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 = list(map(int, input().split()))
V = list(map(int, input().split()))

V = [0] + V + [0]

# 0.前処理
# N+2個の条件をつくる
left = [0]
right = [0]

for i in range(N):
    left.append(right[i])
    right.append(left[-1]+T[i])
    
left.append(right[-1])
right.append(right[-1])

INF = float("inf")
vel = [INF]*(2*right[-1]+1)

# 1.各ステップでの許容測度の最大値の最小値を求める
for t in range(2*right[-1]+1):
    for i in range(N+2):
        if (t/2) < left[i]:
            vel[t] = min(vel[t], -(t/2)+(V[i]+left[i]))
        elif left[i] <= (t/2) and (t/2) <= right[i]:
            vel[t] = min(vel[t], V[i])
        elif (t/2) > right[i]:
            vel[t] = min(vel[t], (t/2)+(V[i]-right[i]))
            

# 2.面積を求める
ans = 0
for t in range(2*right[-1]):
    ans += (1/2)*0.5*(vel[t]+vel[t+1])
    
print(ans)