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(x) for x in input().split()]
vSection = [int(x) for x in input().split()]

vPoint = [0]
for i in range(1, N):
	vPoint.append(min(vSection[i - 1], vSection[i]))
vPoint.append(0)

for i in reversed(range(1, N + 1)):
	if t[i - 1] < vPoint[i - 1] - vPoint[i]:
		vPoint[i - 1] = t[i - 1] + vPoint[i]


ans = 0
VE = 0
for i in range(N):
	VS = VE
	if VS + t[i] <= vPoint[i + 1]:
		VE = VS + t[i]
		ans += VS * t[i] + t[i] ** 2 / 2
	else:
		VE = vPoint[i + 1]
		sumT = sum(t[:i])
		a = VS - (0 if i == 0 else sumT)
		b = VE + sum(t[:i+1])
		y = (a + b) / 2
		if y >= vSection[i]:
			t1 = vSection[i] - VS
			t2 = vSection[i] - VE
			ans += VS * t1 + t1 ** 2 / 2 + vSection[i] * (t[i] - t1 - t2) + VE * t2 + t2 ** 2 / 2
		else:
			x = y - a
			ans += VS * (x - sumT) + (x - sumT) ** 2 / 2 + VE * (t[i] - (x - sumT)) + (t[i] - (x - sumT)) ** 2 / 2
			
print(ans)