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.io.FileInputStream;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.InputMismatchException;

public class Main {
    InputStream is;

    int __t__ = 1;
    int __f__ = 0;
    int __FILE_DEBUG_FLAG__ = __f__;
    String __DEBUG_FILE_NAME__ = "src/D4";

    FastScanner in;
    PrintWriter out;

    public void solve() {
        int N = in.nextInt();
        int[] a = new int[N - 1];
        int[] b = new int[N - 1];
        for (int i = 0; i < N - 1; i++) {
            a[i] = in.nextInt() - 1;
            b[i] = in.nextInt() - 1;
        }
        ArrayList<Integer>[] g = new ArrayList[N];
        for (int i = 0; i < N; i++) {
            g[i] = new ArrayList<>();
        }
        for (int i = 0; i < N - 1; i++) {
            g[a[i]].add(b[i]);
            g[b[i]].add(a[i]);
        }
        int[] depth = new int[N];
        dfs(0, -1, 0, depth, g);
        int l1 = 0;
        for (int u = 0; u < N; u++) {
            if (depth[u] > depth[l1]) l1 = u;
        }
        dfs(l1, -1, 0, depth, g);
        int l2 = 0;
        for (int u = 0; u < N; u++) {
            if (depth[u] > depth[l2]) l2 = u;
        }

        int[] depthL1 = new int[N];
        int[] depthL2 = new int[N];
        dfs(l1, -1, 0, depthL1, g);
        dfs(l2, -1, 0, depthL2, g);

        int diam = depthL1[l2];
        if (diam % 2 != 0) {
            int c1 = -1, c2 = -1;
            for (int u = 0; u < N; u++) {
                if ((diam / 2 == depthL1[u] && (diam + 1) / 2 == depthL2[u]) ||
                        (diam / 2 == depthL2[u] && (diam + 1) / 2 == depthL1[u])) {
                    if (c1 == -1)
                        c1 = u;
                    else if (c2 == -1)
                        c2 = u;
                    else
                        throw new RuntimeException();
                }
            }
            if (c1 == -1 || c2 == -1) {
                throw new RuntimeException();
            }
            System.out.println(((diam + 1) / 2) + " " + calc(c1, c2, g));
        } else {
            int c1 = -1;
            for (int u = 0; u < N; u++) {
                if (depthL1[u] == depthL2[u] && diam / 2 == depthL1[u]) {
                    if (c1 == -1)
                        c1 = u;
                    else
                        throw new RuntimeException();
                }
            }
            if (c1 == -1)
                throw new RuntimeException();

            long res = calc(c1, c1, g);
            for (int c2 : g[c1]) {
                res = Math.min(res, calc(c1, c2, g));
            }
            System.out.println((diam / 2 + 1) + " " + res);
        }
    }

    private long calc(int c1, int c2, ArrayList<Integer>[] g) {
        int N = g.length;
        int[] depthC1 = new int[N];
        int[] depthC2 = new int[N];
        dfs(c1, -1, 0, depthC1, g);
        dfs(c2, -1, 0, depthC2, g);
        int[] deg = new int[N];
        Arrays.fill(deg, 1);
        if (c1 == c2) {
            deg[0] = Math.max(1, g[c1].size());
        } else {
            deg[0] = Math.max(1, Math.max(g[c1].size(), g[c2].size()) - 1);
        }
        for (int u = 0; u < N; u++) {
            if (c1 == u || c2 == u) continue;
            int d = Math.min(depthC1[u], depthC2[u]);
            deg[d] = Math.max(deg[d], g[u].size());
        }
        long res = deg[0];
        for (int u = 1; u < N; u++) {
            res *= Math.max(deg[u] - 1, 1);
        }
        if (c1 != c2) res *= 2;
        return res;
    }

    void dfs(int u, int pre, int d, int[] depth, ArrayList<Integer>[] g) {
        depth[u] = d;
        for (int v : g[u]) {
            if (v == pre) continue;
            dfs(v, u, d + 1, depth, g);
        }
    }

    public void run() {
        if (__FILE_DEBUG_FLAG__ == __t__) {
            try {
                is = new FileInputStream(__DEBUG_FILE_NAME__);
            } catch (FileNotFoundException e) {
                e.printStackTrace();
            }
            System.out.println("FILE_INPUT!");
        } else {
            is = System.in;
        }
        in = new FastScanner(is);
        out = new PrintWriter(System.out);

        solve();
    }

    public static void main(String[] args) {
        new Main().run();
    }

    public void mapDebug(int[][] a) {
        System.out.println("--------map display---------");

        for (int i = 0; i < a.length; i++) {
            for (int j = 0; j < a[i].length; j++) {
                System.out.printf("%3d ", a[i][j]);
            }
            System.out.println();
        }

        System.out.println("----------------------------");
        System.out.println();
    }

    public void debug(Object... obj) {
        System.out.println(Arrays.deepToString(obj));
    }

    class FastScanner {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int curChar;
        private int numChars;

        public FastScanner(InputStream stream) {
            this.stream = stream;
            //stream = new FileInputStream(new File("dec.in"));

        }

        int read() {
            if (numChars == -1)
                throw new InputMismatchException();
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0)
                    return -1;
            }
            return buf[curChar++];
        }

        boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        boolean isEndline(int c) {
            return c == '\n' || c == '\r' || c == -1;
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        int[] nextIntArray(int n) {
            int[] array = new int[n];
            for (int i = 0; i < n; i++)
                array[i] = nextInt();

            return array;
        }

        int[][] nextIntMap(int n, int m) {
            int[][] map = new int[n][m];
            for (int i = 0; i < n; i++) {
                map[i] = in.nextIntArray(m);
            }
            return map;
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        long[] nextLongArray(int n) {
            long[] array = new long[n];
            for (int i = 0; i < n; i++)
                array[i] = nextLong();

            return array;
        }

        long[][] nextLongMap(int n, int m) {
            long[][] map = new long[n][m];
            for (int i = 0; i < n; i++) {
                map[i] = in.nextLongArray(m);
            }
            return map;
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }

        double[] nextDoubleArray(int n) {
            double[] array = new double[n];
            for (int i = 0; i < n; i++)
                array[i] = nextDouble();

            return array;
        }

        double[][] nextDoubleMap(int n, int m) {
            double[][] map = new double[n][m];
            for (int i = 0; i < n; i++) {
                map[i] = in.nextDoubleArray(m);
            }
            return map;
        }

        String next() {
            int c = read();
            while (isSpaceChar(c))
                c = read();
            StringBuilder res = new StringBuilder();
            do {
                res.appendCodePoint(c);
                c = read();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        String[] nextStringArray(int n) {
            String[] array = new String[n];
            for (int i = 0; i < n; i++)
                array[i] = next();

            return array;
        }

        String nextLine() {
            int c = read();
            while (isEndline(c))
                c = read();
            StringBuilder res = new StringBuilder();
            do {
                res.appendCodePoint(c);
                c = read();
            } while (!isEndline(c));
            return res.toString();
        }
    }
}
