Problem:
﻿Score : 400 points

Problem Statement

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}.

Sample Input 1

1
100
30

Sample Output 1

2100.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 30 seconds, it accelerates at a rate of 1m/s^2, covering 450 meters.

In the subsequent 40 seconds, it maintains the velocity of 30m/s, covering 1200 meters.

In the last 30 seconds, it decelerates at the acceleration of -1m/s^2, covering 450 meters.

The total distance covered is 450 + 1200 + 450 = 2100 meters.

Sample Input 2

2
60 50
34 38

Sample Output 2

2632.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 34 seconds, it accelerates at a rate of 1m/s^2, covering 578 meters.

In the subsequent 26 seconds, it maintains the velocity of 34m/s, covering 884 meters.

In the subsequent 4 seconds, it accelerates at a rate of 1m/s^2, covering 144 meters.

In the subsequent 8 seconds, it maintains the velocity of 38m/s, covering 304 meters.

In the last 38 seconds, it decelerates at the acceleration of -1m/s^2, covering 722 meters.

The total distance covered is 578 + 884 + 144 + 304 + 722 = 2632 meters.

Sample Input 3

3
12 14 2
6 2 7

Sample Output 3

76.000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 6 seconds, it accelerates at a rate of 1m/s^2, covering 18 meters.

In the subsequent 2 seconds, it maintains the velocity of 6m/s, covering 12 meters.

In the subsequent 4 seconds, it decelerates at the acceleration of -1m/s^2, covering 16 meters.

In the subsequent 14 seconds, it maintains the velocity of 2m/s, covering 28 meters.

In the last 2 seconds, it decelerates at the acceleration of -1m/s^2, covering 2 meters.

The total distance covered is 18 + 12 + 16 + 28 + 2 = 76 meters.

Sample Input 4

1
9
10

Sample Output 4

20.250000000000000000

The maximum distance is achieved when a train runs as follows:

In the first 4.5 seconds, it accelerates at a rate of 1m/s^2, covering 10.125 meters.

In the last 4.5 seconds, it decelerates at the acceleration of -1m/s^2, covering 10.125 meters.

The total distance covered is 10.125 + 10.125 = 20.25 meters.

Sample Input 5

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

Sample Output 5

20291.000000000000

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace atcoder
{
	class Program
	{
		static int In => int.Parse(Console.ReadLine());
		static int[] Ins => Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
		static void Out(bool b) => Console.WriteLine(b ? "Yes" : "No");

		static void Main(string[] args)
		{
			var n = In;
			var t = Ins;
			var v = Ins;
			var n_sum = new int[n];
			for (int i = 0; i < n; i++)
			{
				if (i == 0) n_sum[i] = t[i];
				else n_sum[i] = n_sum[i - 1] + t[i];
			}
			{
				var tt = new List<int>(v);
				tt.Add(0);
				v = tt.ToArray();
			}
			var sum = t.Sum();
			var a = new int[sum + 1];
			var lim = new int[sum+1];
			{
				int cc = 0;
				for (int i = 0; i < a.Length; i++)
				{
					if (i <= n_sum[cc]) a[i] = v[cc];
					else
					{
						cc++;
						i--;
					}
				}
			}
			int qu = 0;
			{
				int cc = 0;
				for (int i = 0; i < lim.Length; i++)
				{
					if (n_sum[cc] < i) cc++;
					if (i == 0) lim[i] = 0;
					else
					{
						var pre = lim[i - 1];
						lim[i] = Math.Min(lim[i - 1] + 1, a[i]);
						for (int j = cc; j < n; j++)
						{
							if (n_sum[j] - i < pre - v[j + 1])
							{
								lim[i] = pre - 1;
							}
							else if (n_sum[j] - i == pre - v[j + 1])
							{
								lim[i] = pre;
								if (a[i] > lim[i] && a[i - 1] > lim[i - 1]) qu++;
							}
						}
					}
				}
			}
			Console.WriteLine(lim.Sum() + qu * 0.25);
			//Console.ReadLine();
		}
	}
}