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()))
T = [0]
tot = 0
for i in range(len(t)):
    tot += t[i]
    T.append(tot)
T = list(map(lambda x: x*2,T))
v = list(map(int,input().split()))
v.append(0)
curv = 0
dist = 0
curt = 0
#print(T)
#print(v)

for i in range(N):
    for _ in range(T[i], T[i+1]):
        prev = curv
        ok = 1
        #if (0 < T[j+1]-curt <= curv - v[j+1] for j in range(N)):
            #print(T[1],curt,curv,v[1])
        for j in range(N-i):
            if T[i+j+1]/2.0 - curt > curv - v[i+j+1]:
                continue
            else:
                ok = 0
                break
        if ok == 1:
            if curv < v[i]:
                curv += 0.5
        elif ok == 0:
            curv -= 0.5
        curt += 0.5
        dist +=(prev+curv) / 4.0




print(dist)