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:
from fractions import Fraction as F

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

VL = [V[i] for i in xrange(N)]
VL.append(0)
for i in xrange(N-1, -1, -1):
    VL[i] = min(VL[i], VL[i+1] + T[i])

v1 = F(0)
v2 = F(0)
ans = F(0)

for i in xrange(N):
    t1 = F(0)
    v2 = F(v1)

    while t1 < T[i] and v2 < V[i] and v2 - (T[i] - t1) < VL[i+1]:
        t1 += F(1, 2)
        v2 += F(1, 2)
    v2 = v1 + t1
    ans += (v1 + v2) / F(2) * t1

    t2 = F(0)
    while t1 + t2 < T[i] and v2 - (T[i] - t1 - t2) < VL[i+1]:
        t2 += F(1, 2)
    ans += v2 * t2

    v3 = v2 - (T[i] - t1 - t2)
    ans += (v2 + v3) / F(2) * (T[i] - t1 - t2)

    v1 = v3

print "%.09f" % ans