Problem H
Knockout Tournament
Laura is organising a knockout tournament, in which her friend Dale takes part. Laura would like to maximise the probability of Dale winning the tournament by arranging the games in a favourable way. She does not know how to do it, so she asked you for help. Naturally, you refuse to cooperate with such a deplorable act—but then you realise that it is a very nice puzzle!
When the number of players is a power of two, the tournament setup can be described recursively as follows: the players are divided into two equal groups that each play their own knockout tournament, after which the winners of both tournaments play each other. Once a player loses, they are out of the tournament.
When the number of players is not a power of two, some of the last players in the starting lineup advance from the first round automatically so that in the second round the number of players left is a power of two, as shown in Figure 1.
Every player has a rating indicating their strength. A player with rating $a$ wins a game against a player with rating $b$ with probability $\frac{a}{a+b}$ (independently of any previous matches played).
Laura as the organiser can order the starting lineup of players in any way she likes. What is the maximum probability of Dale winning the tournament?
Input
The input consists of:

One line with an integer $n$ ($2 \le n \le 4096$), the total number of players.

$n$ lines, each with an integer $r$ ($1 \le r \le 10^5$), the rating of a player. The first rating given is Dale’s rating.
Output
Output the maximum probability with which Dale can win the tournament given a favourable setup. Your answer should have an absolute or relative error of at most $10^{6}$.
Sample Input 1  Sample Output 1 

4 3 1 2 4 
0.364285714 
Sample Input 2  Sample Output 2 

5 1 1 3 3 3 
0.125 