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())
tt = map(int, input().split())
vv = map(int, input().split())

v_now = 0
v_max = [v_now]
for t, v in reversed(list(zip(tt, vv))):
    for i in range(t * 2):
        if v_now + 0.5 <= v:
            v_now += 0.5
        else:
            v_now = min(v_now, v)
        v_max.append(v_now)
v_max = v_max[::-1]

v = 0
total = 0
for i in range(len(v_max) - 1):
    v0 = v_max[i]
    v1 = v_max[i+1]
    if v + 0.5 <= v1:
        if v < v0:
            total += (v + v + 0.5) * 0.25
            v += 0.5
        else:
            total += v * 0.5
    elif v > v1:
        total += (v + v - 0.5) * 0.25
        v -= 0.5
        assert v == v1, '%s %s'%(v,v1)
    else:
        total += v * 0.5
print(total)