Metro quiz

Two Olympics spectators are waiting in a queue. They each hold a copy of the metro map of Paris, and they devised a little game to kill time. First, player $A$ thinks of a metro line (chosen uniformly at random among all metro lines) that player $B$ will need to guess. In order to guess, player $B$ repeatedly asks whether the line stops at a metro station of her choice, and player $A$ answers truthfully. After enough questions, player $B$ will typically know with certainty which metro line player $A$ had in mind. Of course, player $B$ wants to minimise the number of questions she needs to ask.

You are given the map of the $m$ metro lines (numbered from $1$ to $m$), featuring a total of $n$ metro stations (numbered from $0$ to $n-1$) and indicating, for each line, those stations at which the line stops. Please compute the expected number of questions that player $B$ needs to ask to find the answer, in the optimal strategy.

In other words, given a strategy $S$, note $Q_{S,j}$ the number of questions asked by the strategy if the metro line in the solution is line $j$. Then, note

the expected value of $Q_{S,j}$ assuming that $j$ is uniformly chosen from the set of all metro lines. Your task is to compute $mi{n}_S{E}_S$. If it is not always possible for player $B$ to know which line player $A$ had in mind with certainty, print "not possible".

Input

The first line contains the number $n(1\le n\le 18)$. The second line contains the number $m(1\le m\le 50)$. Then follow $m$ lines: the $k$-th such line contains first a positive integer $p\le n$, then a space, and then $p$ integers $s_1,s_2,...,s_p$; these are the metro stations at which line $k$ stops. A line stops at a given station at most once.

Output

Print one single number — the minimum expected number of questions that player $B$ must ask in order to find the correct metro line, or "not possible" (in lowercase characters). Answers within $10^{-4}$ of the correct answer will be accepted.

Examples

Sample 2 explanation. Ask the first question about station $0$: this is optimal by symmetry of the problem. This lets us distinguish between line $1$, which stops at station $0$, and lines $2$ and $3$, which do not. If needed, ask a second question to distinguish between lines $2$ and $3$.

Player B asks one question if the answer is line $1$, and two questions otherwise. Thus, the expected number of questions she will ask is $(1+2\cdot 2)/3$.