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 PYTHON3):

Solution:
N = int(input())
t = [0] + list(map(int, input().split())) + [0]
v = [0] + list(map(int, input().split())) + [0]
for i in range(N, 0, -1):
    v[i] = min(v[i], v[i + 1] + t[i])
for i in range(1, N + 1):
    v[i] = min(v[i], v[i - 1] + t[i])
ans = 0
v_f = 0
for i in range(1, N + 1):
    v_l = min(v_f + t[i], v[i + 1], v[i])
    if 2 * v[i] - v_f - v_l <= t[i]:
        ans += (v[i] + v_f) * (v[i] - v_f) / 2 + v[i] * (t[i] - 2 * v[i] + v_f + v_l) + (v[i] + v_l) * (v[i] - v_l) / 2
    else:
        t_1 = (t[i] + v_l - v_f) / 2
        t_2 = t[i] - t_1
        ans += (2 * v_f + t_1) * t_1 / 2 + (2 * v_l + t_2) * t_2 / 2
    v_f = v_l
print('{:.10f}'.format(ans))