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:
def main():
	n = int(input())
	*t, = map(int, input().split())
	tot = sum(t)
	*v, = map(int, input().split())
	lim1 = []
	lim2 = [(1, 0, 0), (-1, tot, tot)]
	til = 0
	for t1, t2, v1 in zip([0] + t, t, v):
		lim1.append((til, til + t2, v1))
		lim2.append((-1, til, til + v1))
		lim2.append((1, til + t2, v1 - (til + t2)))
		til += t2

	def mi(time):
		ret = float("inf")
		for l, r, lim in lim1:
			if l <= time <= r:
				ret = min(ret, lim)
		for f, tt, y in lim2:
			if f == -1 and time <= tt:
				ret = min(ret, y - time)
			if f == 1 and time >= tt:
				ret = min(ret, time + y)
		return ret

	print(sum(mi(i / 2) for i in range(tot * 2 + 1)) / 2)

if __name__ == '__main__':
	main()