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 solve(N, T, V):
    from itertools import accumulate
    ds = [0] + list(accumulate(T))
    vs = [0] + [min(V[i], V[i + 1]) for i in range(len(V) - 1)] + [0]
    for i in range(len(vs)):
        d = ds[i]
        vs[i] = min([v + abs(d - ds[j]) for j, v in enumerate(vs)])
    def calc(t, v0, v1, maxv):
        t0 = maxv - v0
        t1 = maxv - v1
        if t0 + t1 < t:
            return t * maxv - t0 * t0 / 2 - t1 * t1 / 2
        else:
            t0 = (t + v1 - v0) / 2
            t1 = (t + v0 - v1) / 2
            v = (t + v0 + v1) / 2
            S = v * t - t0 * t0 / 2 - t1 * t1 / 2
            return S
    r = 0
    for i in range(N):
        c = calc(T[i], vs[i], vs[i + 1], V[i])
        r += c
    return r
    
if __name__ == '__main__':
    N = int(input())
    T = [int(_) for _ in input().split()]
    V = [int(_) for _ in input().split()]
    print(solve(N, T, V))