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B. Graph Subset Problem

time limit per test

1 secondmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou are given an undirected graph with $$$n$$$ vertices and $$$m$$$ edges. Also, you are given an integer $$$k$$$.

Find either a clique of size $$$k$$$ or a non-empty subset of vertices such that each vertex of this subset has at least $$$k$$$ neighbors in the subset. If there are no such cliques and subsets report about it.

A subset of vertices is called a clique of size $$$k$$$ if its size is $$$k$$$ and there exists an edge between every two vertices from the subset. A vertex is called a neighbor of the other vertex if there exists an edge between them.

Input

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases. The next lines contain descriptions of test cases.

The first line of the description of each test case contains three integers $$$n$$$, $$$m$$$, $$$k$$$ ($$$1 \leq n, m, k \leq 10^5$$$, $$$k \leq n$$$).

Each of the next $$$m$$$ lines contains two integers $$$u, v$$$ $$$(1 \leq u, v \leq n, u \neq v)$$$, denoting an edge between vertices $$$u$$$ and $$$v$$$.

It is guaranteed that there are no self-loops or multiple edges. It is guaranteed that the sum of $$$n$$$ for all test cases and the sum of $$$m$$$ for all test cases does not exceed $$$2 \cdot 10^5$$$.

Output

For each test case:

If you found a subset of vertices such that each vertex of this subset has at least $$$k$$$ neighbors in the subset in the first line output $$$1$$$ and the size of the subset. On the second line output the vertices of the subset in any order.

If you found a clique of size $$$k$$$ then in the first line output $$$2$$$ and in the second line output the vertices of the clique in any order.

If there are no required subsets and cliques print $$$-1$$$.

If there exists multiple possible answers you can print any of them.

Example

Input

3 5 9 4 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 10 15 3 1 2 2 3 3 4 4 5 5 1 1 7 2 8 3 9 4 10 5 6 7 10 10 8 8 6 6 9 9 7 4 5 4 1 2 2 3 3 4 4 1 1 3

Output

2 4 1 2 3 1 10 1 2 3 4 5 6 7 8 9 10 -1

Note

In the first test case: the subset $$$\{1, 2, 3, 4\}$$$ is a clique of size $$$4$$$.

In the second test case: degree of each vertex in the original graph is at least $$$3$$$. So the set of all vertices is a correct answer.

In the third test case: there are no cliques of size $$$4$$$ or required subsets, so the answer is $$$-1$$$.

Codeforces (c) Copyright 2010-2021 Mike Mirzayanov

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