# Ticket Completed? Many are familiar with the board game Ticket To Ride1 where players compete to build a railway empire, claiming routes between cities. The game consists of a map of cities and various rail segments each connecting two adjacent cities.

A key way to score points towards winning the game is to complete Destination Tickets. Each ticket specifies two distinct cities. A player earns the points that are indicated on the ticket if they have claimed one or more rail segments that form a path connecting the two cities.

There is one ticket for each distinct unordered pair of cities. In our version of the game, each player is randomly given a ticket and they have an equal probability of receiving any ticket. Given a list of rail segments you have already claimed, determine the probability you earn points from the ticket you are given.

## Input

The first line of input contains two integers $N$ ($2 \leq N \leq 10^5$), which is the number of cities, and $M$ ($0 \leq M \leq 10^6$), which is the number of rail segments you have claimed.

The next $M$ lines describe your claimed rail segments. Each line contains two distinct integers $i$ ($1 \leq i \leq N$) and $j$ ($1 \leq j \leq N$), which are the cities that this rail segment connects.

## Output

Display the probability you earn points from the ticket you are given.

Your answer should have an absolute error of at most $10^{-6}$.

Sample Input 1 Sample Output 1
4 2
1 2
3 4

0.33333333333333333333

Sample Input 2 Sample Output 2
5 4
1 5
2 3
2 4
3 4

0.4

Sample Input 3 Sample Output 3
7 5
1 2
2 3
3 4
5 6
6 7

0.42857142857142857143


Footnotes

1. Ticket To Ride is copyrighted by Days of Wonder, Inc.