Problem Statement

Takahashi is playing a solitaire game similar to the memory-matching game.

There are $2N$ cards. A number is written on the front of each card, and nothing is written on the back.
For each $i$ satisfying $1 \leq i \leq N$, there are exactly two cards on which $A_i$ is written. (The $A_i$ are pairwise distinct.)

Takahashi plays the game with these cards using the following procedure.

• Shuffle the $2N$ cards and lay them out face down.
• Set life to $L$ and score to $0$.
• Repeat the following until life becomes $0$ or there are no cards left on the table:
  • Choose one face-down card on the table, turn it face up, and check the number $X$ written on it.
  • Choose one face-down card on the table, turn it face up, and check the number $Y$ written on it.
  • If $X=Y$, remove those two cards from the table, and increase the score by $X$.
  • If $X\neq Y$, turn the two cards face down again, and decrease life by $1$.

Find the expected value of the score at the end of the game when Takahashi acts optimally to maximize the score at the end of the game.

Below is a more formal description of the game.

• Takahashi knows the rules of this game described below.
• Takahashi knows the values of $A_1,\dots,A_N$.
• Let $B$ be a sequence obtained by choosing a permutation of the length-$2N$ sequence $(A_1,A_1,A_2,A_2,\ldots,A_N,A_N)$ uniformly at random.
• Initially, Takahashi knows none of the values of $B$. Once he learns the value of $B_i$, he remembers it completely thereafter.
• Let life be $L$, score be $0$, and $S$ be $\{1,2,3,\ldots,2N\}$. Takahashi always knows these values.
• Repeat the following until life becomes $0$ or $S$ becomes empty:
  • Based on the information obtained up to this point, Takahashi chooses an element from $S$, and calls it $i$.
  • The value of $B_i$ is revealed, and Takahashi learns it.
  • Based on the information obtained up to this point (including the value of $B_i$), Takahashi chooses an element from $S\setminus\{i\}$, and calls it $j$.
  • The value of $B_j$ is revealed, and Takahashi learns it.
  • If $B_i=B_j$, remove $i$ and $j$ from $S$, and add $B_i$ to the score.
  • If $B_i \neq B_j$, decrease life by $1$.
• Takahashi acts optimally to maximize the expected value of the score at the end of the game.

Constraints

• $1 \leq N \leq 200$
• $1 \leq L \leq 200$
• $1 \leq A_1 < A_2 < \dots < A_N \leq 10^5$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $L$
$A_1$ $A_2$ $\dots$ $A_N$

Output

Output the answer.
Your output will be considered correct if its absolute or relative error from the true answer is at most $10^{-5}$.

3 2
1 2 3

3.8666666667

The game may proceed as follows, for example. To distinguish the six cards, let us call them A, B, C, D, E, F.

• Start the game with life $2$ and score $0$.
• Turn card A face up. $3$ is written on it.
• Turn card B face up. $2$ is written on it.
• Since different numbers are written, turn both cards face down again, and decrease life by $1$ to $1$.
• Turn card C face up. $3$ is written on it.
• Turn card A face up. $3$ is written on it.
• Since the same number is written, remove both cards from the table, and increase the score by $3$ to $3$.
• Turn card D face up. $1$ is written on it.
• Turn card E face up. $2$ is written on it.
• Since different numbers are written, turn both cards face down again, and decrease life by $1$ to $0$.
• Since life has become $0$, the game ends. The score is $3$.

Note that, immediately after turning card C face up in this sequence, Takahashi can make the choice of "turning face up card A, the other already-known card with $3$ written on it, based on the fact that $3$ was written on the front of card C."

5 2
2 3 5 7 101

17.8560846561

20 10
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

770.7122293087

• 来源：AtCoder ABC 470 E