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:
import sys
sys.setrecursionlimit(10000000)
MOD = 10 ** 9 + 7
INF = 10 ** 15

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

def calc_area(v1,v2,t,lim):
    if v1 > v2:
        v1,v2 = v2,v1
    if v2 == lim:
        if lim - v1 < t:
            return t*lim - (lim - v1)*(lim - v1)/2
        else:
            return (v1 + v2)*t/2
    else:
        if 2*lim - v1 - v2 < t:
            return t*lim - (lim - v1)*(lim - v1)/2 - (lim - v2)*(lim - v2)/2
        else:
            mid = (t + v2 - v1)/2
            return (mid + 2*v1)*mid/2 + (t - mid + 2*v2)*(t - mid)/2

def calc(N,T,V,maxv):
    ans = 0
    v = 0
    for i in range(N):
        next_v = min(v + T[i],maxv[i])
        ans += calc_area(v,next_v,T[i],V[i])
        v = next_v
    return ans

maxV = [0] * N
for i in range(N - 2,-1,-1):
    maxV[i] = min(maxV[i + 1] + T[i + 1],V[i],V[i + 1])
ans = calc(N,T,V,maxV) 
print(ans)