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;

template <class T> inline bool chmin(T &a, T b) { /* {{{ */
  if (a > b) {
    a = b;
    return true;
  }
  return false;
} /* }}} */

int main() {
  int N;
  cin >> N;
  vector<int> T(N + 2), TS(N + 2), V(N + 2);
  for (int i = 1; i <= N; i++) {
    cin >> T[i];
    TS[i] = TS[i - 1] + T[i];
  }
  for (int i = 1; i <= N; i++) cin >> V[i];
  vector<double> max_v(40005);
  for (int i = 1; i <= N; i++) {
    for (int t = TS[i - 1] * 2; t <= TS[i] * 2; t++) {
      max_v[t] = V[i];
    }
  }
  for (int i = 0; i <= 40000; i++) {
    double t = (double)i / 2;
    for (int j = 0; j <= N; j++) {
      chmin(max_v[i], V[j] + abs(t - TS[j]));
      chmin(max_v[i], V[j + 1] + abs(t - TS[j]));
    }
  }
  double ans = 0;
  for (int i = 0; i < 40000; i++) {
    ans += (max_v[i] + max_v[i + 1]) / 4;
  }

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

  return 0;
}
/* vim:set fdm=marker: */