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 JAVA):

Solution:
import java.io.*;
import java.util.*;

public class Main {
  private static void execute(ContestReader reader, PrintWriter out) {
    int n = reader.nextInt();
    int m = reader.nextInt();
    int[][] sts = reader.nextIntMatrix(m, 2);
    out.printf("%.20f\n", new Solver(n, m, sts).solve());
  }
  
  public static void main(String[] args) {
    ContestReader reader = new ContestReader(System.in);
    PrintWriter out = new PrintWriter(System.out);
    execute(reader, out);
    out.flush();
  }
}

class Solver {
  final int n, m;
  final int[][] sts;
  List<List<Integer>> graph;
  
  Solver(int n, int m, int[][] sts) {
    this.n = n;
    this.m = m;
    this.sts = sts;
  }
  
  public double solve() {
    graph = new ArrayList<>();
    for (int i = 0; i <= n; i++) {
      graph.add(new ArrayList<>());
    }
    for (int i = 0; i < m; i++) {
      int s = sts[i][0];
      int t = sts[i][1];
      graph.get(s).add(t);
    }
    
    double[] dpcache = new double[n + 1];
    
    List<Integer> removeSrc = new ArrayList<>();
    List<Integer> removeDst = new ArrayList<>();
    for (int i = n - 1; i >= 1; i--) {
      double sum = 0.0;
      double max = 0.0;
      int maxIndex = 0;
      int count = 0;
      for (int next : graph.get(i)) {
        double v = dpcache[next];
        sum += v;
        if (v > max) {
          max = v;
          maxIndex = next;
        }
        count++;
      }
      if (count > 1) {
        removeSrc.add(i);
        removeDst.add(maxIndex);
      }
      dpcache[i] = sum / count + 1.0;
    }
    
    double answer = dpcache[1];
    for (int removeIndex = 0; removeIndex < removeSrc.size(); removeIndex++) {
      double[] dpcache2 = new double[n + 1];
      for (int i = n - 1; i >= 1; i--) {
        double sum = 0.0;
        int count = 0;
        for (int next : graph.get(i)) {
          if (i == removeSrc.get(removeIndex) && next == removeDst.get(removeIndex)) {
            continue;
          }
          sum += dpcache2[next];
          count++;
        }
        dpcache2[i] = sum / count + 1.0;
      }
      answer = Math.min(answer, dpcache2[1]);
    }
    
    
    /*
    double[][] dpcache = new double[2][n + 1];
    Arrays.fill(dpcache[0], Double.MAX_VALUE);
    Arrays.fill(dpcache[1], Double.MAX_VALUE);
    dpcache[0][n] = 0.0;
    dpcache[1][n] = 0.0;
    for (int i = n-1; i >= 1; i--) {
      for (int j = 0; j < 2; j++) {
        double sum = 0.0;
        double max = 0.0;
        int count = 0;
        for (int next : graph.get(i)) {
          double v = dpcache[j][next];
          sum += v;
          max = Math.max(max, v);
          count++;
        }
        dpcache[j][i] = Math.min(dpcache[j][i], sum / count + 1.0);
        if (j == 0 && count > 1) {
          dpcache[1][i] = Math.min(dpcache[1][i], (sum - max) / (count - 1) + 1.0);
        }
      }
    }
    */
    return answer;
  }
}

class ContestReader {
  private BufferedReader reader;
  private StringTokenizer tokenizer;
  
  ContestReader(InputStream in) {
    reader = new BufferedReader(new InputStreamReader(in));
  }
  
  public String next() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
      try {
        tokenizer = new java.util.StringTokenizer(reader.readLine());
      } catch (Exception e) {
        throw new RuntimeException(e);
      }
    }
    return tokenizer.nextToken();
  }
  
  public int nextInt() {
    return Integer.parseInt(next());
  }
  
  public long nextLong() {
    return Long.parseLong(next());
  }
  
  public double nextDouble() {
    return Double.parseDouble(next());
  }
  
  public String[] nextArray(int n) {
    String[] array = new String[n];
    for (int i = 0; i < n; i++) {
      array[i] = next();
    }
    return array;
  }
  
  public int[] nextIntArray(int n) {
    int[] array = new int[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextInt();
    }
    return array;
  }
  
