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:

```
$N$ $M$
$u_1$ $v_1$
$u_2$ $v_2$
$\vdots$
$u_M$ $v_M$
```

Print the answer.

-   $2 \leq N \leq 17$
-   $N-1 \leq M \leq \frac{N(N-1)}{2}$
-   $1 \leq u_i, v_i \leq N$
-   The given graph is simple and connected.
-   All values in input are integers.

14

• 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 11.
• 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.