Problem:
Score : 600 points

Problem Statement

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}.

Sample Input 1

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

Sample Output 1

1.5000000000

If Aoki blocks the passage from Room 1 to Room 2, Takahashi will go along the path 1 → 3 → 4 with probability \frac{1}{2} and 1 → 4 with probability \frac{1}{2}. E = 1.5 here, and this is the minimum possible value of E.

Sample Input 2

3 2
1 2
2 3

Sample Output 2

2.0000000000

Blocking any one passage makes Takahashi unable to reach Room N, so Aoki cannot block a passage.

Sample Input 3

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

Sample Output 3

3.0133333333

Solution in C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

/*
 * じょえチャンネル
 * 高評価・チャンネル登録よろしくおねがいします！
 * https://www.youtube.com/channel/UCRXsI3FL_kvaVL9zoolBfbQ
 */

#include <bits/stdc++.h>

//#pragma GCC target("avx2")
//#pragma GCC optimize("O3")
//#pragma GCC optimize("unroll-loops")




//here!!!
// define int long long !!!!!

#define int long long

// define int long long !!!!!


#define mod (int)((1e9)+7)
//constexpr int mod = 998244353ll;


#ifdef int
#define inf (int)(3e18)
#else
#define inf (int)(5e8)
#endif

#define intt long long
#define itn long long
#define P pair<long long,long long>

#define rep(i,n) for(int i=0;i<n;i++)
#define REP(i,n) for(int i=1;i<=n;i++)
#define rev_rep(i,n) for(int i=n-1;i>=0;i--)
#define REV_REP(i,n) for(int i=n;i>=1;i--)

#define ALL(v) v.begin(),v.end()

#define smallpriority_queue(x) priority_queue<x,vector<x>,greater<x>>

using namespace std;

//Library
//モッドパウ
inline int modpow(int x, int y, int m = mod) {
    int res = 1;
    while (y) {
        if (y & 1) {
            res *= x;
            res %= m;
        }
        x = x * x % m;
        y /= 2;
    }
    return res;
}

int mypow(int x, int y) {
    int res = 1;
    while (y) {
        if (y % 2) {
            res *= x;
        }
        x = x * x;
        y /= 2;
    }
    return res;
}
//is the number (x) a prime number?
bool prime(int x) {
    if (!x || x == 1) {
        return false;
    }
    for (int i = 2; i * i <= x; i++) {
        if (!(x % i)) {
            return false;
        }
    }
    return true;
}


//saidai-kouyakusuu
inline int gcd(int x, int y) {
    if (!y) {
        return x;
    }
    return gcd(y, x % y);
}


//number of keta
int keta(int x) {
    int ans = 0;
    while (x) {
        x /= 10;
        ans++;
    }
    return ans;
}

//sum of keta
int ketasum(int x) {
    int ans = 0;
    while (x) {
        ans += x % 10;
        x /= 10;
    }
    return ans;
}

inline int lcm(int x, int y) {
    int ans = x / gcd(x, y) * y;
    return ans;
}
int twobeki(int x) {
    int ans = 0;
    while (1) {
        if (!(x & 1)) {
            ans++;
            x >>= 1;
        }
        else {
            break;
        }
    }
    return ans;
}


template <class T, class U>
inline bool chmax(T& lhs, const U& rhs) {
    if (lhs < rhs) {
        lhs = rhs;
        return 1;
    }
    return 0;
}
template <class T, class U>
inline bool chmin(T& lhs, const U& rhs) {
    if (lhs > rhs) {
        lhs = rhs;
        return 1;
    }
    return 0;
}
void Yes(){
    cout<<"Yes"<<endl;
}
void No(){
    cout<<"No"<<endl;
}
void YES(){
    cout<<"YES"<<endl;
}
void NO(){
    cout<<"NO"<<endl;
}


#define fin(i) scanf("%lld",&i)
#define fout(i) printf("%lld",i)
#define fendl printf("\n")

int kai(int x, int y) {
    int res = 1;
    for (int i = x - y + 1; i <= x; i++) {
        res *= i; res %= mod;
    }
    return res;
}

int comb(int x, int y) {
    if (y > x)return 0;
    //    cout<<kai(x, y)<<' '<<modpow(kai(y, y), mod - 2)<<endl;
    return kai(x, y) * modpow(kai(y, y), mod - 2) % mod;
}
//Library-End


