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:
import sys

stdin = sys.stdin

ni = lambda: int(ns())
na = lambda: list(map(int, stdin.readline().split()))
ns = lambda: stdin.readline()

n = ni()
t = na()
v = na()

bound = [0] * (n+1)
for i in range(n-1):
    bound[i+1] = min(v[i], v[i+1], bound[i] + t[i])
for i in range(n-1,0,-1):
    bound[i] = min(bound[i], v[i], v[i-1], bound[i+1] + t[i])

ans = 0
for i in range(n):
    l = bound[i]
    r = bound[i+1]
    # l + x = r + t - x
    # x = (r-l+t)/2
    x = (r-l+t[i])/2
    if l + x <= v[i]:
        ans += (l + (l + x)) * x / 2 + (l + x + r) * (t[i] - x) / 2
    else:
        ans += (v[i] + l) * (v[i] - l) / 2 + (v[i] + r) * (v[i] - r) / 2
        ans += (t[i] - (v[i] - l) - (v[i] - r)) * v[i]
print("{:.14f}".format(ans))