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 CPP):

Solution:
#include <iostream>
#include <iomanip>
using namespace std;

int main()
{
    int n;
    cin >> n;
    int t[102], v[102];
    t[0] = 0;
    for(int i = 0; i < n; i++){
        cin >> t[i + 1];
        t[i + 1] *= 2;
        t[i + 1] += t[i];
    }
    for(int i = 0; i < n; i++){
        cin >> v[i];
        v[i] *= 2;
    }
    int b[50000];
    for(int i = 0; i <= t[n]; i++) b[i] = min(i, t[n] - i);
    for(int i = 0; i < n; i++){
        for(int j = 0; j < t[i]; j++) b[j] = min(b[j], v[i] + t[i] - j);
        for(int j = t[i]; j < t[i + 1]; j++) b[j] = min(b[j], v[i]);
        for(int j = t[i + 1]; j <= t[n]; j++) b[j] = min(b[j], v[i] + j - t[i + 1]);
    }
    int ans = 0;
    for(int i = 0; i < t[n]; i++) ans += b[i] + b[i + 1];
    cout << fixed << setprecision(15) << ans / 8.0 << endl;
}