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:
#!/usr/bin/env python

from collections import deque
import itertools as it
import sys
import math

sys.setrecursionlimit(10000000)

N = input()
T = map(int, raw_input().split())
V = map(int, raw_input().split())

upper = []

Ts = 0
for i in range(N):
    upper.append((Ts, V[i]))
    Ts += T[i]
    upper.append((Ts, V[i]))

S = sum(T)
upper.append((0, 0))
upper.append((S, 0))

ans = [0]
Ts = 0
for i in range(N):
    dt = 0
    while True:
        dt += 0.5
        t = Ts + dt
        v = V[i]
        for p in upper:
            v = min(v, p[1] + abs(p[0] - t))
        ans.append(v)
        if dt == T[i]:
            break
    Ts += T[i]

print (sum(ans[1:]) + sum(ans[:-1])) / 4