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:
ADJ = 10**2
n = int(input())
ts = list(map(int, input().split()))
vs = list(map(int, input().split()))
for i in range(n):
    ts[i] *= ADJ
    vs[i] *= ADJ
tt = sum(ts)
vss = [0]*tt
j = 0
cnt = 0
for i in range(tt):
    vss[i] = vs[j]
    cnt += 1
    if cnt == ts[j]:
        j += 1
        cnt = 0
#print(vss)
max_speed = 1
max_speeds = [0]*tt
for i in reversed(range(tt-1)):
    max_speeds[i] = min(max_speed,vss[i],vss[i+1],max_speeds[i+1]+1)
    max_speed += 1
#print(max_speeds)
cv = 0
for i in range(tt):
    if cv < max_speeds[i]:
        cv += 1
    elif cv == max_speeds[i]:
        pass
    else:
        cv -= 1
    vss[i] = cv
#print(vss)
res = 0
for i in range(tt):
    if i == 0:
        res += vss[i] / 2
    else:
        if vss[i] >= vss[i-1]:
            res += vss[i-1] + (vss[i] - vss[i-1]) / 2
        else:
            res += (vss[i-1] - vss[i]) / 2 + vss[i]
    #print()
    #print(str(i)+" "+str(res))
print(res/ADJ/ADJ)