int binarySearch(vector<T> vec, int beginIndex, int endIndex, function<bool(T)> confilm) {
		// 解が両端にある場合や解なしの判定のために、両端の１つ外側から始める
		int lb = beginIndex - 1; // lower bound
		int ub = endIndex + 1; // upper bound

		while (ub - lb > 1) {
			int mid = (ub + lb) / 2;
			if (confilm(vec[mid])) {
				ub = mid;
			} else {
				lb = mid;
			}
		}

		if (ub == endIndex + 1) {
			// 解なし
			return -1;
		}

		return ub;
	}

}

using namespace ValLib;

int main()
{
	std::ios::sync_with_stdio(false);

	int N;
	cin >> N;
	vector<int> t(N), v(N);
	for (int i = 0; i < N; ++i) {
		cin >> t[i];
	}
	for (int i = 0; i < N; ++i) {
		cin >> v[i];
	}

	int T = accumulate(t);

	vector<double> maxy(T * 2 - 1);
	int time = 0;
	double vel = 0.0;
	for (int i = 0; i < N; ++i) { // 区間ごと
		for (double j = 0; j < t[i]; j = j + 0.5) { // 区間内で0.5s刻み
			if (time >= maxy.size()) {
				break;
			}
			maxy[time] = min((double)v[i], vel + 0.5);
			vel = min(vel + 0.5, (double)v[i]);
			++time;
		}
	}
	time = T * 2 - 2;
	vel = 0.0;
	for (int i = N - 1; i >= 0; --i) { // 区間ごと
		for (double j = 0; j < t[i]; j = j + 0.5) { // 区間内で0.5s刻み
			if (time < 0) {
				break;
			}
			maxy[time] = min(maxy[time], min((double)v[i], vel + 0.5));
			vel = min(vel + 0.5, (double)v[i]);
			--time;
		}
	}

	double result = 0.0;
	if (maxy[0] > 0) {
		result += 0.125;
	}
	for (int i = 1; i < maxy.size(); ++i) {
		double val = 0.0;
		if (maxy[i] > maxy[i - 1]) {
			val = (maxy[i] - 0.5) * 0.5 + 0.125;
		} else if (maxy[i] < maxy[i - 1]) {
			val = maxy[i] * 0.5 + 0.125;
		} else {
			val = maxy[i] * 0.5;
		}
		result += val;
	}
	if (maxy[T * 2 - 2] > 0) {
		result += 0.125;
	}


	cout << std::setprecision(18) << result << endl;

	getchar();
	getchar();
}


Measured memory_bytes: 512