#define vecin(v) for(int i=0;i<v.size();i++)scanf("%lld",&v[i]);
#define vecout(v) {for(int i=0;i<(int)v.size();i++)printf("%lld ",v[i]);printf("\n");}


template<typename T>
class kaageSegTree {
protected:
    unsigned int n = 1, rank = 0;
    std::vector<T> node;
    T nodee;
    virtual T nodef(const T&, const T&)const = 0;
public:
    kaageSegTree(unsigned int m, T init, T nodee):nodee(nodee) {
        while (n < m) {
            n *= 2;
            rank++;
        }
        node.resize(2 * n);
        for (unsigned int i = n; i < 2 * n; i++)node[i] = init;
    }
    kaageSegTree(const std::vector<T>& initvec, T nodee):nodee(nodee) {
        unsigned int m = initvec.size();
        while (n < m) {
            n *= 2;
            rank++;
        }
        node.resize(2 * n);
        for (unsigned int i = n; i < 2 * n; i++) {
            if (i - n < m)node[i] = initvec[i - n];
        }
    }
    virtual void update(int i, T x) {
        i += n;
        node[i] = x;
        while (i != 1) {
            i >>= 1;
            node[i] = nodef(node[2 * i], node[2 * i + 1]);
        }
    }
    virtual T query(int l, int r) {
        l += n; r += n;
        T ls = nodee, rs = nodee;
        while (l < r) {
            if (l & 1) {
                ls = nodef(ls, node[l]);
                l++;
            }
            if (r & 1) {
                r--;
                rs = nodef(node[r], rs);
            }
            l >>= 1; r >>= 1;
        }
        return nodef(ls, rs);
    }
    virtual T operator[](const int& x) {
        return node[n + x];
    }
    void fill(T x) {
        std::fill(ALL(node), x);
    }
    void print() {
        rep(i, n)std::cout << operator[](i) << " ";
        std::cout << std::endl;
    }
};
class RSQ :public kaageSegTree<int> {
    int nodef(const int& lhs,const int& rhs)const{return lhs+rhs;}
public:
    RSQ(int size, const int& def = 0) :kaageSegTree<int>(size, def, 0) {}
    RSQ(const std::vector<int>& initvec) :kaageSegTree<int>(initvec, 0) {}
};
class RMiQ :public kaageSegTree<int> {
    int nodef(const int& lhs,const int& rhs)const{return std::min(lhs,rhs);}
public:
    RMiQ(int size, const int& def = 0) :kaageSegTree<int>(size, def, inf) {}
    RMiQ(const std::vector<int>& initvec) :kaageSegTree<int>(initvec, inf) {}
};
class RMaQ :public kaageSegTree<int> {
    int nodef(const int& lhs,const int& rhs)const{return std::max(lhs,rhs);}
public:
    RMaQ(int size, const int& def = 0) :kaageSegTree<int>(size, def, -inf) {}
    RMaQ(const std::vector<int>& initvec) :kaageSegTree<int>(initvec, -inf) {}
};
template<typename T, typename U>
class IntervalSegTree :public kaageSegTree<T> {
protected:
    using kaageSegTree<T>::n;
    using kaageSegTree<T>::rank;
    using kaageSegTree<T>::node;
    using kaageSegTree<T>::nodef;
    using kaageSegTree<T>::nodee;
    std::vector<U> lazy;
    std::vector<bool> lazyflag;
    std::vector<int> width;
    virtual void lazyf(U&, const U&) = 0;
    virtual void updf(T&, const U&, const unsigned int&) = 0;
    void eval(int k) {
        for (int i = rank; i > 0; i--) {
            int nk = k >> i;
            if (lazyflag[nk]) {
                updf(node[2 * nk], lazy[nk], width[2 * nk]);
                updf(node[2 * nk + 1], lazy[nk], width[2 * nk + 1]);
                if (lazyflag[2 * nk])lazyf(lazy[2 * nk], lazy[nk]);
                else lazy[2 * nk] = lazy[nk];
                if (lazyflag[2 * nk + 1])lazyf(lazy[2 * nk + 1], lazy[nk]);
                else lazy[2 * nk + 1] = lazy[nk];
                lazyflag[2 * nk] = lazyflag[2 * nk + 1] = true;
                lazyflag[nk] = false;
            }
        }
    }
public:
    IntervalSegTree(unsigned int m, T init, T nodee) :kaageSegTree<T>(m, init, nodee) {
        lazy.resize(2 * n); lazyflag.resize(2 * n); width.resize(2 * n);
        width[1] = n;
        for (unsigned int i = 2; i < 2 * n; i++) {
            width[i] = width[i >> 1] >> 1;
        }
    }
    IntervalSegTree(T nodee, const std::vector<T>& initvec) :kaageSegTree<T>(initvec, nodee) {
        lazy.resize(2 * n); lazyflag.resize(2 * n); width.resize(2 * n);
        width[1] = n;
        for (unsigned int i = 2; i < 2 * n; i++) {
            width[i] = width[i >> 1] >> 1;
        }
    }
    void update(int i, U x) {
        i += n;
        eval(i);
        updf(node[i], x, width[i]);
        if (lazyflag[i])lazyf(lazy[i], x);
        else {
            lazyflag[i] = true;
            lazy[i] = x;
        }
        while (i /= 2)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    void update(int l, int r, U x) {
        l += n; r += n;
        int nl = l, nr = r;
        while (!(nl & 1))nl >>= 1;
        while (!(nr & 1))nr >>= 1;
        nr--;
        eval(nl); eval(nr);
        while (l < r) {
            if (l & 1) {
                updf(node[l], x, width[l]);
                if (lazyflag[l])lazyf(lazy[l], x);
                else {
                    lazyflag[l] = true;
                    lazy[l] = x;
                }
                l++;
            }
            if (r & 1) {
                r--;
                updf(node[r], x, width[r]);
                if (lazyflag[r])lazyf(lazy[r], x);
                else {
                    lazyflag[r] = true;
                    lazy[r] = x;
                }
            }
            l >>= 1; r >>= 1;
        }
        while (nl >>= 1)node[nl] = nodef(node[2 * nl], node[2 * nl + 1]);
        while (nr >>= 1)node[nr] = nodef(node[2 * nr], node[2 * nr + 1]);
    }
    T query(int l, int r) {
        l += n; r += n;
        eval(l); eval(r - 1);
        int ls = nodee, rs = nodee;
        while (l < r) {
            if (l & 1) {
                ls = nodef(ls, node[l]);
                l++;
            }
            if (r & 1) {
                r--;
                rs = nodef(node[r], rs);
            }
            l >>= 1; r >>= 1;
        }
        return nodef(ls, rs);
    }
    T operator[](const int& x) {
        eval(n + x);
        return node[n + x];
    }
    T queryForAll() {
        return node[1];
    }
};
class RAQRSQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return a + b; }
    void lazyf(int& a, const int& b) { a += b; }
    void updf(int& a, const int& b, const unsigned int& width) { a += width * b; }
public:
    RAQRSQ(int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, 0) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RAQRSQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>((int)0, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};
class RAQRMiQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return std::min(a, b); }
    void lazyf(int& a, const int& b) { a += b; }
    void updf(int& a, const int& b, const unsigned int& width) { a += b; }
public:
    RAQRMiQ(int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, inf) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RAQRMiQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>(inf, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};
class RAQRMaQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return std::max(a, b); }
    void lazyf(int& a, const int& b) { a += b; }
    void updf(int& a, const int& b, const unsigned int& width) { a += b; }
public:
    RAQRMaQ(unsigned int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, -inf) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RAQRMaQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>(-inf, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};
class RUQRSQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return a + b; }
    void lazyf(int& a, const int& b) { a = b; }
    void updf(int& a, const int& b, const unsigned int& width) { a = width * b; }
public:
    RUQRSQ(int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, 0) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RUQRSQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>((int)0, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};
class RUQRMiQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return std::min(a, b); }
    void lazyf(int& a, const int& b) { a = b; }
    void updf(int& a, const int& b, const unsigned int& width) { a = b; }
public:
    RUQRMiQ(int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, inf) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RUQRMiQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>(inf, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};
class RUQRMaQ :public IntervalSegTree<int, int> {
    int nodef(const int& a, const int& b)const { return std::max(a, b); }
    void lazyf(int& a, const int& b) { a = b; }
    void updf(int& a, const int& b, const unsigned int& width) { a = b; }
public:
    RUQRMaQ(int size, const int& def = 0) :IntervalSegTree<int, int>(size, def, -inf) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
    RUQRMaQ(const std::vector<int>& initvec) :IntervalSegTree<int, int>(-inf, initvec) {
        for (int i = n - 1; i > 0; i--)node[i] = nodef(node[2 * i], node[2 * i + 1]);
    }
};

