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 f(x,l,r,v):
    if 0<=x<=l:
        return v+(l-x)
    if l<=x<=r:
        return v
    if r<=x:
        return v+x-r
def speed(t):
    num=10**9
    for l,r,v in table:
        num=min(num,f(t,l,r,v))
    return num
N=int(input())
T=[int(i) for i in input().split()]
V=[int(i) for i in input().split()]
X=[0]*(N+1)
for i in range(1,N+1):
    X[i]=X[i-1]+T[i-1]
table=[(X[i] , X[i+1],V[i]) for i in range(N)]
table.append((0,0,0))
table.append((X[-1],X[-1],0))
#print(table)
s=speed(0)
meter=0
for i in range(1,2*X[-1]+1):
    t=speed(i*0.5)
    meter+=(s+t)*(0.25)
    s=t
print(meter)