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(int, input().split()))
V = list(map(int, input().split()))

tmax = sum(T)
Vmax = max(V)
dp = [[-1 * float('inf') for _ in range(Vmax + 1)] for _ in range(tmax)]
dp[0][0] = 0
i = 0
j = 0
vmax = V[0]
for t in range(1, tmax):
    i += 1
    if i == T[j] + 1:
        i = 1
        j += 1
        vmax = V[j]
    for v in range(1, vmax + 1):
        acc = dp[t - 1][v - 1] + v - 0.5 if v != 0 else 0.5
        if v < vmax:
            same = dp[t - 1][v] + v + 0.25
            bra = dp[t - 1][v + 1] + v + 0.5
        else:
            same = dp[t - 1][v] + v
            bra = -float('inf')
        dp[t][v] = max(acc, same, bra)

if tmax == 1:
    print(0.25)
else:
    print(dp[-1][1] + 0.5)