Problem:
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.

Sample Input 1

5
1 2
2 3
3 4
3 5

Sample Output 1

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.

Sample Input 2

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

Sample Output 2

3 4

Sample Input 3

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

Sample Output 3

4 6

Sample Input 4

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

Sample Output 4

4 12

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
// #include <boost/multiprecision/cpp_int.hpp>
// using i128 = boost::multiprecision::int128_t;
#define _GLIBCXX_DEBUG
using namespace std;
using ll = long long;
using ld = long double;
using V = vector<int>;
using Vll = vector<ll>;
using Vld = vector<ld>;
using Vbo = vector<bool>;
using VV = vector<V>;
using VVll = vector<Vll>;
using VVld = vector<Vld>;
using VVbo = vector<Vbo>;
using VVV = vector<VV>;
using VVVll = vector<VVll>;
using P = pair<int, int>;
using Pll = pair<ll, ll>;
using Pld = pair<ld, ld>;
#define rep2(i, m, n) for(int i=int(m); i<int(n); ++i)
#define drep2(i, m, n) for(int i=int(m)-1; i>=int(n); --i)
#define rep(i, n) rep2(i, 0, n)
#define drep(i, n) drep2(i, n, 0)
#define all(a) a.begin(), a.end()
struct fast_ios { fast_ios(){ cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(20); }; } fast_ios_;
template<typename T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; } return false; }
template<typename T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; } return false; }
template<typename T1, typename T2> istream &operator>>(istream &is, pair<T1, T2> &p) { is >> p.first >> p.second; return is; }
template<typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &p) { os << p.first << " " << p.second; return os; }
template<typename T> istream &operator>>(istream &is, vector<T> &v) { for (auto &e : v) is >> e; return is; }
template<typename T> ostream &operator<<(ostream &os, const vector<T> &v) { for (auto &e : v) os << e << " "; return os; }
template<typename T> inline int count_between(vector<T> &a, T l, T r) { return lower_bound(all(a), r) - lower_bound(all(a), l); } // [l, r)
inline int Log2(ll x) { int k; for (k = 0; x > 0; ++k) x >>= 1; return k; } // number of binary digits
const int INF  = 1<<30;
const ll INFll = 1ll<<62;
const ld EPS   = 1e-10;
const ld PI    = acos(-1.0);
const int MOD  = int(1e9)+7;
// const int MOD  = 998244353;


VV G;
V deg;

void dfs(int u, int p=-1, int d=0) {
  int cnt = 0;
  for (int v : G[u]) if (v != p) {
    dfs(v, u, d+1);
    ++cnt;
  }
  chmax(deg[d], cnt);
}


int main() {
  int n; cin >> n;
  G.assign(n, V());
  vector<P> E(n-1);
  rep(i, n-1) {
    int a, b; cin >> a >> b;
    --a; --b;
    G[a].push_back(b);
    G[b].push_back(a);
    E[i] = {a, b};
  }

  Pll ans = {INFll, INFll};
  rep(i, n) {
    deg.assign(n, -1);
    dfs(i);
    int x = 0;
    ll y = 1;
    for (; deg[x]>0; ++x) y *= deg[x];
    chmin(ans, {x+1, y});
  }
  for (auto [u, v] : E) {
    deg.assign(n, -1);
    dfs(u, v);
    dfs(v, u);
    int x = 0;
    ll y = 2;
    for (; deg[x]>0; ++x) y *= deg[x];
    chmin(ans, {x+1, y});
  }
  
  cout << ans << "\n";
  return 0;
}