Andrey and Problem

B. Andrey and Problem

time limit per test

2 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

Andrey needs one more problem to conduct a programming contest. He has n friends who are always willing to help. He can ask some of them to come up with a contest problem. Andrey knows one value for each of his fiends — the probability that this friend will come up with a problem if Andrey asks him.

Help Andrey choose people to ask. As he needs only one problem, Andrey is going to be really upset if no one comes up with a problem or if he gets more than one problem from his friends. You need to choose such a set of people that maximizes the chances of Andrey not getting upset.

Input

The first line contains a single integer n (1 ≤ n ≤ 100) — the number of Andrey's friends. The second line contains n real numbers p_i (0.0 ≤ p_i ≤ 1.0) — the probability that the i-th friend can come up with a problem. The probabilities are given with at most 6 digits after decimal point.

Output

Print a single real number — the probability that Andrey won't get upset at the optimal choice of friends. The answer will be considered valid if it differs from the correct one by at most 10^- 9.

Examples

Input

4
0.1 0.2 0.3 0.8

Output

0.800000000000

Input

2
0.1 0.2

Output

0.260000000000

Note

In the first sample the best strategy for Andrey is to ask only one of his friends, the most reliable one.

In the second sample the best strategy for Andrey is to ask all of his friends to come up with a problem. Then the probability that he will get exactly one problem is 0.1·0.8 + 0.9·0.2 = 0.26.

概率计算：P(某set) = 

令：  和  

现在考虑：

1.考虑某个集合，再加一个概率为Pi的朋友后能不能使总概率提高。

即： 由公式可知， 如果 S < 1，则delta > 0，则可以加入这个朋友。

2.如果要加一个朋友有两个候选的，其概率分别为Pi,Pj，（设Pi < Pj）那么加哪个会更优呢？

Δi - Δj = P·pi·(1 - S) - P·pj·(1 - S) = P·(1 - S)·(pi - pj) > 0.  如果 S < 1 那么 pi > pj 时可以使 Δi - Δj  > 0，即P较大的那个带来的价值更高，所以优先选P大的那个。虽然这个

只是局部更优，但是用反证法可以证明是全局最优的。

所以由上述分析，以概率从大到小的方式逐步加入元素，因为由1，很可能S<1，即可能加一个会更优，所以我们逐步加入，又由2，优先加最大的，所以从大到小加入能保证最优。

然后每加入一个都计算该种情况下的总概率，更新答案。

上面的题解来自http://www.cnblogs.com/whatbeg/p/3799957.html；

 1 #include<stdio.h>
 2 #include<algorithm>
 3 #include<iostream>
 4 #include<stdlib.h>
 5 #include<queue>
 6 #include<string.h>
 7 #include<map>
 8 #include<vector>
 9 using namespace std;
10 typedef long long LL;
11 double  dp[106][106];
12 double ans[106];
13 bool cmp(double x,double  y)
14 {
15         return x>y;
16 }
17 int main(void)
18 {
19         int n;
20         while(scanf("%d",&n)!=EOF)
21         {
22                 int i,j;
23                 for(i = 0; i < n; i++)
24                 {
25                         scanf("%lf",&ans[i]);
26                 }
27                 sort(ans,ans+n,cmp);
28                 double maxx = ans[0];
29                 double x = (1-ans[0]);
30                 if(ans[0]!=1)
31                 {
32                         double y = ans[0]/(1-ans[0]);
33                         for(i = 1; i < n; i++)
34                         {
35                                 double d = -x*y+x*(1-ans[i])*(y+ans[i]/(1-ans[i]));
36                                 x*=(1-ans[i]);
37                                 y+=ans[i]/(1-ans[i]);
38                                 if(d <= 0)break;
39                                 else maxx = maxx+d;
40                         }
41                 }
42                 printf("%.10f\n",maxx);
43         }
44         return 0;
45 }