Score : $1100$ points

Problem Statement

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.

Notes

The phrase "picking $u$ as the root and picking $v$ as the root would result in isomorphic rooted trees" for a tree $G$ means that there exists a bijective function $f_{uv}$ from the vertex set of $G$ to itself such that both of the following conditions are met:

• $f_{uv}(u)=v$
• For every pair of two vertices $(a,b)$, edge $(a,b)$ exists if and only if edge $(f_{uv}(a),f_{uv}(b))$ exists.

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.

5
1 2
2 3
3 4
3 5

2 4

If we connect a new vertex $6$ to vertex $2$, painting the vertices $(1,4,5,6)$ red and painting the vertices $(2,3)$ blue is a good coloring. Since painting all the vertices in a single color is not a good coloring, we can see that the colorfulness of this tree is $2$. This is actually the optimal solution. There are four leaves, so we should print $2$ and $4$.

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

3 4

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

4 6

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

4 12