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 = [int(x) for x in input().split()]
V = [int(x) for x in input().split()]
T.append(0)
V.append(0)
Vt = [0 for i in range(N + 1)]
r = sum(T)
R = [0 for i in range(N)]
for i in range(N - 1):
  r -= T[i]
  R[i] = r
for i in range(N - 1, -1, -1):
  Vt[i] = min(V[i], V[i + 1], Vt[i + 1] + T[i + 1])
L = 0
v0 = 0
for i in range(N):
  ta = min(V[i] - v0, (T[i] - (v0 - Vt[i])) / 2, T[i])
  tb = max(v0 + ta - Vt[i], 0)
  L += (2 * v0 + ta) * ta / 2 + (v0 + ta) * (T[i] - ta - tb) + (2 * (v0 + ta) - tb) * tb / 2
  v0 = v0 + ta - tb
print(L)