Score : $600$ points

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 $10^{-6}$.

3 2 3
1 2 3

8.00000000000000000000

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

3 3 2
5 1 1

4.66666666666666666667

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

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

676780145098.25000000000000000000