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 = [int(x) for x in input().split()]
V = [int(x) for x in input().split()]

import itertools
tv = [0] + list(itertools.chain(*[[v] * t * 2 for t, v in zip(T, V)])) + [0]
for i in itertools.accumulate(T):
    tv[i * 2] = min(tv[i * 2], tv[i * 2 + 1])
for i in range(len(tv)-1, 0, -1):
    if tv[i] < tv[i-1] - 0.5:
        tv[i-1] = tv[i] + 0.5
    elif tv[i] > tv[i-1] + 0.5:
        for j in range(i, len(tv) - 1):
            if tv[j-1] + 0.5 < tv[j]:
                tv[j] = tv[j-1] + 0.5
            else:
                break
print(sum([(v1 + v2) * 0.5 * 0.5 for v1, v2 in zip(tv[1:], tv[:-1])]))