Problem: Btoh yuo adn yuor roomatme lhoate wianshg disehs, btu stlil sdmoeboy msut peorrfm tihs cohre dialy. Oen dya yuo decdie to idourtcne smoe syestm. Yuor rmmotaoe sstgegus teh fooniwllg dael. Yuo argee on tow arayrs of ientgres M adn R, nmebur upmicnog dyas (induiclng teh cunrret oen) wtih sicsescuve irnegets (teh ceurrnt dya is zreo), adn yuo wsah teh diehss on dya D if adn olny if terhe etsixs an iednx i scuh taht D mod M[i] = R[i], otwsehrie yuor rmootmae deos it. Yuo lkie teh cncepot, btu yuor rmotaome's cuinnng simle meaks yuo ssecupt sthnoemig, so yuo itennd to vefriy teh fnerisas of teh aemnrgeet.

Yuo aer geivn ayarrs M adn R. Cuaclatle teh pceanregte of dyas on wchih yuo edn up dnoig teh wisahng. Amsuse taht yuo hvae iiiftlneny mnay dyas aehad of yuo. 

Input

The first line of input contains a single integer N (1 ≤ N ≤ 16).

The second and third lines of input contain N integers each, all between 0 and 16, inclusive, and represent arrays M and R, respectively. All M[i] are positive, for each i R[i] < M[i].

Output

Output a single real number. The answer is considered to be correct if its absolute or relative error does not exceed 10 - 4.

Examples

Input

1
2
0


Output

0.500000


Input

2
2 3
1 0


Output

0.666667

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
struct outFloat {
  int precision;
  long double value;
  outFloat(long double v, int p) : precision(p), value(v) {}
};
const int inf = 123456789;
const long long llInf = 123456789012345678;
const long double pi = 3.14159265358979323846;
const int maxN = 2000000;
int init() {
  ios_base::sync_with_stdio(0);
  srand(time(NULL));
  return 0;
}
int initializer = init();
template <typename T1, typename T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
  is >> p.first >> p.second;
  return is;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &p) {
  for (int i = 0; i < p.size(); i++) is >> p[i];
  return is;
}
class inputer {
 private:
  istream *istr;

 public:
  inputer(istream &is) : istr(&is) {}
  template <typename T>
  void get(T &n) {
    (*istr) >> n;
  }
  template <typename T, typename... Args>
  void get(T &n, Args &...args) {
    get(n);
    get(args...);
  }
  template <typename... Args>
  void getLn(Args &...args) {
    get(args...);
    getLn();
  }
  template <typename T>
  operator T() {
    T n;
    get(n);
    return n;
  }
  void getLn() {
    string s;
    getline(cin, s);
  }
  template <typename T>
  void getv(vector<T> &v, int size) {
    v.resize(size);
    (*istr) >> v;
  }
};
string delim = " ";
template <typename T1, typename T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
  os << p.first << delim << p.second;
  return os;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &p) {
  for (int i = 0; i < p.size(); i++) {
    if (i != 0) os << delim;
    os << p[i];
  }
  return os;
}
ostream &operator<<(ostream &os, const outFloat &of) {
  os << setprecision(of.precision) << of.value;
  return os;
}
class outputer {
 private:
  ostream *ostr;

 public:
  outputer(ostream &os) : ostr(&os) {}
  template <typename T>
  void put(const T &n) {
    (*ostr) << n;
  }
  template <typename T, typename... Args>
  void put(const T &n, const Args &...args) {
    put(n);
    put(args...);
  }
  template <typename... Args>
  void putLn(const Args &...args) {
    put(args...);
    putLn();
  }
  template <typename T>
  outputer &operator=(const T &n) {
    put(n);
    return *this;
  }
  void putLn() { (*ostr) << "\n"; }
  template <typename T>
  void putv(vector<T> v, int size, string delim) {
    for (int i = 0; i < size; i++) {
      if (i != 0) (*ostr) << delim;
      (*ostr) << v[i];
    }
  }
};
inputer in(cin);
outputer out(cout);
char getChar() {
  char c = 0;
  while (c <= ' ') cin >> c;
  return c;
}
bool willItBeAC() {
  return true;
  ;
}
int getMyRating() { return 2121; }
bool prime(long long n) {
  for (long long i = 2; i * i <= n; i++)
    if (n % i == 0) return false;
  return true;
}
vector<long long> divs(long long n) {
  vector<long long> v;
  long long i = 2;
  while (i * i <= n)
    if (n % i == 0) {
      n /= i;
      v.push_back(i);
    } else
      i++;
  v.push_back(n);
  return v;
}
long long gcd(long long a, long long b) { return b ? gcd(b, a % b) : a; }
long long extGcd(long long a, long long b, long long &x, long long &y) {
  if (a == 0) {
    x = 0;
    y = 1;
    return b;
  }
  long long x1, y1, d = extGcd(b % a, a, x1, y1);
  x = y1 - (b / a) * x1;
  y = x1;
  return d;
}
template <typename T>
class fenwick {
 private:
  vector<T> t;
  int n;

 public:
  fenwick(int nn) { init(nn); }
  void init(int nn) {
    n = nn;
    t.assign(n, 0);
  }
  T sum(int r) {
    T result = 0;
    for (; r >= 0; r = (r & (r + 1)) - 1) result += t[r];
    return result;
  }
  void inc(int i, T delta) {
    for (; i < n; i = (i | (i + 1))) t[i] += delta;
  }
  T sum(int l, int r) { return sum(r) - sum(l - 1); }
  T get(int i) { return sum(i, i); }
  void put(int i, T v) { inc(i, v - sum(i, i)); }
  int size() { return n; }
};
int main() {
  int n = in;
  vector<int> m, r;
  int ans = 0;
  in.getv(m, n);
  in.getv(r, n);
  for (int i = 0; i < 720720; i++) {
    bool ok = false;
    for (int j = 0; j < n; j++)
      if (i % m[j] == r[j]) ok = true;
    if (ok) ans++;
  }
  out = outFloat(1.0 * ans / 720720, 10);
  return 0;
}