Problem: Given a connected undirected graph with n vertices and an integer k, you have to either:

  * either find an independent set that has exactly ⌈k/2⌉ vertices.
  * or find a simple cycle of length at most k. 



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 that for any input you can always solve at least one of these problems, but it's left as an exercise for the reader.

Input

The first line contains three integers n, m, and k (3 ≤ k ≤ n ≤ 10^5, n-1 ≤ m ≤ 2 ⋅ 10^5) — the number of vertices and edges in the graph, and the parameter k from the statement.

Each of the next m lines contains two integers u and v (1 ≤ u,v ≤ 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.

Output

If you choose to solve the first problem, then on the first line print 1, followed by a line containing ⌈k/2⌉ 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 not exceeding n, the vertices in the desired cycle, in the order they appear in the cycle.

Examples

Input


4 4 3
1 2
2 3
3 4
4 1


Output


1
1 3 

Input


4 5 3
1 2
2 3
3 4
4 1
2 4


Output


2
3
2 3 4 

Input


4 6 3
1 2
2 3
3 4
4 1
1 3
2 4


Output


2
3
1 2 3 

Input


5 4 5
1 2
1 3
2 4
2 5


Output


1
1 4 5 

Note

In the first sample:

<image>

Notice that printing the independent set \{2,4\} is also OK, but printing the cycle 1-2-3-4 isn't, because its length must be at most 3.

In the second sample:

<image>

Notice that printing the independent set \{1,3\} or printing the cycle 2-1-4 is also OK.

In the third sample:

<image>

In the fourth sample:

<image>

Solution (in JAVA):

Solution:
import javax.swing.*;
import java.io.*;
import java.lang.reflect.Array;
import java.math.*;
import java.security.*;
import java.util.*;

public class Main {
    private static int n, m, k;
    private static int [] depth;
    private static ArrayList<Integer> cycle;
    private static LinkedList<Integer> [] adj;

    private static int dfs(int u, int p, int d){
        if(cycle != null) return -1;
        if (depth[u] != 0) {
            cycle = new ArrayList<>();
            return u;
        }
        depth[u] = d;
        for (int v : adj[u]) if (v != p) {
            int t = dfs(v, u, d + 1);
            if (t != -1) {
                cycle.add(u);
                if (t == u) return -1;
                else return t;
            }
        }
        return -1;
    }


    public static void main(String[] args) throws NumberFormatException, IOException {
        IO io = new IO(null, null);
        n = io.getNextInt();
        m = io.getNextInt();
        k = io.getNextInt();
        depth = new int[n+1];
        adj = new LinkedList[n+1];
        for (int i = 0; i <= n; i++)
            adj[i] = new LinkedList<>();
        for (int e = 0; e < m; e++) {
            int a = io.getNextInt();
            int b = io.getNextInt();
            adj[a].add(b);
            adj[b].add(a);
        }
        dfs(1, 0, 1);
//        System.err.println(cycle);
        if(cycle != null) {
            Collections.reverse(cycle);
            boolean [] vis = new boolean[n+1];
            int S = 0, E = n;
            int prv = -1;
            for (int u : cycle) {
                int h = 0;
                for(int t : adj[u])
                    if (t != prv && vis[t]){
                        h = Math.max(h, depth[t]);
                    }
                prv = u;
                if (h != 0) {
                    S = h;
                    E = depth[u];
                    break;
                }
                vis[u] = true;
            }
            if (S != 0) {
                ArrayList<Integer> aux = new ArrayList<>();
                for (int x : cycle)
                    if (S <= depth[x] && depth[x] <= E)
                        aux.add(x);
                cycle = aux;
            }
        }
        if (cycle == null) {
            LinkedList<Integer> [] V = new LinkedList[2];
            for (int c = 0; c < 2; c++)
                V[c] = new LinkedList<>();
            k = (k + 1)/2;
            for (int u = 1; u <= n; u++) {
                int d = depth[u] & 1;
                V[d].add(u);
                if (V[d].size() == k) break;
            }
            int c = (V[1].size() == k) ? 1 : 0;
            io.println(1);
            for (int x : V[c])
                io.print(x + " ");
            io.println("");
        } else if (cycle.size() <= k){
            io.println("2");
            io.println(cycle.size());
            for (int x : cycle)
                io.print(x + " ");
            io.println("");
        } else {
            io.println(1);
            k = (k + 1)/2;
            for (int i = 0; k > 0; k--, i += 2)
                io.print(cycle.get(i) + " ");
            io.println("");
        }
        io.close();
    }

}



class IO{
    private BufferedReader br;
    private StringTokenizer st;
    private PrintWriter writer;
    private String inputFile,outputFile;

    public boolean hasMore() throws IOException{
        if(st != null && st.hasMoreTokens()) return true;
        if(br != null && br.ready()) return true;
        return false;
    }

    public String getNext() throws FileNotFoundException, IOException{
        while(st == null || !st.hasMoreTokens()) st = new StringTokenizer(br.readLine());
        return st.nextToken();
    }

    public String getNextLine() throws FileNotFoundException, IOException{
        return br.readLine().trim();
    }

    public int getNextInt() throws FileNotFoundException, IOException{
        return Integer.parseInt(getNext());
    }
    public long getNextLong() throws FileNotFoundException, IOException{
        return Long.parseLong(getNext());
    }

    public void print(double x,int num_digits) throws  IOException{
        writer.printf("%." + num_digits + "f" ,x);
    }
    public void println(double x,int num_digits) throws  IOException{
        writer.printf("%." + num_digits + "f\n" ,x);
    }
    public void print(Object o) throws  IOException{
        writer.print(o.toString());
    }

    public void println(Object o) throws  IOException{
        writer.println(o.toString());
    }
    public IO(String x,String y) throws FileNotFoundException, IOException{
        inputFile = x;
        outputFile = y;
        if(x != null) br = new BufferedReader(new FileReader(inputFile));
        else br = new BufferedReader(new InputStreamReader(System.in));
        if(y != null) writer = new PrintWriter(new BufferedWriter(new FileWriter(outputFile)));
        else writer = new PrintWriter(new OutputStreamWriter(System.out));
    }

    protected void close() throws IOException{
        br.close();
        writer.close();
    }
    public void outputArr(Object [] A) throws IOException{
        int L = A.length;
        for (int i = 0;i < L;i++) {
            if(i > 0) writer.print(" ");
            writer.print(A[i]);
        }
        writer.print("\n");
    }
}