Problem: In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.

In the plan developed by the president Takahashi, the trains will run as follows:

* A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.
* In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.



According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.

Find the maximum possible distance that a train can cover in the run.

Constraints

* 1 \leq N \leq 100
* 1 \leq t_i \leq 200
* 1 \leq v_i \leq 100
* All input values are integers.

Input

Input is given from Standard Input in the following format:


N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N


Output

Print the maximum possible that a train can cover in the run.
Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

Examples

Input

1
100
30


Output

2100.000000000000000


Input

2
60 50
34 38


Output

2632.000000000000000


Input

3
12 14 2
6 2 7


Output

76.000000000000000


Input

1
9
10


Output

20.250000000000000000


Input

10
64 55 27 35 76 119 7 18 49 100
29 19 31 39 27 48 41 87 55 70


Output

20291.000000000000

Solution (in JAVA):

Solution:
import java.lang.reflect.Array;
import java.util.ArrayDeque;
import java.util.Arrays;
import java.util.Scanner;

public class Main implements Runnable {
	public static void main(String[] args) {
		// long t = System.currentTimeMillis();
		new Thread(null, new Main(), "", Runtime.getRuntime().maxMemory()).start();
		// System.err.println(System.currentTimeMillis() - t);
	}

	class State {
		protected double srct;
		protected double dstt;
		protected int a;
		protected double srcv;
		protected double dstv;

		public State(double srct_, double dstt_, int a_, double srcv_) {
			srct = srct_;
			dstt = dstt_;
			a = a_;
			srcv = srcv_;
			dstv = srcv + a * (dstt - srct);
		}
	}

	void solve(int n, double[] t, double[] dt, double[] v) {
		ArrayDeque<State> dq = new ArrayDeque<>();
		dq.add(new State(-1, 0, 0, 0));
		for (int i = 0; i < n; ++i) {
			while (i > 0 && true) {
				State s = dq.peekLast();
				if (s.srcv - (t[i - 1] - s.srct) <= v[i])
					break;
				else {
					dq.pollLast();
				}
			}
			State ps = dq.peekLast();
			if (ps.dstv > v[i]) {
				dq.pollLast();
				double T = t[i - 1] - ps.srct;
				double ddt = (v[i] + T - ps.srcv) / (1 + ps.a);
				ps.dstt = ps.srct + ddt;
				ps.dstv = ps.srcv + (ps.dstt - ps.srct) * ps.a;
				dq.addLast(ps);
				dq.add(new State(ps.dstt, t[i - 1], -1, ps.dstv));
			}
			ps = dq.peekLast();
			if (ps.dstv + dt[i] > v[i]) {
				dq.addLast(new State(ps.dstt, ps.dstt + v[i] - ps.dstv, 1, ps.dstv));
				dq.addLast(new State(ps.dstt + v[i] - ps.dstv, t[i], 0, v[i]));
			} else {
				dq.addLast(new State((i > 0 ? t[i - 1] : 0), t[i], 1, ps.dstv));
			}
		}

		double ans = 0;
		for (State state : dq) {
			double ddt = state.dstt - state.srct;
			ans += ddt * state.srcv + ddt * ddt * state.a * 0.5;
		}
		System.out.println(ans);
	}

	public void run() {
		Scanner sc = new Scanner(System.in);
		int n = sc.nextInt();
		double[] t = new double[n + 1];
		double[] dt = new double[n + 1];
		double[] v = new double[n + 1];
		for (int i = 0; i < n; ++i) {
			t[i] = sc.nextDouble();
			dt[i] = t[i];
			if (i > 0)
				t[i] = t[i - 1] + t[i];
		}
		for (int i = 0; i < n; ++i) {
			v[i] = sc.nextDouble();
		}
		++n;
		v[n - 1] = 0;
		t[n - 1] = t[n - 2];
		dt[n - 1] = 0;
		solve(n, t, dt, v);
	}

	static void tr(Object... objects) {
		System.out.println(Arrays.deepToString(objects));
	}

}