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 PYTHON):

Solution:
N=input()
t=map(int,raw_input().split())
v=map(int,raw_input().split())

T=sum(t)
t_len=len(t)

maxv=[0]
for ind_i,i in enumerate(t):
	for j in range(i*2):
		if j==i*2-1:
			if ind_i==t_len-1:
				maxv.append(0)
			else:	
				maxv.append( min(v[ind_i],v[ind_i+1]) )
		else:
			maxv.append(v[ind_i])


for i in range(1,2*T):
	maxv[i]=min(maxv[i],maxv[i-1]+0.5)


for i in reversed(range(1,2*T)):
	maxv[i]=min(maxv[i],maxv[i+1]+0.5)
#print maxv

#for ind,i in enumerate(maxv):
#	print ind,i


cnt=0
for i in maxv:
	cnt+=0.5

print sum(maxv)/2