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()))
max_t = sum(T)
vl = [0]*(max_t+1)
vr = [0]*(max_t+1)
vm = [0]*(max_t+1)
t = 0
for s, u in zip(T,V):
  for i in range(t,t+s):
    vm[t] = u
    t += 1
t = 1
for s, u in zip(T,V):
  for i in range(t, t+s):
    vl[t] = min(u, vl[t-1]+1)
    t += 1
t = max_t
for s, u in zip(T[::-1],V[::-1]):
  for i in range(t-s,t)[::-1]:
    vr[t-1] = min(u,vr[t]+1)
    t -= 1
v = [min(vl[i],vr[i]) for i in range(max_t+1)]
#print(vm)
#print(vl)
#print(vr)
#print(v)
ans = 0
for i in range(max_t):
  ans += (v[i] + v[i+1]) / 2
  if v[i] == v[i+1] and v[i] != vm[i]:
    ans += 0.25
print("{:.10f}".format(ans))