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:
from itertools import accumulate

N = int(input())
T = [int(s) for s in input().split()]
V = [int(s) for s in input().split()] + [0]
lt = sum(T)
S = list(accumulate(T))

for i in range(len(T), 0, -1):
    V[i-1] = min(V[i-1], V[i] + T[i-1])

vel2 = [0] * (2 * lt + 1)
for i in range(len(T)):
    ti = T[i]
    tr = S[i]
    for ts in range(tr - ti, tr):
        for t in (2 * ts + 1, 2 * ts + 2):
            if vel2[t-1] - (2 * tr - (t - 1)) == 2 * V[i+1]:
                vel2[t] = vel2[t-1] - 1
            elif vel2[t-1] == 2 * V[i]:
                vel2[t] = vel2[t-1]
            else:
                vel2[t] = vel2[t-1] + 1
ans = sum([(vel2[t-1] + vel2[t]) / 8 for t in range(1, 2 * lt + 1)])
print(ans)