Problem: In computer science, there is a method called "Divide And Conquer By Node" to solve some hard problems about paths on a tree. Let's desribe how this method works by function:

solve(t) (t is a tree): 

  1. Chose a node x (it's common to chose weight-center) in tree t. Let's call this step "Line A". 
  2. Deal with all paths that pass x. 
  3. Then delete x from tree t. 
  4. After that t becomes some subtrees. 
  5. Apply solve on each subtree. 



This ends when t has only one node because after deleting it, there's nothing. 

Now, WJMZBMR has mistakenly believed that it's ok to chose any node in "Line A". So he'll chose a node at random. To make the situation worse, he thinks a "tree" should have the same number of edges and nodes! So this procedure becomes like that.

Let's define the variable totalCost. Initially the value of totalCost equal to 0. So, solve(t) (now t is a graph): 

  1. totalCost = totalCost + (size of t). The operation "=" means assignment. (Size of t) means the number of nodes in t. 
  2. Choose a node x in graph t at random (uniformly among all nodes of t). 
  3. Then delete x from graph t. 
  4. After that t becomes some connected components. 
  5. Apply solve on each component. 



He'll apply solve on a connected graph with n nodes and n edges. He thinks it will work quickly, but it's very slow. So he wants to know the expectation of totalCost of this procedure. Can you help him?

Input

The first line contains an integer n (3 ≤ n ≤ 3000) — the number of nodes and edges in the graph. Each of the next n lines contains two space-separated integers ai, bi (0 ≤ ai, bi ≤ n - 1) indicating an edge between nodes ai and bi.

Consider that the graph nodes are numbered from 0 to (n - 1). It's guaranteed that there are no self-loops, no multiple edges in that graph. It's guaranteed that the graph is connected.

Output

Print a single real number — the expectation of totalCost. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6.

Examples

Input

5
3 4
2 3
2 4
0 4
1 2


Output

13.166666666666666


Input

3
0 1
1 2
0 2


Output

6.000000000000000


Input

5
0 1
1 2
2 0
3 0
4 1


Output

13.166666666666666

Note

Consider the second example. No matter what we choose first, the totalCost will always be 3 + 2 + 1 = 6.

