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(" "))
tsa = [0]
for t in ts:
    tsa.append(tsa[-1]+t)

T = 0
X = 0
V = 0
def acc():
    ppp = False
    for j in range(N):
        if tsa[j] <= T:
            continue
        if vs[j] > V:
            continue
        c = tsa[j] - T - (V - vs[j])
        if c <= 0:
            return -1
        d = tsa[j] - (T+1) - (V+1 - vs[j])
        if d < 0:
            ppp = True
    if tsa[-1] - T - (V - 0)<=0:
        return -1
    if tsa[-1] - (T+1) - (V+1 - 0) < 0:
        ppp = True
    if ppp:
        return 0
    return 1

def tind():
    for i in range(len(ts)):
        if tsa[i] <= T and  T < tsa[i+1]:
            return i

for i in range(sum(ts)):
    a = acc()
    ti  = tind()
    if a == 1:
        if V == vs[ti]:
            X += V
        else:
            X += V + 0.5
            V += 1
    elif a == 0:
        if V == vs[ti]:
            X += V
        else:
            X += V + 0.25
    else:
        X += V - 0.5
        V -= 1
    T=i+1

print X