////SegTree
//template <class T>
//class SegTree {
//    int n;                       // 葉の数
//    vector<T> node;              // データを格納するvector
//    T def;                       // 初期値かつ単位元
//    function<T(T, T)> operation; // 区間クエリで使う処理
//    function<T(T, T)> update;    // 点更新で使う処理
//
//    // 区間[a,b)の総和。ノードk=[l,r)に着目している。
//    T _query(int a, int b, int k, int l, int r) {
//        if (r <= a || b <= l) return def; // 交差しない
//        if (a <= l && r <= b)
//            return node[k]; // a,l,r,bの順で完全に含まれる
//        else {
//            T c1 = _query(a, b, 2 * k + 1, l, (l + r) / 2); // 左の子
//            T c2 = _query(a, b, 2 * k + 2, (l + r) / 2, r); // 右の子
//            return operation(c1, c2);
//        }
//    }
//
//public:
//    // _n:必要サイズ, _def:初期値かつ単位元, _operation:クエリ関数,
//    // _update:更新関数
//    SegTree(size_t _n, T _def, function<T(T, T)> _operation,
//            function<T(T, T)> _update)
//    : def(_def), operation(_operation), update(_update) {
//        n = 1;
//        while (n < _n) {
//            n *= 2;
//        }
//        node = vector<T>(2 * n , def);
//    }
//
//    // 場所i(0-indexed)の値をxで更新
//    void change(int i, T x) {
//        i += n - 1;
//        node[i] = update(node[i], x);
//        while (i > 0) {
//            i = (i - 1) / 2;
//            node[i] = operation(node[i * 2 + 1], node[i * 2 + 2]);
//        }
//    }
//
//    // [a, b)の区間クエリを実行
//    T query(int a, int b) {
//        return _query(a, b, 0, 0, n);
//    }
//
//    // 添字でアクセス
//    T operator[](int i) {
//        return node[i + n - 1];
//    }
//};

