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()))
def exe(x, y, z, t):
  ans = t * y
  if x < y:
    ans -= (y - x) * (y - x) / 2
  if y > z:
    ans -= (y - z) * (y - z) / 2
  return ans

def upd(x, y, z, t):
  a = y - x
  b = y - z
  if y > z:
    if t < a:
      y = x + t
    elif t < a + b:
      y = x + (t + (z - x)) / 2
  else:
    if t < a:
      y = x + t
  return y

v = [0] + v + [0]
L = [0] * len(v)
v[1] = upd(v[0], v[1], v[2], t[0])
v[-2] = upd(v[-1], v[-2], v[-3], t[-1])
for i in range(1, N):
  k = -1
  for j in range(1, N + 1):
    if L[j] == 0:
      if k == -1:
        k = j
      else:
        if v[j] < v[k]:
          k = j
  L[k] = 1
  if k != 1:
    v[k - 1] = upd(min(v[k], v[k - 2]), v[k - 1], max(v[k], v[k - 2]), t[k - 2])
  if k != N:
    v[k + 1] = upd(min(v[k], v[k + 2]), v[k + 1], max(v[k], v[k + 2]), t[k])

ans = 0
for i in range(N):
  ans += exe(v[i], v[i + 1], v[i + 2], t[i])

print(ans)