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.

Input

Input is given from Standard Input in the following format:
N M S
$A_1 A_2 \cdots A_N$

Sample Output 1

8.00000000000000000000

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

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.

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

见题面 Input 部分

见题面 Output 部分