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:
# 19/4/17再戦
# -x + a = x + b だと x = (sum)/2で~.5になる可能性がある
# う～～～ん

n = int(input())
ts = list(map(int, input().split()))
vs = list(map(int, input().split()))
sum_t = sum(ts)
n = 2*sum_t+1
max_v = [float("inf")]*n
# 0から右上、-1から左上
for i in range(n):
    max_v[i] = min(i/2, (sum_t-i/2))

# intでtimeは管理するか...
# 0→0, 0.5→1, 1→2 ... i/2→i
t_cnt = 0
for t, v in zip(ts, vs):
    for j in range(2*t+1):
        max_v[t_cnt + j] = min(max_v[t_cnt + j], v)

    # 左側に伸ばしていく
    left = t_cnt
    while left >= 0:
        v_left = v + (t_cnt - left)/2
        if max_v[left] < v_left:
            break
        max_v[left] = v_left
        left -= 1
    # 右側に伸ばしていく
    t_cnt += 2*t
    right = t_cnt
    while right < n:
        v_right = v + (right - t_cnt)/2
        if max_v[right] < v_right:
            break
        max_v[right] = v_right
        right += 1

#print(max_v)
ans = sum(max_v) - (max_v[0]+max_v[-1])/2
print(ans/2)