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.util.*;
import java.io.*;

public class Main {
	static int mod = (int) 1e9 + 7;
	static int DX[] = { -1, 0, 1, 0 }, DY[] = { 0, -1, 0, 1 };

	public static void main(String[] args) {
		FastScanner fs = new FastScanner(System.in);
		int n = fs.nextInt();
		int t[] = new int[n];
		int v[] = new int[n];
		for(int i=0;i<n;i++)t[i] = fs.nextInt();
		for(int i=0;i<n;i++)v[i] = fs.nextInt();
		
		int tsum[] = new int[n+1];
		for(int i=0;i<n;i++)tsum[i+1]=tsum[i]+t[i];
		int total = 0;
		for(int i=0;i<n;i++) total += t[i];
		//0.5ずつ速度を求める。
		
		double maxV[] = new double[2*total+3];
		Arrays.fill(maxV, 1e9);
		for(int i=0;i<n;i++) {
			if(i==0) {
				for(int j=0;j<tsum[i+1];j++) {
					maxV[2*j] = Math.min(maxV[2*j],v[i]);
					maxV[2*j+1] = Math.min(maxV[2*j+1],v[i]);
					maxV[2*j+2] = Math.min(maxV[2*j+2],v[i]);
				}
			}
			else {
				for(int j=tsum[i];j<tsum[i+1];j++) {
					maxV[2*j] = Math.min(maxV[2*j],v[i]);
					maxV[2*j+1] = Math.min(maxV[2*j+1],v[i]);
					maxV[2*j+2] = Math.min(maxV[2*j+2],v[i]);
				}
			}
		}
		
		
		
		maxV[0]=0;
		for(int i=1;i<=2*total;i++) {
			maxV[i] = Math.min(maxV[i], maxV[i-1]+0.5); 
		}
		maxV[2*total]=0;
		for(int i=2*total;i>0;i--) {
			maxV[i-1] = Math.min(maxV[i-1], maxV[i]+0.5);  
		}
		
		double ans = 0;
		for(int i=0;i<2*total;i++) {
			double max = Math.max(maxV[i], maxV[i+1]);
			double min = Math.min(maxV[i], maxV[i+1]);
			double square = 0.5 * min;
			double triangle = 0.5 * 0.5 * (max - min);
			double s = square + triangle; //台形の面積
			ans += s;
		}
		System.out.println(ans);
		
		
		
		System.out.println();
		
	}
}

//高速なScanner
class FastScanner {
	private BufferedReader reader = null;
	private StringTokenizer tokenizer = null;

	public FastScanner(InputStream in) {
		reader = new BufferedReader(new InputStreamReader(in));
		tokenizer = null;
	}

	public String next() {
		if (tokenizer == null || !tokenizer.hasMoreTokens()) {
			try {
				tokenizer = new StringTokenizer(reader.readLine());
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
		}
		return tokenizer.nextToken();
	}

	public String nextLine() {
		if (tokenizer == null || !tokenizer.hasMoreTokens()) {
			try {
				return reader.readLine();
			} catch (IOException e) {
				throw new RuntimeException(e);
			}
		}
		return tokenizer.nextToken("\n");
	}

	public long nextLong() {
		return Long.parseLong(next());
	}

	public int nextInt() {
		return Integer.parseInt(next());
	}

	public double nextDouble() {
		return Double.parseDouble(next());
	}

	public int[] nextIntArray(int n) {
		int[] a = new int[n];
		for (int i = 0; i < n; i++)
			a[i] = nextInt();
		return a;
	}

	public long[] nextLongArray(int n) {
		long[] a = new long[n];
		for (int i = 0; i < n; i++)
			a[i] = nextLong();
		return a;
	}
}