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:
# 写経AC
N = int(input())
T = [int(i) for i in input().split()]
V = [int(i) for i in input().split()]

Ts = sum(T)

# Vmax[t]: t秒での最大速度 (0.5秒間隔)
Vmax = [float("inf")] * (Ts * 2 + 1)

ts = 0
for i in range(N):
    for t in range(T[i]):
        t1 = (ts + t) * 2
        t2 = (ts + t) * 2 + 1
        Vmax[t1] = min(Vmax[t1], V[i])
        Vmax[t2] = min(Vmax[t2], V[i])

    # 時間の区切りの部分は閉区間が重なっている
    Vmax[(ts + T[i]) * 2] = min(Vmax[(ts + T[i]) * 2], V[i])

    ts += T[i]

# 0秒とT秒
Vmax[0] = Vmax[Ts * 2] = 0
# 左から右に
for t in range(Ts * 2):
    Vmax[t + 1] = min(Vmax[t + 1], Vmax[t] + 0.5)
# 右から左に
for t in range(Ts * 2 - 1, 0, -1):
    Vmax[t] = min(Vmax[t], Vmax[t + 1] + 0.5)

ans = 0.0
for i in range(Ts * 2):
    # 台形の公式
    ans += (Vmax[i] + Vmax[i + 1]) * 0.5 / 2

print(ans)