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()))
V = list(map(int,input().split()))

b = 2
bg = 1.0 / b

M = [] #t = i秒の最高速度
for i in range(n) :
    for j in range(T[i] * b) :
        M.append(V[i])

R = [0] + T[:]
for i in range(n) :
    R[i+1] += R[i]


for i in range(n) :

    le = R[i] * b
    ri = R[i+1] * b

    
    j = 0
    if i-1 >= 0 and V[i-1] > V[i] :
        while le >= 0 :
            M[le] = min(M[le], V[i] + j*bg)
            #print(M[le])
            j += 1
            le -= 1
    j = 0
    if i+1 < n and V[i+1] > V[i] :
        while ri < len(M) :
            M[ri] = min(M[ri], V[i] + j*bg)
            j += 1
            ri += 1

for i in range(len(M)) :
    M[i] = min(i*bg, (len(M)-i)*bg, M[i])

M += [0]

ans = 0
for i in range(len(M)-1) :
    ans += (M[i] + M[i+1]) * bg * 0.5

print(ans)