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:
def inpl(): return [int(i) for i in input().split()]

def trapezoid(St, En, H, i):
    if St+H[i][0] >= En-H[i][0]:
        return (En-St)**2/4
    result = H[i][0]*(En-St-H[i][0])
    if H[i][1] > St + H[i][0]:
        result += trapezoid(St+H[i][0],min(En-H[i][0],H[i][1]),H,i+1)
    if H[i][2] < En - H[i][0]: 
        result += trapezoid(max(St+H[i][0],H[i][2]),En-H[i][0],H,i+1)       
    return result

N = int(input())
t = inpl()
v = inpl()
H = [[v[0], 0, t[0]]]
for i in range(1,N):
    H.append([v[i], H[i-1][2], H[i-1][2]+t[i]])

H.append([float('inf'), 0, 0])
H.sort(key = lambda x: x[0])
for i in range(N,0,-1):
    H[i][0] -= H[i-1][0]    
print(trapezoid(0,sum(t), H, 0))