Coloring of the vertices of a tree G is called a good coloring when, for every pair of two vertices u and v painted in the same color, picking u as the root and picking v as the root would result in isomorphic rooted trees.

Also, the colorfulness of G is defined as the minimum possible number of different colors used in a good coloring of G.

You are given a tree with N vertices. The vertices are numbered 1 through N, and the i-th edge connects Vertex a_i and Vertex b_i. We will construct a new tree T by repeating the following operation on this tree some number of times:

* Add a new vertex to the tree by connecting it to one of the vertices in the current tree with an edge.



Find the minimum possible colorfulness of T. Additionally, print the minimum number of leaves (vertices with degree 1) in a tree T that achieves the minimum colorfulness.

Constraints

* 2 \leq N \leq 100
* 1 \leq a_i,b_i \leq N(1\leq i\leq N-1)
* The given graph is a tree.

Input

Input is given from Standard Input in the following format:


N
a_1 b_1
:
a_{N-1} b_{N-1}


Output

Print two integers with a space in between. First, print the minimum possible colorfulness of a tree T that can be constructed. Second, print the minimum number of leaves in a tree that achieves it.

It can be shown that, under the constraints of this problem, the values that should be printed fit into 64-bit signed integers.

Examples

Input

5
1 2
2 3
3 4
3 5


Output

2 4


Input

8
1 2
2 3
4 3
5 4
6 7
6 8
3 6


Output

3 4


Input

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


Output

4 6


Input

13
5 6
6 4
2 8
4 7
8 9
3 2
10 4
11 10
2 4
13 10
1 8
12 1


Output

4 12

import java.util.Arrays;

public class Main {
  private static void solve() {
    int n = ni();
    int[] from = new int[n - 1];
    int[] to = new int[n - 1];
    
    for (int i = 0; i < n - 1; i ++) {
      from[i] = ni() - 1;
      to[i] = ni() - 1;
    }
    int[][] g = packU(n, from, to);
    int[] center = center(g);

    long[] ret;
    if (center.length == 1) {
      int v = center[0];
      ret = f1(v, g);
      for (int next : g[v]) {
        long[] now = f2(v, next, g);
        if (now[0] < ret[0] || now[0] == ret[0] && now[1] < ret[1]) {
          ret = now;
        }
      }
    } else {
      ret = f2(center[0], center[1], g);
    }
    System.out.println(ret[0] + " " + ret[1]);
  }

  private static long[] f2(int root1, int root2, int[][] g) {
    int n = g.length;
    int[] deg1 = new int[n + 1];
    int[] deg2 = new int[n + 1];
    Arrays.fill(deg1, -1);
    Arrays.fill(deg2, -1);
    dfs(root1, root2, g, deg1, 0);
    dfs(root2, root1, g, deg2, 0);
    
    int[] deg = new int[n + 1];
    int maxDepth = 0;
    for (int i = 0; i < deg.length; i ++) {
      deg[i] = Math.max(deg1[i], deg2[i]);
      if (deg[i] < 0) {
        maxDepth = i;
        break;
      }
    }

    long leaf = 2;
    for (int i = 0; i < maxDepth - 1; i ++) {
      leaf *= deg[i];
    }
    return new long[] {maxDepth, leaf};
  }

  private static void dfs(int now, int pre, int[][] g, int[] deg, int depth) {

    for (int next : g[now]) {
      if (next == pre) continue;
      
      dfs(next, now, g, deg, depth + 1);
    }
    
    deg[depth] = Math.max(deg[depth], g[now].length - 1);
  }

  private static long[] f1(int root, int[][] g) {
    int[][] p3 = parents(g, root);
    int[] depth = p3[2];
    int maxDepth = Arrays.stream(depth).max().getAsInt() + 1;
    int[] deg = new int[maxDepth];
    int n = g.length;
    
    for (int i = 0; i < n; i ++) {
      int x = i == root ? 0 : -1;
      deg[depth[i]] = Math.max(deg[depth[i]], g[i].length + x);
    }
    
    long leaf = 1;
    for (int i = 0; i < maxDepth - 1; i ++) {
      leaf *= deg[i];
    }
    return new long[] {maxDepth, leaf};
  }
  
  public static int[][] parents(int[][] g, int root)
  {
      int n = g.length;
      int[] par = new int[n];
      Arrays.fill(par, -1);
      
      int[] depth = new int[n];
      depth[0] = 0;
      
      int[] q = new int[n];
      q[0] = root;
      for(int p = 0, r = 1;p < r;p++) {
          int cur = q[p];
          for(int nex : g[cur]){
              if(par[cur] != nex){
                  q[r++] = nex;
                  par[nex] = cur;
                  depth[nex] = depth[cur] + 1;
              }
          }
      }
      return new int[][] {par, q, depth};
  }
  
