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(lambda t: int(t) * 10, input().split()))
V = list(map(int, input().split())) + [0]
R = sum(T)

maxSpeed = [10**18] * (R + 1)
maxSpeed[0] = 0
maxSpeed[R] = 0
now = 0
for i, t in enumerate(T):
    now += t
    maxSpeed[now] = min(V[i], V[i + 1])

now = 0
for v, t in zip(V, T):
    for i in range(t + 1):
        maxSpeed[now + i] = min(maxSpeed[now + i], v)
    now += t

for t in range(1, R + 1):
    maxSpeed[t] = min(maxSpeed[t], maxSpeed[t - 1] + 0.1)
for t in range(R)[::-1]:
    maxSpeed[t] = min(maxSpeed[t], maxSpeed[t + 1] + 0.1)

base = 0
upper = 0
for v1, v2 in zip(maxSpeed, maxSpeed[1:]):
    base += min(v1, v2)
    upper += abs(v1 - v2)
ans = base + upper / 2
print(ans / 10)