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 PYTHON):

Solution:
N = int(raw_input())
ts = map(int, raw_input().split())
vs = map(int, raw_input().split())

vs.append(0)

for i in xrange(N-1, -1, -1):
  vs[i] = min(vs[i], vs[i+1] + ts[i])

def dist(vstart, vend, time):
  return time * (min(vstart, vend) + (abs(vstart - vend) / 2.0))

v = 0
ans = 0.0
for i in xrange(N):
  a = min(ts[i], vs[i] - v)
  vs[i] = min(vs[i], v + a)
  b = max(0, vs[i] - vs[i+1]) # <= ts[i]
  c = ts[i] - a - b
  #print i, v, vs[i], ts[i], '   ', a, c, b

  if c >= 0:
    ans += dist(v, vs[i], a) + dist(vs[i], vs[i], c) + dist(vs[i], vs[i+1], b)
  else:
    a, b = (a-b+ts[i])/2.0, (b-a+ts[i])/2.0
    ans += dist(v, v+a, a) + dist(v+a, v+a-b, b)

  v = v + a - b

print ans