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

T_p = [0,0]
temp = 0
for tt in T:
    temp += tt
    T_p.append(temp)
T_p.append(temp)

Tmax = temp

def vv(i,tt):
    a = T_p[i]
    b = T_p[i + 1]
    if i == 0 or i == N + 1:
        vmax = 0
    else:
        vmax = V[i - 1]
    
    if tt >= b:
        res = tt - b + vmax
    elif tt <= a:
        res = a - tt + vmax
    else:
        res = vmax

    return res

mints = [min([vv(i, tt/2) for i in range(N + 2)]) for tt in range(2 * Tmax + 1)]

res = 0
for i in range(len(mints) -1 ):
    res += (mints[i] + mints[i + 1])/4

print(res)