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
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 9)
MOD = 10 ** 9 + 7

N = int(input())
T = [0] + list(map(int, input().split()))
V = [0] + list(map(int, input().split()))

left = [0] * (N + 1)
right = [0] * (N + 1)

for i in range(1, N):
    start = left[i - 1]
    left[i] = min(start + T[i], V[i], V[i + 1])

for i in range(N - 1, 0, -1):
    start = right[i + 1]
    right[i] = min(start + T[i + 1], V[i], V[i + 1])

total = [min(left[i], right[i]) for i in range(N + 1)]
ans = 0
for i in range(1, N + 1):
    x = (V[i] - total[i - 1]) + (V[i] - total[i])
    if x >= T[i]:
        V[i] = (total[i - 1] + T[i] + total[i]) / 2
        x = T[i]
    d =(total[i - 1] + V[i]) * (V[i] - total[i - 1]) / 2 #台形左側
    d += (total[i] + V[i]) * (V[i] - total[i]) / 2 #台形右側
    d += V[i] * (T[i] - x) #真ん中の定速部分

    ans += d

print (ans)