Problem Statement

Given are integers $N$, $M$, $S$, and a sequence of $N$ integers $A=(A_1,A_2,\\cdots,A_N)$.

Consider making a sequence of $N$ non-negative real numbers $x=(x_1,x_2,\\cdots,x_N)$ satisfying all of the following conditions:

• $0 \\leq x_1 \\leq x_2 \\leq \\cdots \\leq x_N \\leq M$,
• $\\sum_{1 \\leq i \\leq N} x_i=S$.

Now, let us define the score of $x$ as $\\sum_{1 \\leq i \\leq N} A_i \\times x_i$. Find the maximum possible score of $x$.

Constraints

• $1 \\leq N \\leq 5000$
• $1 \\leq M \\leq 10^6$
• $1 \\leq S \\leq \\min(N \\times M,10^6)$
• $1 \\leq A_i \\leq 10^6$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $S$ $A_1$ $A_2$ $\\cdots$ $A_N$

Constraints

Print the answer. Your output will be considered correct if its absolute or relative error is at most $10^{-6}$.

Sample Input 1

3 2 3
1 2 3

Sample Output 1

8.00000000000000000000

The optimal choice is $x=(0,1,2)$.

Sample Input 2

3 3 2
5 1 1

Sample Output 2

4.66666666666666666667

The optimal choice is $x=(2/3,2/3,2/3)$.

Sample Input 3

10 234567 1000000
353239 53676 45485 617014 886590 423581 172670 928532 312338 981241

Sample Output 3

676780145098.25000000000000000000