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:
'''
38分あたり
https://www.youtube.com/watch?v=T1zXzsoK268
'''
def input_num_list(): return list(map(int, input().split()))

n = int(input())
t = input_num_list()
T = sum(t)
v = input_num_list()
maxv = [100 for _ in range(2 * T + 1)]
maxv[0] = 0
maxv[2 * T] = 0

t_l, t_r = 0, 0
for i in range(n):
    t_l = t_r
    t_r += t[i]
    vv = v[i]
    for j in range(2 * t_l, 2 * t_r + 1):
        maxv[j] = min(maxv[j], vv)

#print(maxv)
i = 1
while i <= 2 * T:
    maxv[i] = min(maxv[i], maxv[i - 1] + 0.5)
    i += 1
#print(maxv)
i = 2 * T - 1
while i >= 0:
    maxv[i] = min(maxv[i], maxv[i + 1] + 0.5)
    i -= 1
#print(maxv)
ans = 0
i = 0
while i <= 2 * T - 1:
    ans += 0.25 * (maxv[i] + maxv[i + 1])
    i += 1
    
print(ans)
#print(maxv)