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
using namespace std;
const double inf = 1e121;
const double eps = 1e-10;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
long long randint(long long l, long long r) {
  long long out = rng() % (r - l + 1) + l;
  return out >= l ? out : out + r - l + 1;
}
template <typename A, typename B>
string to_string(pair<A, B> p);
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);
string to_string(const string& s) { return '"' + s + '"'; }
string to_string(const char* s) { return to_string((string)s); }
string to_string(bool b) { return (b ? "true" : "false"); }
string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (long long i = 0; i < static_cast<long long>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}
template <size_t N>
string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}
template <typename A>
string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto& x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}
template <typename A, typename B>
string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ")";
}
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}
template <typename A, typename B, typename C, typename D, typename E>
string to_string(tuple<A, B, C, D, E> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + "," +
         to_string(get<4>(p)) + ")";
}
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}
template <typename T>
struct is_pair {
  static const bool value = false;
};
template <typename T, typename U>
struct is_pair<std::pair<T, U>> {
  static const bool value = true;
};
const long long INF = 0x3f3f3f3f3f3f3f3fll;
template <typename T>
typename enable_if<!is_integral<T>::value && !is_pair<T>::value, void>::type
read(T& x) {
  cin >> x;
}
long long read() {
  char c;
  long long out = 0, f = 1;
  for (c = getchar(); !isdigit(c) && c != '-'; c = getchar()) {
  }
  if (c == '-') {
    f = -1;
    c = getchar();
  }
  for (; isdigit(c); c = getchar()) out = (out << 3) + (out << 1) + c - '0';
  return out * f;
}
template <typename T>
typename enable_if<is_integral<T>::value, T>::type read(T& x) {
  char c;
  T f = 1;
  x = 0;
  for (c = getchar(); !isdigit(c) && c != '-'; c = getchar()) {
  }
  if (c == '-') {
    f = -1;
    c = getchar();
  }
  for (; isdigit(c); c = getchar()) x = (x << 3) + (x << 1) + c - '0';
  return x *= f;
}
char read(char& x) {
  for (x = getchar(); isspace(x); x = getchar()) {
  }
  return x;
}
double read(double& x) {
  scanf("%lf", &x);
  return x;
}
template <typename T>
typename enable_if<!is_integral<T>::value && !is_pair<T>::value, void>::type
write(const T& x) {
  cout << x;
}
template <typename T>
typename enable_if<is_integral<T>::value, void>::type write(const T& x) {
  if (x < 0) {
    putchar('-');
    write(-x);
    return;
  }
  if (x > 9) write(x / 10);
  putchar(x % 10 + '0');
}
void write(const char& x) { putchar(x); }
void write(const double& x) { printf("%.12lf", x); }
template <typename T>
typename enable_if<is_pair<T>::value, void>::type read(T& x) {
  read(x.first);
  read(x.second);
}
template <typename T>
typename enable_if<is_pair<T>::value, void>::type write(const T& x) {
  write(x.first);
  putchar(' ');
  write(x.second);
}
template <typename T, typename... Args>
void read(T& x, Args&... args) {
  read(x);
  read(args...);
}
template <
    typename OutputIt,
    typename = typename enable_if<
        is_same<output_iterator_tag,
                typename iterator_traits<OutputIt>::iterator_category>::value ||
        (is_base_of<forward_iterator_tag, typename iterator_traits<OutputIt>::
                                              iterator_category>::value &&
         !is_const<OutputIt>::value)>::type>
void read(OutputIt __first, OutputIt __last) {
  for (; __first != __last; ++__first) read(*__first);
}
template <
    typename InputIt,
    typename = typename enable_if<is_base_of<
        input_iterator_tag,
        typename iterator_traits<InputIt>::iterator_category>::value>::type>
void wts(InputIt __first, InputIt __last) {
  bool isFirst = true;
  for (; __first != __last; ++__first) {
    if (isFirst)
      isFirst = false;
    else
      putchar(' ');
    write(*__first);
  }
  putchar('\n');
}
template <
    typename InputIt,
    typename = typename enable_if<is_base_of<
        input_iterator_tag,
        typename iterator_traits<InputIt>::iterator_category>::value>::type>
void wtb(InputIt __first, InputIt __last) {
  for (; __first != __last; ++__first) {
    write(*__first);
    putchar('\n');
  }
}
template <typename T>
void wts(const T& x) {
  write(x);
  putchar(' ');
}
template <typename T>
void wtb(const T& x) {
  write(x);
  putchar('\n');
}
template <typename T>
void wte(const T& x) {
  write(x);
  exit(0);
}
template <typename T, typename... Args>
void wts(const T& x, Args... args) {
  wts(x);
  wts(args...);
}
template <typename T, typename... Args>
void wtb(const T& x, Args... args) {
  wts(x);
  wtb(args...);
}
template <typename T, typename... Args>
void wte(const T& x, Args... args) {
  wts(x);
  wte(args...);
}
template <typename T1, typename T2>
inline bool up(T1& x, const T2& y) {
  return x < y ? x = y, 1 : 0;
}
template <typename T1, typename T2>
inline bool dn(T1& x, const T2& y) {
  return y < x ? x = y, 1 : 0;
}
template <typename T1, typename T2, typename T3>
inline bool inRange(const T1& x, const T2& l, const T3& r) {
  return !(x < l) && !(r < x);
}
template <typename T1, typename T2>
inline auto minOfDifferentTypes(const T1& x, const T2& y)
    -> decltype(x < y ? x : y) {
  return x < y ? x : y;
}
template <typename T1, typename T2>
inline auto maxOfDifferentTypes(const T1& x, const T2& y)
    -> decltype(x < y ? y : x) {
  return x < y ? y : x;
}
template <typename T1, typename T2, typename T3>
inline T1& madd(T1& x, const T2& y, const T3& modulo) {
  return x = (long long)(x + y + modulo) % modulo;
}
template <typename T1, typename T2, typename T3>
inline T1& mmul(T1& x, const T2& y, const T3& modulo) {
  return x = (long long)x * y % modulo;
}
inline long long modadd(long long x, long long y, long long modulo) {
  return (x + y) >= modulo ? x + y - modulo : x + y;
}
inline long long isinf(long long x) { return x < INF ? x : -1; }
inline void yesno(bool x) { wtb(x ? "Yes" : "No"); }
signed main() {
  long long n = read();
  vector<vector<pair<long long, long long>>> g(n + 1);
  for (long long i = (1), iend = (n); i <= iend; ++i) {
    long long u = read() + 1;
    long long v = read() + 1;
    g[u].emplace_back(v, i);
    g[v].emplace_back(u, i);
  }
  long long dfntot = 0;
  vector<long long> dfn(n + 1);
  stack<long long> stk;
  vector<bool> oncycle(n + 1);
  vector<long long> cycle;
  function<void(long long, long long)> findCycle = [&](long long u,
                                                       long long pa) {
    dfn[u] = ++dfntot;
    stk.push(u);
    for (auto x : g[u]) {
      long long v = x.first;
      long long id = x.second;
      if (id == pa) continue;
      if (!dfn[v])
        findCycle(v, id);
      else if (dfn[v] < dfn[u]) {
        while (1) {
          long long t = stk.top();
          stk.pop();
          cycle.push_back(t);
          oncycle[t] = true;
          if (t == v) break;
        }
      }
    }
    if (cycle.empty()) stk.pop();
  };
  findCycle(1, 0);
  vector<long long> dep(n + 1), treeId(n + 1);
  vector<vector<long long>> dis(n + 1, vector<long long>(n + 1)),
      tree(cycle.size());
  function<void(long long, long long, long long)> getTree =
      [&](long long u, long long pa, long long id) {
        tree[id].push_back(u);
        treeId[u] = id;
        for (auto x : g[u]) {
          long long v = x.first;
          if (oncycle[v] || v == pa) continue;
          dep[v] = dep[u] + 1;
          getTree(v, u, id);
        }
      };
  for (long long i = (0), iend = (cycle.size() - 1); i <= iend; ++i)
    getTree(cycle[i], 0, i);
  double ans = 0;
  function<void(long long, long long, long long)> calcTree =
      [&](long long u, long long pa, long long d) {
        ans += 1.0 / d;
        for (auto x : g[u]) {
          long long v = x.first;
          if (v == pa || treeId[v] != treeId[u]) continue;
          calcTree(v, u, d + 1);
        }
      };
  for (long long i = (0), iend = (cycle.size() - 1); i <= iend; ++i) {
    for (auto u : tree[i]) calcTree(u, 0, 1);
    for (long long j = (i + 1), jend = (cycle.size() - 1); j <= jend; ++j) {
      for (auto u : tree[i]) {
        for (auto v : tree[j]) {
          long long x = dep[u] + dep[v];
          long long y = j - i;
          long long z = cycle.size() - y;
          ans += 2.0 / (x + y + 1) + 2.0 / (x + z + 1) - 2.0 / (x + y + z);
        }
      }
    }
  }
  wtb(ans);
  return 0;
}