Problem: There is a cave consisting of N rooms and M one-directional passages. The rooms are numbered 1 through N.

Takahashi is now in Room 1, and Room N has the exit. The i-th passage connects Room s_i and Room t_i (s_i < t_i) and can only be traversed in the direction from Room s_i to Room t_i. It is known that, for each room except Room N, there is at least one passage going from that room.

Takahashi will escape from the cave. Each time he reaches a room (assume that he has reached Room 1 at the beginning), he will choose a passage uniformly at random from the ones going from that room and take that passage.

Aoki, a friend of Takahashi's, can block one of the passages (or do nothing) before Takahashi leaves Room 1. However, it is not allowed to block a passage so that Takahashi is potentially unable to reach Room N.

Let E be the expected number of passages Takahashi takes before he reaches Room N. Find the value of E when Aoki makes a choice that minimizes E.

Constraints

* 2 \leq N \leq 600
* N-1 \leq M \leq \frac{N(N-1)}{2}
* s_i < t_i
* If i != j, (s_i, t_i) \neq (s_j, t_j). (Added 21:23 JST)
* For every v = 1, 2, ..., N-1, there exists i such that v = s_i.

Input

Input is given from Standard Input in the following format:


N M
s_1 t_1
:
s_M t_M


Output

Print the value of E when Aoki makes a choice that minimizes E. Your output will be judged as correct when the absolute or relative error from the judge's output is at most 10^{-6}.

Examples

Input

4 6
1 4
2 3
1 3
1 2
3 4
2 4


Output

1.5000000000


Input

3 2
1 2
2 3


Output

2.0000000000


Input

10 33
3 7
5 10
8 9
1 10
4 6
2 5
1 7
6 10
1 4
1 3
8 10
1 5
2 6
6 9
5 6
5 8
3 6
4 8
2 7
2 9
6 7
1 2
5 9
6 8
9 10
3 9
7 8
4 5
2 10
5 7
3 5
4 7
4 9


Output

3.0133333333

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define rep(i, n) for (ll i = 0; i < (ll)(n); i++)
double ans = 0;

int main()
{
ll n , m ;
cin >> n >> m ;
vector<vector<ll>> g(n);
rep(i, m)
{
  ll a, b;
  cin >> a >> b;
  g[a - 1].push_back(b - 1);
}
vector<double> dp(n);
for (ll i = n-2; i >= 0; i--)
{
  for(auto r:g[i]){
    dp[i]+=(dp[r]+1)/g[i].size();
  }
}
ans=dp[0];
rep(i,n-1){
  ll to;
  double e=0;
  if(g[i].size()==1)continue;
  for(auto r:g[i]){
    if(dp[r]>e){
      e=dp[r];
      to=r;
    }
  }
  vector<double> dpp(n);
  for (ll j = n-2; j >= 0; j--)
  {
    if(i==j)for(auto r:g[j]){
      if(to==r)continue;
      dpp[j]+=(dpp[r]+1)/(g[j].size()-1);
    }
    else for(auto r:g[j]){
      dpp[j]+=(dpp[r]+1)/g[j].size();
    }
  }
  ans=min(ans,dpp[0]);
}
cout<<fixed<<setprecision(12)<<ans<<endl;
return 0;
}