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:
import sys
def I(): return int(sys.stdin.readline())
def LI(): return list(map(int,sys.stdin.readline().split()))
mod = 10**9 + 7
inf = float('inf')

N = I()
t = LI()
v = LI()
T = sum(t)*2
vcandi =[[0]*(N+2) for _ in range(T+1)]
sumt = int(0)
for i in range(N):
    time = t[i]*2
    for j in range(sumt):
        vcandi[j][i] = v[i]+(sumt-j)*0.5
    for j in range(time+1):
        vcandi[sumt+j][i] = v[i]
    sumt = sumt+time
    for j in range(T-sumt+1):
        vcandi[sumt+j][i] = v[i]+j*0.5
for i in range(T+1):
    vcandi[i][N] = i*0.5
    vcandi[i][N+1] = (T-i)*0.5
ansV =[0]*(T+1)
for i in range(T+1):
    ansV[i] = min(vcandi[i])

#print(ansV)
cur = float(0)
for i in range(T):
    cur += (ansV[i]+ansV[i+1])/2*0.5

ans = cur
print(ans)