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; input = sys.stdin.buffer.readline
from collections import defaultdict
con = 10 ** 9 + 7; INF = float("inf")

def getlist():
	return list(map(int, input().split()))

def dist(i, j, lim, t):
	if abs(i - j) > t:
		dist = -INF
	else:
		x = (t + j - i) / 2
		y = (t + j + i) / 2
		if  y <= lim:
			dist = (i + y) * x / 2 + (j + y) * (t - x) / 2
		else:
			dist = (i + lim) * (lim - i) / 2 + lim * (t - (lim - i) - (lim - j)) + (j + lim) * (lim - j) / 2
			# print(dist)

	return dist

#処理内容
def main():
	N = int(input())
	T = getlist()
	V = getlist()
	DP = [[-INF] * 101 for i in range(N + 1)]
	DP[0][0] = 0
	for i in range(N - 1):
		lim = V[i]
		t = T[i]
		for p in range(lim + 1):
			for q in range(lim + 1):
				d = dist(p, q, lim, t)
				DP[i + 1][q] = max(DP[i + 1][q], DP[i][p] + d)

	lim = V[-1]
	t = T[-1]
	for p in range(lim + 1):
		d = dist(p, 0, lim, t)
		DP[N][0] = max(DP[N][0], DP[N - 1][p] + d)


	print(DP[N][0])

if __name__ == '__main__':
	main()