template <class T>
class SegTree {
    int n;
    vector<T> node;
    T def;
    function<T(T,T)> operation;
    function<T(T,T)> update;
    
public:
    SegTree(unsigned int _n, T _def, function<T(T, T)> _operation,
            function<T(T, T)> _update)
    : def(_def), operation(_operation), update(_update) {
        n=1;
        while (n < _n) {
            n *= 2;
        }
        node = vector<T>(n * 2, def);
    }
    SegTree(vector<int>& initvec, function<T(T, T)> _operation,
            function<T(T, T)> _update)
    :  operation(_operation), update(_update) {
        n=1;
        while (n < initvec.size()) {
            n *= 2;
        }
        node = vector<T>(n * 2, def);
        for(int i=n;i<n+initvec.size();i++){
            node[i]=initvec[i-n];
        }
        for(int i=n-1;i>=1;i--){
            node[i]=operation(node[i*2],node[i*2+1]);
        }
    }
    void change(int i,T x){
        i+=n;
        node[i]=update(node[i],x);
        while (i>=1) {
            i>>=1;
            node[i]=operation(node[i*2],node[i*2+1]);
        }
    }
    T query(int l, int r){
        l+=n;
        r+=n;
        T rx=def,lx=def;
        while(l<r){
            if (l&1) {
                lx=operation(lx,node[l]);
                l++;
            }
            if (r&1) {
                r--;
                rx=operation(node[r],rx);
            }
            l>>=1; r>>=1;
        }
        return operation(lx,rx);
    }
    T operator [] (const int& x){
        return node[x-n];
    }
    void fill(T x) {
        std::fill(ALL(node), x);
    }
    void print() {
        rep(i, n)std::cout << operator[](i) << " ";
        std::cout << std::endl;
    }
    
};

#define R_MIN ([](long long a, long long b) { return min(a, b); })
#define R_MAX ([](long long a, long long b) { return max(a, b); })
#define R_SUM ([](long long a, long long b) { return a + b; })

#define NORMAL_UPDATE ([](long long a, long long b) { return b; })
#define ADD_UPDATE ([](long long a, long long b) { return a + b; })
#define MINUS_UPDATE ([](long long a, long long b) { return a - b; }

class Union_Find {
    vector<int> par;
    vector<int> rankmy;
    vector<int> ookisa;
    
public:
    Union_Find(int size) {
        par = vector<int>(size);
        rankmy = vector<int>(size);
        ookisa=vector<int>(size);
        for (int i = 0; i < size; i++) {
            par[i] = i;
            ookisa[i]=1;
        }
    }
    
    int find(int x) {
        if (par[x] == x) {
            return x;
        }
        return par[x] = find(par[x]);
    }
    
    void unite(int x, int y) {
        x = find(x);
        y = find(y);
        if (x == y) {
            return;
        }
        if (rankmy[x] < rankmy[y]) {
            par[x] = y;
            ookisa[y]+=ookisa[x];
            ookisa[x]=0;
        }
        else {
            par[y] = x;
            ookisa[x]+=ookisa[y];
            ookisa[y]=0;
            if (rankmy[x] == rankmy[y]) {
                rankmy[x]++;
            }
        }
    }
    int size(int i){
        i=find(i);
        return ookisa[i];
    }
    bool same(int x, int y){
        return find(x) == find(y);
    }
    
