CF1325F.Ehab's Last Theorem

It's the year 5555. You have a graph, and you want to find a long cycle and a huge independent set, just because you can. But for now, let's just stick with finding either.

Given a connected graph with $n$ vertices, you can choose to either:

• find an independent set that has exactly $\lceil\sqrt{n}\rceil$ vertices.
• find a simple cycle of length at least $\lceil\sqrt{n}\rceil$.

An independent set is a set of vertices such that no two of them are connected by an edge. A simple cycle is a cycle that doesn't contain any vertex twice. I have a proof you can always solve one of these problems, but it's too long to fit this margin.

现在是 5555 年。你有一个图，你想找一个很长的环，或者一个很大的独立集——仅仅因为你能做到。但目前，我们只需专注于找出其中任意一个即可。

给定一个包含 $n$ 个顶点的连通图，你可以选择以下任一任务完成：

• 找出一个恰好包含 $\lceil\sqrt{n}\rceil$ 个顶点的独立集；
• 找出一个长度至少为 $\lceil\sqrt{n}\rceil$ 的简单环。

独立集是指一组顶点，其中任意两个顶点之间均无边相连；简单环是指不重复经过任何顶点的环。我有一个证明，表明上述两个问题中总有一个可解，但该证明太长，无法写在此处的页边空白中。

The first line contains two integers $n$ and $m$ ($5 \le n \le 10^5$, $n-1 \le m \le 2 \cdot 10^5$) — the number of vertices and edges in the graph.

Each of the next $m$ lines contains two space-separated integers $u$ and $v$ ($1 \le u,v \le n$) that mean there's an edge between vertices $u$ and $v$. It's guaranteed that the graph is connected and doesn't contain any self-loops or multiple edges.

第一行包含两个整数 $n$ 和 $m$（$5 \le n \le 10^5$，$n-1 \le m \le 2 \cdot 10^5$），分别表示图中的顶点数和边数。

接下来的 $m$ 行中，每行包含两个用空格分隔的整数 $u$ 和 $v$（$1 \le u,v \le n$），表示顶点 $u$ 和 $v$ 之间存在一条边。保证该图是连通的，且不包含自环或重边。

If you choose to solve the first problem, then on the first line print "1", followed by a line containing $\lceil\sqrt{n}\rceil$ distinct integers not exceeding $n$, the vertices in the desired independent set.

If you, however, choose to solve the second problem, then on the first line print "2", followed by a line containing one integer, $c$, representing the length of the found cycle, followed by a line containing $c$ distinct integers integers not exceeding $n$, the vertices in the desired cycle, in the order they appear in the cycle.

如果你选择解决第一个问题，则在第一行输出“1”，然后在下一行输出 $\lceil\sqrt{n}\rceil$ 个互不相同的整数，且每个整数均不超过 $n$，这些整数即为所求独立集中的顶点。

然而，如果你选择解决第二个问题，则在第一行输出“2”，然后在下一行输出一个整数 $c$，表示所找到环的长度，再在下一行输出 $c$ 个互不相同的整数（每个整数均不超过 $n$），这些整数即为所求环中的顶点，且按其在环中出现的顺序排列。

In the first sample:

Notice that you can solve either problem, so printing the cycle $2-4-3-1-5-6$ is also acceptable.

In the second sample:

Notice that if there are multiple answers you can print any, so printing the cycle $2-5-6$, for example, is acceptable.

In the third sample:

在第一个样例中：

注意，你可以求解任一问题，因此输出环 $2-4-3-1-5-6$ 也是可接受的。

在第二个样例中：

注意，若存在多个答案，你可以输出任意一个，例如输出环 $2-5-6$ 即可接受。

在第三个样例中：