  public static int[][] packU(int n, int[] from, int[] to){ return packU(n, from, to, from.length); }
  public static int[][] packU(int n, int[] from, int[] to, int sup)
  {
      int[][] g = new int[n][];
      int[] p = new int[n];
      for(int i = 0;i < sup;i++)p[from[i]]++;
      for(int i = 0;i < sup;i++)p[to[i]]++;
      for(int i = 0;i < n;i++)g[i] = new int[p[i]];
      for(int i = 0;i < sup;i++){
          g[from[i]][--p[from[i]]] = to[i];
          g[to[i]][--p[to[i]]] = from[i];
      }
      return g;
  }
  
  public static int[] center(int[][] g)
  {
      int n = g.length;
      int start = 0;
      int[] d1 = dist(g, start);
      int maxd1 = -1;
      int argmaxd1 = -1;
      for(int i = 0;i < n;i++){
          if(d1[i] > maxd1){
              maxd1 = d1[i];
              argmaxd1 = i;
          }
      }
      
      int[] d2 = dist(g, argmaxd1);
      int maxd2 = -1;
      int argmaxd2 = -1;
      for(int i = 0;i < n;i++){
          if(d2[i] > maxd2){
              maxd2 = d2[i];
              argmaxd2 = i;
          }
      }
      
      if(maxd2 % 2 == 0){
          int h = maxd2/2;
          int j = argmaxd2;
          outer:
          for(int i = maxd2;i > h;i--){
              for(int e : g[j]){
                  if(d2[e] == i-1){
                      j = e;
                      continue outer;
                  }
              }
              throw new RuntimeException();
          }
          return new int[]{j};
      }else{
          int h = maxd2/2;
          int j = argmaxd2;
          int[] cs = new int[2];
          int p = 0;
          outer:
          for(int i = maxd2;i >= h;i--){
              if(i <= h+1)cs[p++] = j;
              if(i > h){
                  for(int e : g[j]){
                      if(d2[e] == i-1){
                          j = e;
                          continue outer;
                      }
                  }
                  throw new RuntimeException();
              }
          }
          return cs;
      }
  }
  
  public static int[] dist(int[][] g, int start)
  {
      int n = g.length;
      int[] d = new int[n];
      Arrays.fill(d, n+3);
      int[] q = new int[n];
      int p = 0;
      q[p++] = start;
      d[start] = 0;
      for(int r = 0;r < p;r++){
          int cur = q[r];
          for(int e : g[cur]){
              if(d[e] > d[cur] + 1){
                  d[e] = d[cur] + 1;
                  q[p++] = e;
              }
          }
      }
      return d;
  }
  
  public static void main(String[] args) {
    new Thread(null, new Runnable() {
      @Override
      public void run() {
        long start = System.currentTimeMillis();
        String debug = args.length > 0 ? args[0] : null;
        if (debug != null) {
          try {
            is = java.nio.file.Files.newInputStream(java.nio.file.Paths.get(debug));
          } catch (Exception e) {
            throw new RuntimeException(e);
          }
        }
        reader = new java.io.BufferedReader(new java.io.InputStreamReader(is), 32768);
        solve();
        out.flush();
        tr((System.currentTimeMillis() - start) + "ms");
      }
    }, "", 64000000).start();
  }

  private static java.io.InputStream is = System.in;
  private static java.io.PrintWriter out = new java.io.PrintWriter(System.out);
  private static java.util.StringTokenizer tokenizer = null;
  private static java.io.BufferedReader reader;

  public static String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }

  private static double nd() {
    return Double.parseDouble(next());
  }

  private static long nl() {
    return Long.parseLong(next());
  }

  private static int[] na(int n) {
    int[] a = new int[n];
    for (int i = 0; i < n; i++)
      a[i] = ni();
    return a;
  }

  private static char[] ns() {
    return next().toCharArray();
  }

  private static long[] nal(int n) {
    long[] a = new long[n];
    for (int i = 0; i < n; i++)
      a[i] = nl();
    return a;
  }

  private static int[][] ntable(int n, int m) {
    int[][] table = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[i][j] = ni();
      }
    }
    return table;
  }

  private static int[][] nlist(int n, int m) {
    int[][] table = new int[m][n];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        table[j][i] = ni();
      }
    }
    return table;
  }

  private static int ni() {
    return Integer.parseInt(next());
  }

  private static void tr(Object... o) {
    if (is != System.in)
      System.out.println(java.util.Arrays.deepToString(o));
  }
}


