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()))
V1 = list(map(int, input().split()))
V1.append(0)

ST = [0] * (N + 1)
for i in range(N):
    ST[i + 1] = ST[i] + T[i]

V2 = [0] * N
for i in range(N):
    V2[i] = V1[i + 1] + ST[i + 1]
z = V2[-1]
for i in range(N - 1, -1, -1):
    if V2[i] > z:
        V2[i] = z
    else:
        z = V2[i]


Vmax = [0] * (ST[-1] * 2 + 1)
j = 0
t = 0
for i, Tn in enumerate(ST):
    while t < Tn or j == 0:
        Vmax[2 * t + j] = min(V1[i - 1], V2[i - 1])
        V2 = [x - 0.5 for x in V2]
        t += j
        j ^= 1

v, cnt = 0, 0
for i in range(ST[-1] * 2):
    nVmax = Vmax[i + 1]
    nv = min(v + 0.5, nVmax)
    cnt += (v + nv) * 0.5 / 2
    v = nv

print(cnt)