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

T.append(0)
V.append(0)
time = [0]*(N+1)
time[0] = T[0]
for i in range(N):
    time[i+1] = time[i]+T[i+1]
    
L = [0]*(2*time[-1]+1)
i = 0
t = 0.5
while t <= time[-1]:
    num = V[i]
    if t == time[i]:
        i += 1
        num = min(num, V[i])
        
    v = L[int(2*t)-1]+0.5
    while v > num:
        v -= 0.5
        
    for j in range(int(i)+1, N+1):
        if v-0.5-V[j] > time[j-1]-t:
            v -= 1
        elif v-V[j] > time[j-1]-t:
            v -= 0.5
        
    L[int(2*t)] = v
    t += 0.5

ans = 0
for i in range(len(L)-1):
    if L[i+1] > L[i]:
        ans += L[i]*0.5+0.25
    elif L[i+1] == L[i]:
        ans += L[i]*0.5
    else:
        ans += L[i]*0.5-0.25
        
print(ans)