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.

Kid2Will - Fire Aura

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你有 $n$ 盏灯，编号从 $1$ 到 $n$。初始时，所有灯均处于关闭状态。

你还有 $n$ 个按钮。第 $i$ 个按钮会切换所有下标为 $i$ 的倍数的灯的状态：若灯当前为关闭，则变为开启；若为开启，则变为关闭。

你需要按照以下规则按下若干按钮：

• 至少要按下其中一个按钮；
• 同一个按钮不能被多次按下；
• 给定 $m$ 对 $(u_i, v_i)$。若你按下了按钮 $u_i$，则你也必须按下按钮 $v_i$（可在任意时刻按下，不一定要在按下 $u_i$ 之后立即按下）。注意：若你按下了按钮 $v_i$，则不一定需要按下按钮 $u_i$。

你不希望消耗过多电能。请找出一种按钮按下方案，使得最终最多有 $\lfloor n/5 \rfloor$ 盏灯处于开启状态；若不存在这样的方案，请输出 $-1$。

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 $2 \cdot 10^5$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1 \le t \le 10^4$）。随后是各测试用例的描述。

每个测试用例的第一行包含两个整数 $n$ 和 $m$（$1 \leq n \leq 2 \cdot 10^5$，$0 \leq m \leq 2 \cdot 10^5$），分别表示灯的数量和配对数量。

接下来的 $m$ 行中，每行包含两个整数 $u_i$、$v_i$（$1 \leq u_i, v_i \leq n$，$u_i \neq v_i$）：若按下按钮 $u_i$，则也必须按下按钮 $v_i$。保证所有配对 $(u_i, v_i)$ 互不相同。

保证所有测试用例的 $n$ 之和以及 $m$ 之和均不超过 $2 \cdot 10^5$。

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.

对于每个测试用例：

• 如果不存在一种按钮选择方案，使得最终亮着的灯的数量不超过 $\lfloor n/5 \rfloor$，则输出一行，包含单个整数 $-1$。
• 否则，输出两行。第一行应包含一个整数 $k$（$1 \le k \le n$）—— 按下的按钮数量；第二行应包含 $k$ 个整数 $b_1, b_2, \dots, b_k$（$1 \le b_i \le n$）—— 按下的按钮的编号（顺序任意）。这些 $b_i$ 必须互不相同，且最终亮着的灯的数量不得超过 $\lfloor n/5 \rfloor$。

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.

在第一个测试用例中，你最多需要打开 $\lfloor 4/5 \rfloor$ 盏灯，即不允许打开任何一盏灯。你可以验证：无论选择至少一个按钮进行按压，都无法使恰好 $0$ 盏灯被打开。

在第二个测试用例中，你可以依次按压按钮 $3$、$5$、$1$、$2$。

• 初始时，所有灯均处于关闭状态；
• 按压按钮 $3$ 后，所有下标为 $3$ 的倍数的灯（即灯 $3$）被翻转，因此灯 $3$ 被打开；
• 按压按钮 $5$ 后，所有下标为 $5$ 的倍数的灯（即灯 $5$）被翻转，因此灯 $3$ 和灯 $5$ 处于开启状态；
• 按压按钮 $1$ 后，所有下标为 $1$ 的倍数的灯（即灯 $1$、$2$、$3$、$4$、$5$）被翻转，因此灯 $1$、$2$、$4$ 被打开；
• 按压按钮 $2$ 后，所有下标为 $2$ 的倍数的灯（即灯 $2$、$4$）被翻转，因此最终仅有灯 $1$ 处于开启状态。

该方案是合法的，因为：

• 你按压了至少一个按钮；
• 每个按钮至多被按压一次；
• 你按压了按钮 $u_2 = 5$，这意味着你必须也按压按钮 $v_2 = 1$；事实上，按钮 $1$ 确实已被按压；
• 最终，仅灯 $1$ 处于开启状态。

在第三个测试用例中，按压按钮 $8$、$9$、$10$ 后，仅有灯 $8$、$9$、$10$ 被打开。