You are given a simple connected undirected graph with $N$ vertices and $M$ edges. (A graph is said to be simple if it has no multi-edges and no self-loops.) For $i = 1, 2, \ldots, M$, the $i$-th edge connects Vertex $u_i$ and Vertex $v_i$. A sequence $(A_1, A_2, \ldots, A_k)$ is said to be a path of length $k$ if both of the following two conditions are satisfied: - For all $i = 1, 2, \dots, k$, it holds that $1 \leq A_i \leq N$. - For all $i = 1, 2, \ldots, k-1$, Vertex $A_i$ and Vertex $A_{i+1}$ are directly connected by an edge. An empty sequence is regarded as a path of length $0$. Let $S = s_1s_2\ldots s_N$ be a string of length $N$ consisting of $0$ and $1$. A path $A = (A_1, A_2, \ldots, A_k)$ is said to be a good path with respect to $S$ if the following conditions are satisfied: - For all $i = 1, 2, \ldots, N$, it holds that: - if $s_i = 0$, then $A$ has even number of $i$'s. - if $s_i = 1$, then $A$ has odd number of $i$'s. There are $2^N$ possible $S$ (in other words, there are $2^N$ strings of length $N$ consisting of $0$ and $1$). Find the sum of "the length of the shortest good path with respect to $S$" over all those $S$. Under the Constraints of this problem, it can be proved that, for any string $S$ of length $N$ consisting of $0$ and $1$, there is at least one good path with respect to $S$.

Input is given from Standard Input in the following format: >$N$ $M$\ >$u_1$ $v_1$\ >$u_2$ $v_2$\ >$\cdots$\ >$u_M$ $v_M$ #### Constraints - $2 \leq N \leq 17$ - $N-1 \leq M \leq \frac{N(N-1)}{2}$ - $1 \leq u_i, v_i \leq N$ - The given graph is simple and connected. - All values in input are integers.

Print the answer.

- For $S=000$, the empty sequence $()$ is the shortest good path with respect to $S$, whose length is $0$. - For $S=100$, $(1)$ is the shortest good path with respect to $S$, whose length is $1$. - For $S=010$, $(2)$ is the shortest good path with respect to $S$, whose length is $1$. - For $S=110$, $(1,2)$ is the shortest good path with respect to $S$, whose length is $2$. - For $S=001$, $(3)$ is the shortest good path with respect to $S$, whose length is $1$. - For $S=101$, $(1,2,3,2)$ is the shortest good path with respect to $S$, whose length is $4$. - For $S=011$, $(2,3)$ is the shortest good path with respect to $S$, whose length is $2$. - For $S=111$, $(1,2,3)$ is the shortest good path with respect to $S$, whose length is $3$. Therefore, the sought answer is $0+1+1+2+1+4+2+3=14$.

AtCoder [ABC244F]