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()))
v = list(map(int,input().split()))

v.append(0)
interval = 0.5
total_t = sum(t)
now_t = 0
now_v = 0
T = [sum(t[:i]) for i in range(1,N+1)]
now_n = 0
d = 0
d_int = 0
while now_t < total_t:
	nxt_t, nxt_v = T[now_n], v[now_n + 1]
	delta_t, delta_v = nxt_t - now_t, nxt_v - now_v
	rest_t = [T[i]-now_t for i in range(N)]
	if now_v < v[now_n]:
		if delta_t + delta_v == 0:
			a = -1
		else:
			a = 1
	elif now_v == v[now_n]: 
		if delta_t + delta_v == 0:
			a = -1
		else:
			a = 0
	for i in range(N):
		rest = rest_t[i]
		if rest > 0:
			d_v = now_v - v[i+1]
			if rest == d_v:
				a = -1

	d_int += 2 * int(now_v * 2) + a
	d += now_v * interval + 1/2 * a * interval * interval
	now_t += interval
	now_v += a * interval
	if now_t == T[now_n]:
		now_n += 1

print(d)