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<bits/stdc++.h>
using namespace std;

int main() {
    int n;
    cin >> n;
    vector<int> t(n+3), v(n+2);
    for (int i = 2; i <= n+1; i++) cin >> t[i];
    for (int i = 1; i <= n; i++) cin >> v[i];
    partial_sum(begin(t), end(t), begin(t));

    double p = 0, ans = 0;
    for (double a = 0; a <= t.back(); a += .5) {
        priority_queue<double, vector<double>, greater<double>> pq;
        for (int i = 0; i < n+2; i++) pq.push(v[i] + max({ t[i] - a, a - t[i+1], .0 }));
        ans += 0.25 * (p + pq.top());
        p = pq.top();
    }

    cout << fixed << setprecision(10) << ans << endl;
}