Score : $475$ points

### Problem Statement

There are $N$ roulette wheels.The $i$-th $(1\leq i\leq N)$ wheel has $P _ i$ integers $S _ {i,1},S _ {i,2},\ldots,S _ {i,P _ i}$ written on it, and you can play it once by paying $C _ i$ yen.When you play the $i$-th wheel once, an integer $j$ between $1$ and $P _ i$, inclusive, is chosen uniformly at random, and you earn $S _ {i,j}$ points.

The points you earn from the wheels are determined independently of past results.

Takahashi wants to earn at least $M$ points.Takahashi will act to minimize the amount of money he pays before he earns at least $M$ points.After each play, he can choose which wheel to play next based on the previous results.

Find the expected amount of money Takahashi will pay before he earns at least $M$ points.

## More formal definition

Here is a more formal statement.For a strategy that Takahashi can adopt in choosing which wheel to play, the expected amount of money $E$ that he pays before he earns at least $M$ points with that strategy is defined as follows.

- For a natural number $X$, let $f(X)$ be the expected amount of money Takahashi pays before he earns at least $M$ points or plays the wheels $X$ times in total according to that strategy. Let $E=\displaystyle\lim _ {X\to+\infty}f(X)$.

Under the conditions of this problem, it can be proved that $\displaystyle\lim _ {X\to+\infty}f(X)$ is finite no matter what strategy Takahashi adopts.Find the value of $E$ when he adopts a strategy that minimizes $E$.

### Constraints

- $1\leq N\leq 100$
- $1\leq M\leq 100$
- $1\leq C _ i\leq 10 ^ 4\ (1\leq i\leq N)$
- $1\leq P _ i\leq 100\ (1\leq i\leq N)$
- $0\leq S _ {i,j}\leq M\ (1\leq i\leq N,1\leq j\leq P _ i)$
- $\displaystyle\sum _ {j=1}^{P _ i}S _ {i,j}\gt0\ (1\leq i\leq N)$
- All input values are integers.

### Input

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

$N$ $M$$C _ 1$ $P _ 1$ $S _ {1,1}$ $S _ {1,2}$ $\ldots$ $S _ {1,P _ 1}$$C _ 2$ $P _ 2$ $S _ {2,1}$ $S _ {2,2}$ $\ldots$ $S _ {2,P _ 2}$$\vdots$$C _ N$ $P _ N$ $S _ {N,1}$ $S _ {N,2}$ $\ldots$ $S _ {N,P _ N}$

### Output

Print the expected amount of money Takahashi will pay until he earns at least $M$ points in a single line.Your output will be considered correct when the relative or absolute error from the true value is at most $10 ^ {-5}$.

### Sample Input 1

3 14100 2 5 950 4 1 2 4 870 5 2 4 2 8 8

### Sample Output 1

215.913355350494384765625

For instance, Takahashi can play the wheels as follows.

- Pay $50$ yen to play roulette $2$ and earn $S _ {2,4}=8$ points.
- Pay $50$ yen to play roulette $2$ and earn $S _ {2,1}=1$ point.
- Pay $100$ yen to play roulette $1$ and earn $S _ {1,1}=5$ points. He has earned a total of $8+1+5\geq14$ points, so he quits playing.

In this case, he pays $200$ yen before earning $14$ points.

Your output will be considered correct when the relative or absolute error from the true value is at most $10 ^ {-5}$, so outputs such as `215.9112`

and `215.9155`

would also be considered correct.

### Sample Input 2

2 1001 2 1 210 6 0 0 0 0 0 100

### Sample Output 2

60

It is optimal to keep spinning roulette $2$ until you get $100$ points.

### Sample Input 3

20 903252 9 0 4 2 7 3 2 3 2 42147 1 14033 8 0 4 1 7 5 2 5 03795 6 6 6 2 3 2 23941 7 2 4 4 7 2 0 52815 6 2 1 0 5 2 23020 2 3 63858 9 4 2 7 3 0 4 4 6 54533 10 3 6 4 0 6 4 4 2 7 74198 8 6 7 0 6 3 6 5 63739 8 2 7 1 5 1 4 4 72465 4 1 4 0 14418 9 7 6 2 4 6 1 5 0 75450 12 0 4 4 7 7 4 4 5 4 5 3 74196 9 1 6 5 5 7 2 3 6 34776 9 2 2 7 3 6 6 1 6 62286 3 3 5 63152 3 4 1 53509 7 0 6 7 0 1 0 32913 6 0 1 5 0 5 6

### Sample Output 3

45037.072314895291126319493887599716