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 java.util.*;
import java.io.*;

public class CFA {
  BufferedReader br;
  PrintWriter out;
  StringTokenizer st;
  boolean eof;
  private static final long MOD = 1000L * 1000L * 1000L + 7;
  private static final int[] dx = {0, -1, 0, 1};
  private static final int[] dy = {1, 0, -1, 0};
  private static final String yes = "Yes";
  private static final String no = "No";

  void solve() {
//    int T = nextInt();
    int T = 1;
    for (int i = 0; i < T; i++) {
      helper();
    }
  }

  int n;
  List<List<Integer>> graph;
  int[] depth;
  boolean[] vis;
  int[] prev;
  void helper() {
    n = nextInt();
    int m = nextInt();
    graph = new ArrayList<>();
    for (int i = 0; i < n; i++) {
      graph.add(new ArrayList<>());
    }
    for (int i = 0; i < m; i++) {
      int from = nextInt() - 1;
      int to = nextInt() - 1;
      graph.get(from).add(to);
      graph.get(to).add(from);
    }

    int sq = 1;
    while(sq * sq < n) {
      sq++;
    }

    // Find cycle.
    depth = new int[n];
    vis = new boolean[n];
    prev = new int[n];
    Arrays.fill(prev, -1);
    dfs(-1, 0, 0);
    for (int from = 0; from < n; from++) {
      for (int to : graph.get(from)) {
        int diff = Math.abs(depth[from] - depth[to]);
        if (diff >= sq - 1) {
          // Find a cycle with size >= sq.
          int start;
          int end;
          if (depth[from] < depth[to]) {
            start = to;
            end = from;
          } else {
            start = from;
            end = to;
          }
          List<Integer> cycle = new ArrayList<>();
          while (start != end) {
            cycle.add(start);
            start = prev[start];
          }
          cycle.add(end);
          outln(2);
          outln(cycle.size());
          for (int v : cycle) {
            out((v + 1) + " ");
          }
          outln("");
          return;
        }
      }
    }

    // Find independent set.
    int[] degree = new int[n];
    for (int i = 0; i < n; i++) {
      for (int j : graph.get(i)) {
        if (i < j) {
          degree[i]++;
          degree[j]++;
        }
      }
    }
    TreeSet<Pair> pairs = new TreeSet<>();
    for (int i = 0; i < n; i++) {
      pairs.add(new Pair(i, degree[i]));
    }

    List<Integer> points = new ArrayList<>();
    boolean[] alreadyRemoved = new boolean[n];
    while (!pairs.isEmpty()) {
      Pair cur = pairs.pollFirst();
      points.add(cur.idx);
      if (points.size() >= sq) {
        break;
      }

      List<Pair> remove = new ArrayList<>();
      remove.add(new Pair(cur.idx, cur.cnt));
      for (int nxt : graph.get(cur.idx)) {
        if (!alreadyRemoved[nxt]) {
          remove.add(new Pair(nxt, degree[nxt]));
        }
      }
      for (Pair p : remove) {
        pairs.remove(p);
        alreadyRemoved[p.idx] = true;
      }
      for (Pair p : remove) {
        for (int neigh : graph.get(p.idx)) {
          boolean good = pairs.remove(new Pair(neigh, degree[neigh]));
          if (good) {
            degree[neigh]--;
            pairs.add(new Pair(neigh, degree[neigh]));
          }
        }
      }
    }

    outln(1);
    for (int v : points) {
      out((v + 1) + " ");
    }
    outln("");
  }

  class Pair implements Comparable<Pair> {
    int idx;
    int cnt;
    public Pair(int idx, int cnt) {
      this.idx = idx;
      this.cnt = cnt;
    }

    @Override
    public int compareTo(Pair o) {
      if (cnt != o.cnt) {
        return Integer.compare(cnt, o.cnt);
      }
      return Integer.compare(idx, o.idx);
    }

    @Override
    public String toString() {
      return idx + " " + cnt;
    }
  }

  void dfs(int from, int to, int d) {
    depth[to] = d;
    vis[to] = true;
    prev[to] = from;
    for (int nxt : graph.get(to)) {
      if (nxt == from || vis[nxt]) {
        continue;
      }
      dfs(to, nxt, d + 1);
    }
  }

  void shuffle(long[] a) {
    int n = a.length;
    for(int i = 0; i < n; i++) {
      int r = i + (int) (Math.random() * (n - i));
      long tmp = a[i];
      a[i] = a[r];
      a[r] = tmp;
    }
  }

  long gcd(long a, long b) {
    while(a != 0 && b != 0) {
      long c = b;
      b = a % b;
      a = c;
    }
    return a + b;
  }
  private void outln(Object o) {
    out.println(o);
  }
  private void out(Object o) {
    out.print(o);
  }
  private void formatPrint(double val) {
    outln(String.format("%.9f%n", val));
  }
  public CFA() {
    br = new BufferedReader(new InputStreamReader(System.in));
    out = new PrintWriter(System.out);
    solve();
    out.close();
  }
  public static void main(String[] args) {
    new CFA();
  }

  public long[] nextLongArr(int n) {
    long[] res = new long[n];
    for(int i = 0; i < n; i++)
      res[i] = nextLong();
    return res;
  }
  public int[] nextIntArr(int n) {
    int[] res = new int[n];
    for(int i = 0; i < n; i++)
      res[i] = nextInt();
    return res;
  }
  public String nextToken() {
    while (st == null || !st.hasMoreTokens()) {
      try {
        st = new StringTokenizer(br.readLine());
      } catch (Exception e) {
        eof = true;
        return null;
      }
    }
    return st.nextToken();
  }
  public String nextString() {
    try {
      return br.readLine();
    } catch (IOException e) {
      eof = true;
      return null;
    }
  }
  public int nextInt() {
    return Integer.parseInt(nextToken());
  }
  public long nextLong() {
    return Long.parseLong(nextToken());
  }
  public double nextDouble() {
    return Double.parseDouble(nextToken());
  }
}