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:
INF = 10 ** 20
n = int(input())
tlst = list(map(int, input().split()))
vlst = list(map(int, input().split()))
dp = [[-INF] * 201 for _ in range(n + 1)]
dp[0][0] = 0
for i in range(n):
    t, v = tlst[i], vlst[i]
    for v_fr in range(v + 1):
        pre_score = dp[i][v_fr]
        for v_to in range(v + 1):
            diff = abs(v_to - v_fr)
            if diff > t:continue
            max_speed = min((v_to + v_fr + t) / 2, v)
            dp[i + 1][v_to] = max(dp[i + 1][v_to], pre_score + max_speed * t - (max_speed - v_to) ** 2 / 2 - (max_speed - v_fr) ** 2 / 2)
print(dp[n][0])