1250N · Wires

Problem: Polycarpus has a complex electronic device. The core of this device is a circuit board. The board has $10^9$ contact points which are numbered from $1$ to $10^9$. Also there are $n$ wires numbered from $1$ to $n$, each connecting two distinct contact points on the board. An electric signal can pass between wires $A$ and $B$ if:

• either both wires share the same contact point;
• or there is a sequence of wires starting with $A$ and ending with $B$, and each pair of adjacent wires in the sequence share a contact point.

The picture shows a circuit board with $5$ wires. Contact points with numbers $2, 5, 7, 8, 10, 13$ are used. Here an electrical signal can pass from wire $2$ to wire $3$, but not to wire $1$.

Currently the circuit board is broken. Polycarpus thinks that the board could be fixed if the wires were re-soldered so that a signal could pass between any pair of wires.

It takes $1$ minute for Polycarpus to re-solder an end of a wire. I.e. it takes one minute to change one of the two contact points for a wire. Any contact point from range $[1, 10^9]$ can be used as a new contact point. A wire's ends must always be soldered to distinct contact points. Both wire's ends can be re-solded, but that will require two actions and will take $2$ minutes in total.

Find the minimum amount of time Polycarpus needs to re-solder wires so that a signal can pass between any pair of wires. Also output an optimal sequence of wire re-soldering.

Input Format: The input contains one or several test cases. The first input line contains a single integer $t$ — number of test cases. Then, $t$ test cases follow.

The first line of each test case contains a single integer $n$ ($1 \le n \le 10^5$) — the number of wires. The following $n$ lines describe wires, each line containing two space-separated integers $x_i, y_i$ ($1 \le x_i, y_i \le 10^9$, $x_i \neq y_i$) — contact points connected by the $i$-th wire. A couple of contact points can be connected with more than one wire.

Sum of values of $n$ across all test cases does not exceed $10^5$.

Output Format: For each test case first print one line with a single integer $k$ — the minimum number of minutes needed to re-solder wires so that a signal can pass between any pair of wires. In the following $k$ lines print the description of re-solderings. Each re-soldering should be described by three integers $w_j, a_j, b_j$ ($1 \le w_j \le n$, $1 \le a_j, b_j \le 10^9$). Such triple means that during the $j$-th re-soldering an end of the $w_j$-th wire, which was soldered to contact point $a_j$, becomes soldered to contact point $b_j$ instead. After each re-soldering of a wire it must connect two distinct contact points. If there are multiple optimal re-solderings, print any of them.