Cactus Eater

As is well known, a cactus is a connected undirected graph without loops or multiple edges, in which every edge belongs to at most one simple cycle.

A microorganism called cactofagi, the cactus eater, was recently discovered. When it lands on a connected component, it does the following:

1. It releases $t$ joules of energy, where $t$ is the number of vertices in the component.
2. It chooses a random vertex in the component and eats it along with all edges incident to it.
3. It moves to every connected component that remains.

Find the expected total energy released when the microorganism lands on a given cactus and the cactus is fully destroyed.

The first line contains two integers $n$ and $m$, the number of vertices and edges of the cactus ($1 \le n \le 400$; $1 \le m \le n(n-1)/2$).

Each of the next $m$ lines contains two integers $a$ and $b$, the vertices joined by an edge ($1 \le a,b \le n$). It is guaranteed that the given graph is a cactus.

It can be shown that the answer is a rational number. Write it as an irreducible fraction $p/q$ and print the value of $pq^{-1} \bmod 998244353$.

In the second example the answer is 13.6, or $68/5$ as an irreducible fraction.