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 atcoder_express(N: int, T: list, V: list)->float:
    MAXT = sum(T)
    graph = [0] * (2*MAXT + 1)

    T = [0, 0] + T + [0]
    for i in range(N+2):
        T[i+1] += T[i]

    V = [0] + V + [0]

    conditions = []
    for i in range(N+2):
        v, l, r = V[i], T[i], T[i+1]
        conditions.append((l, r, v))

    total = 0.0
    prev_v = 0.0
    for t in range(MAXT*2+1):
        min_v = float('inf')
        x = t / 2
        for l, r, v in conditions:
            if x < l:
                min_v = min(min_v, v + (l - x))
            elif r < x:
                min_v = min(min_v, v + (x - r))
            else:
                min_v = min(min_v, v)
        total += (prev_v + min_v) * 0.5 / 2
        prev_v = min_v

    return total


if __name__ == "__main__":
    N = int(input())
    T = [int(s) for s in input().split()]
    V = [int(s) for s in input().split()]
    ans = atcoder_express(N, T, V)
    print(ans)