E. Making Connections
time limit per test
5 seconds
memory limit per test
1024 megabytes
input
standard input
output
standard output

Given that everything is online these days, connectivity is a must. A computer network can be modeled as a graph, where each computer is a vertex and each direct network connection between pairs of computers is an undirected edge.

Consider the process of building a computer network. At the very beginning there will be $$$n$$$ computers, with no connections between any of them. Then, as time goes on, pairs of computers are chosen, one pair at a time, and a direct network connection between them is added. In the middle of such a process, we might get the following graph modeling the current connections:

This network currently has three groups: the first group has 4 computers that can communicate with each other directly or indirectly, the second group consists of 1 computer by itself, and the third group has 2 computers that can communicate with each other. So, a group is each of the separate components of the graph.

We can define the average connectivity of a network as the sum of the group sizes squared, divided by the number of components. For the example graph shown above, the current connectivity equals $$$(4^2 + 1^2 + 2^2)/3 = 21/3 = 7$$$.

As a network is being built, the project manager would like to know the average connectivity of the network at that given snapshot of time. Write a program to handle the queries as the network is being constructed.

Given a network of $$$n$$$ initially separate computers, along with a sequence of steps, where each step is either a pair of computers being connected or a query about the average connectivity as of that moment, process each step (i.e., connect the two computers or answer the query).

Input

The first input line contains two space separated integers: $$$n$$$ $$$(1 ≤ n ≤ 10^5)$$$, representing the number of computers, and $$$m$$$ $$$(1 ≤ m ≤ 3×10^5)$$$, representing the total number of steps (each step being either connect two computers or a query on the average connectivity as of that moment). Assume that the computers are numbered $$$1$$$ through $$$n$$$, inclusive.

Each of the following $$$m$$$ input lines contains information about one operation, in the order that they occur. Each of these lines will start with a single integer, either $$$1$$$ or $$$2$$$, providing the step type. If the step type is $$$1$$$, it means that a pair of computers is being connected with a direct connection. The value of $$$1$$$ will be followed by $$$u$$$ and $$$v$$$ $$$(1 ≤ u, v ≤ n, u ≠ v)$$$, representing the pair of computers being connected with a direct connection. If the step type is $$$2$$$, this is a query and no other information will be on that input line.

Output

For each query, output the average connectivity of the network at that point in time as a fraction in lowest terms on a line by itself. Specifically, output two integers, $$$x$$$ and $$$y$$$, with the character '/' in between, indicating that the average connectivity of the network at the time is $$$x$$$ divided by $$$y$$$ such that $$$x$$$ and $$$y$$$ share no common factors, e.g., $$$21/12$$$ is not correct (should be $$$7/4$$$).

Examples
Input
7 9
2
1 1 2
1 1 3
2
1 3 4
1 2 3
2
1 6 7
2
Output
1/1
13/5
19/4
7/1
Input
4 9
1 1 2
2
1 3 4
2
1 2 3
2
1 1 4
1 2 4
2
Output
2/1
4/1
16/1
16/1
Note

It's possible that the input contains multiple direct connections for the same pair of computers; the extra direct connections between the same two computers do not have any effects. It's also possible that, at the end of the process, not all $$$n$$$ computers are connected in the same component, i.e., there may be more than one component even at the end of the process.