};

class BIT {
    vector<int> data;
    int size=0;
public:
    BIT(int _size){
        data=vector<int>(_size+1);
        size=_size;
    }
    void add(int i,int x){
        while (i<=size) {
            data[i]+=x;
            i += i & -i;
        }
    }
    int sum(int i){
        assert(i<=size);
        int s=0;
        while(i>0){
            s+=data[i];
            i -= i & -i;
        }
        return s;
    }
    int lower_bound(int x){
        if(x<=0){
            return 0;
        }else{
            int i=0;int r=1;
            while(r<size) r=r<<1;
            for(int len=r;len>0;len=len>>1) {
                if(i+len<size && data[i+len]<x){
                    x-=data[i+len];
                    i+=len;
                }
            }
            return i+1;
        }
    }
};

//Union-Find-End

int perm[2000005];
void init_perm() {
    perm[0] = 1;
    REP(i, 2000004) {
        perm[i] = perm[i - 1] * i % mod;
    }
}

int nCk(int x, int y) {
    if (y>x) {
        return 0;
    }
    if (x<0) {
        return 0;
    }
    return perm[x] * modpow(perm[x - y], mod - 2) % mod * modpow(perm[y], mod - 2) % mod;
}

double kyori(pair<int, int> f, pair<int, int> s) {
    double ans = 0;
    double t = fabs(f.first - s.first);
    double y = fabs(f.second - s.second);
    ans = sqrt(t * t + y * y);
    return ans;
}

inline string stringmax(string& x,string& y){
    if (x.size()>y.size()) {
        return x;
    }
    if (x.size()<y.size()) {
        return y;
    }
    rep(i,x.size()){
        if (x[i]>y[i]) {
            return x;
        }
        if (x[i]<y[i]) {
            return y;
        }
    }
    return x;
}

vector<int>  RollingHash(string &s, string& t){
    vector<int> ans;
    __int128 h=0,hamod=0,ki=0,kim=0,hikaku=0,one=0;
    one=1;
    ki=1000000007ll;
    hamod=(one<<61)-1;
    kim=1;
    rep(i,t.size()){
        hikaku*=ki;
        h*=ki;
        kim*=ki;
        hikaku+=t[i];
        h+=s[i];
        hikaku%=hamod;
        h%=hamod;
        kim%=hamod;
    }
    rep(i,(int)s.size()-(int)t.size()+1){
        if (h==hikaku) {
            ans.emplace_back(i);
        }
        h*=ki;
        h%=hamod;
        h+=s[i+(int)t.size()];
        h%=hamod;
        h+=hamod;
        h-=s[i]*kim%hamod;
        h%=hamod;
    }
    return ans;
}
struct edge {
    int to;
    int length;
    edge(int _to, int _length){
        to=_to;
        length=_length;
    }
};
vector<int> djkstra(vector<vector<edge>> &road,int start){
    vector<int> kyo(road.size(),inf);
    smallpriority_queue(P) q;
    q.push({0,start});
    kyo[start]=0;
    while (q.size()) {
        int x=q.top().second;
        itn now=q.top().first;
        q.pop();
        if (kyo[x]<now) {
            continue;
        }
        for(auto&i:road[x]){
            if (kyo[i.to]>now+i.length) {
                kyo[i.to]=now+i.length;
                q.push({kyo[i.to],i.to});
            }
        }
    }
    return kyo;
}

#define endl "\n" //interactive の時に注意！！！
#define printd cout<<fixed<<setprecision(10)


double n,m,s[600],t[600],ans;
vector<pair<double, double>> road[604];
signed main() {
    ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    cin>>n>>m;
    rep(i,m){
        cin>>s[i]>>t[i];
        road[(int)s[i]].emplace_back(t[i],i);
    }
    ans=inf;
    rep(i,m+1){
        double kitaichi[604];
        fill(kitaichi, kitaichi+(int)n, 0.0);
        REV_REP(j,n-1){
            double cnt=0;
            for(auto&k:road[j]){
                if (k.second==i) {
                    continue;
                }
                cnt+=1;
                kitaichi[j]+=kitaichi[(int)k.first]+1.0;
            }
            kitaichi[j]/=cnt;
        }
        chmin(ans, kitaichi[1]);
    }
    printd<<ans<<endl;
}