CF1557A.Ezzat and Two Subsequences

Ezzat has an array of $n$ integers (maybe negative). He wants to split it into two non-empty subsequences $a$ and $b$ , such that every element from the array belongs to exactly one subsequence, and the value of $f(a) + f(b)$ is the maximum possible value, where $f(x)$ is the average of the subsequence $x$ .

A sequence $x$ is a subsequence of a sequence $y$ if $x$ can be obtained from $y$ by deletion of several (possibly, zero or all) elements.

The average of a subsequence is the sum of the numbers of this subsequence divided by the size of the subsequence.

For example, the average of $[1,5,6]$ is $(1+5+6)/3 = 12/3 = 4$ , so $f([1,5,6]) = 4$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^3$ )— the number of test cases. Each test case consists of two lines.

The first line contains a single integer $n$ ( $2 \le n \le 10^5$ ).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $-10^9 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $3\cdot10^5$ .

For each test case, print a single value — the maximum value that Ezzat can achieve.

Your answer is considered correct if its absolute or relative error does not exceed $10^{-6}$ .

Formally, let your answer be $a$ , and the jury's answer be $b$ . Your answer is accepted if and only if $\frac{|a - b|}{\max{(1, |b|)}} \le 10^{-6}$ .

In the first test case, the array is $[3, 1, 2]$ . These are all the possible ways to split this array:

• $a = [3]$ , $b = [1,2]$ , so the value of $f(a) + f(b) = 3 + 1.5 = 4.5$ .
• $a = [3,1]$ , $b = [2]$ , so the value of $f(a) + f(b) = 2 + 2 = 4$ .
• $a = [3,2]$ , $b = [1]$ , so the value of $f(a) + f(b) = 2.5 + 1 = 3.5$ .

Therefore, the maximum possible value $4.5$ .In the second test case, the array is $[-7, -6, -6]$ . These are all the possible ways to split this array:

• $a = [-7]$ , $b = [-6,-6]$ , so the value of $f(a) + f(b) = (-7) + (-6) = -13$ .
• $a = [-7,-6]$ , $b = [-6]$ , so the value of $f(a) + f(b) = (-6.5) + (-6) = -12.5$ .

Therefore, the maximum possible value $-12.5$ .