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())) + [0]
V = list( map( int, input().split())) + [0]
now = 0
end = min( V[0], now + T[0])
ans = 0
for i in range(N):
    # print(i)
    t = T[i]
    v = V[i]
    end = min( v, now + t)
    plus = 0
    for j in range(i, N):
        end = min( end, V[j+1]+plus)
        plus += T[j+1]
    x, y = (end + t - now)/2, (end + t + now)/2
    # print(now, end,v,t,x,y)
    if y <= v:
        # print("T")
        # print((now+y)*x/2, (end+y)*(t-x)/2)
        ans += (now+y)*x/2 + (end+y)*(t-x)/2
        # print((now+y)*x/2 + (end+y)*(t-x)/2)
    else:
        # print("D")
        # print((v+now)*(v-now)/2, (v+end)*(v-end)/2, v*(end-2*v+t+now))
        ans += (v+now)*(v-now)/2 + (v+end)*(v-end)/2+v*(end-2*v+t+now)
        # print((v+now)*(v-now)/2 + (v+end)*(v-end)/2+v*(end-2*v+t+now))
    now = end
print(ans)