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 main():
    N = int(input())
    T = list(map(int, input().split()))
    V = list(map(int, input().split()))
    T = [t * 2 for t in T]
    V = [v * 2 for v in V]
    Tall = sum(T)
    R = [0] * (Tall + 1)
    j = N - 1
    tc, vc, vn = T[-1], V[-1], V[-1]
    R[-1] = 0
    for i in reversed(range(Tall)):
        tc -= 1
        if tc == 0:
            j -= 1
            tc, vn = T[j], V[j]
        R[i] = min(R[i + 1] + 1, vc, vn)
        vc = vn
    v = 0
    d = 0.0
    j = 0
    tc, vc, vn = T[j], V[j], V[j]
    T.append(0)
    V.append(0)
    for i in range(1, Tall + 1):
        tc -= 1
        if tc == 0:
            j += 1
            tc, vn = T[j], V[j]
        vnn = min(v + 1, R[i], vc, vn)
        d += (v + vnn) / 2
        vc = vn
        v = vnn
    print(d / 4)

main()