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()))
MAX_T = 200
MAX_V = 100
INF = 1e9
index = 0
tsum = 0
T = sum(t)
dp = [[-INF] * (2*MAX_V+1) for i in range(2*T+1)]
dp[0][0] = 0.0
for i in range(2*T):
    if i+1 > 2 * (tsum + t[index]):
        tsum += t[index]
        index += 1
    for j in range(2*MAX_V+1):
        jj = 0.5 * j
        if jj > v[index]:
            continue
        if jj < v[index]:
            dp[i+1][j+1] = max((dp[i+1][j+1], dp[i][j] + 0.5 * jj + 0.125))
        if jj > 0:
            dp[i+1][j-1] = max((dp[i+1][j-1], dp[i][j] + 0.5 * jj - 0.125))
        dp[i+1][j] = max((dp[i+1][j], dp[i][j] + 0.5 * jj))

print(dp[2*T][0])