CF1909E.Multiple Lamps

Kid2Will - Fire Aura

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You have $n$ lamps, numbered from $1$ to $n$ . Initially, all the lamps are turned off.

You also have $n$ buttons. The $i$ -th button toggles all the lamps whose index is a multiple of $i$ . When a lamp is toggled, if it was off it turns on, and if it was on it turns off.

You have to press some buttons according to the following rules.

• You have to press at least one button.
• You cannot press the same button multiple times.
• You are given $m$ pairs $(u_i, v_i)$ . If you press the button $u_i$ , you also have to press the button $v_i$ (at any moment, not necessarily after pressing the button $u_i$ ). Note that, if you press the button $v_i$ , you don't need to press the button $u_i$ .

You don't want to waste too much electricity. Find a way to press buttons such that at the end at most $\lfloor n/5 \rfloor$ lamps are on, or print $-1$ if it is impossible.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^4$ ). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ( $1 \leq n \leq 2 \cdot 10^5$ , $0 \leq m \leq 2 \cdot 10^5$ ) — the number of lamps and the number of pairs, respectively.

Each of the next $m$ lines contains two integers $u_i$ , $v_i$ ( $1 \leq u_i, v_i \leq n$ , $u_i \neq v_i$ ). If you press the button $u_i$ , you also have to press the button $v_i$ . It is guaranteed that the pairs $(u_i, v_i)$ are distinct.

It is guaranteed that the sum of $n$ and the sum of $m$ over all test cases do not exceed $2 \cdot 10^5$ .

For each test case:

• If there is no choice of buttons that makes at most $\lfloor n/5 \rfloor$ lamps on at the end, output a single line containing $-1$ .
• Otherwise, output two lines. The first line should contain an integer $k$ ( $1 \le k \le n$ ) — the number of pressed buttons. The second line should contain $k$ integers $b_1, b_2, \dots, b_k$ ( $1 \le b_i \le n$ ) — the indices of the pressed buttons (in any order). The $b_i$ must be distinct, and at the end at most $\lfloor n/5 \rfloor$ lamps must be turned on.

In the first test case, you need to turn at most $\lfloor 4/5 \rfloor$ lamps on, which means that no lamp can be turned on. You can show that no choice of at least one button turns $0$ lamps on.

In the second test case, you can press buttons $3$ , $5$ , $1$ , $2$ .

• Initially, all the lamps are off;
• after pressing button $3$ , the lamps whose index is a multiple of $3$ (i.e., $3$ ) are toggled, so lamp $3$ is turned on;
• after pressing button $5$ , the lamps whose index is a multiple of $5$ (i.e., $5$ ) are toggled, so lamps $3$ , $5$ are turned on;
• after pressing button $1$ , the lamps whose index is a multiple of $1$ (i.e., $1$ , $2$ , $3$ , $4$ , $5$ ) are toggled, so lamps $1$ , $2$ , $4$ are turned on;
• after pressing button $2$ , the lamps whose index is a multiple of $2$ (i.e., $2$ , $4$ ) are toggled, so lamp $1$ is turned on.

This is valid because

• you pressed at least one button;
• you pressed all the buttons at most once;
• you pressed button $u_2 = 5$ , which means that you had to also press button $v_2 = 1$ : in fact, button $1$ has been pressed;
• at the end, only lamp $1$ is on.

In the third test case, pressing the buttons $8$ , $9$ , $10$ turns only the lamps $8$ , $9$ , $10$ on.