  public long[] nextLongArray(int n) {
    long[] array = new long[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextLong();
    }
    return array;
  }
  
  public double[] nextDoubleArray(int n) {
    double[] array = new double[n];
    for (int i = 0; i < n; i++) {
      array[i] = nextDouble();
    }
    return array;
  }
  
  public int[][] nextIntMatrix(int n, int m) {
    int[][] matrix = new int[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextInt();
      }
    }
    return matrix;
  }
  
  public long[][] nextLongMatrix(int n, int m) {
    long[][] matrix = new long[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextLong();
      }
    }
    return matrix;
  }
  
  public double[][] nextDoubleMatrix(int n, int m) {
    double[][] matrix = new double[n][m];
    for (int i = 0; i < n; i++) {
      for (int j = 0; j < m; j++) {
        matrix[i][j] = nextDouble();
      }
    }
    return matrix;
  }
}

class ModCalculator {
  private final long mod;
  private final ModCombinationCache modCombinationCache;
  
  ModCalculator(long mod) {
    this.mod = mod;
    this.modCombinationCache = new ModCombinationCache();
  }
  
  public long add(long a, long b) {
    return (a + b) % mod;
  }
  
  public long sub(long a, long b) {
    return (a - b + mod) % mod;
  }
  
  public long mul(long a, long b) {
    return (a * b) % mod;
  }
  
  public long pow(long a, long b) {
    if (b == 0) {
      return 1;
    }
    long v = pow(mul(a, a), b / 2);
    if (b % 2 == 1) {
      return mul(v, a);
    } else {
      return v;
    }
  }
  
  public long inverse(long a) {
    return pow(a, mod - 2);
  }
  
  public long div(long a, long b) {
    return mul(a, inverse(b));
  }
  
  public long getF(int n) {
    return modCombinationCache.getF(n);
  }
  
  public long getP(int n, int r) {
    return modCombinationCache.getP(n, r);
  }
  
  public long getC(int n, int k) {
    return modCombinationCache.getC(n, k);
  }
  
  class ModCombinationCache {
    private final List<Long> factorialCache;
    private final List<Long> factorialInverseCache;
    
    public ModCombinationCache() {
      factorialCache = new ArrayList<>();
      factorialCache.add(1L);
      factorialInverseCache = new ArrayList<>();
      factorialInverseCache.add(1L);
    }
    
    private void resize(int n) {
      for (int i = factorialCache.size() - 1; i < n; i++) {
        long v = mul(factorialCache.get(i), i + 1);
        factorialCache.add(v);
        factorialInverseCache.add(inverse(v));
      }
    }
    
    long getF(int n) {
      resize(n);
      return factorialCache.get(n);
    }
    
    long getP(int n, int r) {
      resize(n);
      return mul(factorialCache.get(n), factorialInverseCache.get(n - r));
    }
    
    long getC(int n, int k) {
      resize(n);
      return mul(factorialCache.get(n), mul(factorialInverseCache.get(k), factorialInverseCache.get(n-k)));
    }
  }
}
 
class Algorithm {
  private static void swap(Object[] list, int a, int b) {
    Object tmp = list[a];
    list[a] = list[b];
    list[b] = tmp;
  }
  
  public static <T extends Comparable<? super T>> boolean nextPermutation(T[] ts) {
    int rightMostAscendingOrderIndex = ts.length - 2;
    while (rightMostAscendingOrderIndex >= 0 &&
        ts[rightMostAscendingOrderIndex].compareTo(ts[rightMostAscendingOrderIndex + 1]) >= 0) {
      rightMostAscendingOrderIndex--;
    }
    if (rightMostAscendingOrderIndex < 0) {
      return false;
    }
    
    int rightMostGreatorIndex = ts.length - 1;
    while (ts[rightMostAscendingOrderIndex].compareTo(ts[rightMostGreatorIndex]) >= 0) {
      rightMostGreatorIndex--;
    }
    
    swap(ts, rightMostAscendingOrderIndex, rightMostGreatorIndex);
    for (int i = 0; i < (ts.length - rightMostAscendingOrderIndex - 1) / 2; i++) {
      swap(ts, rightMostAscendingOrderIndex + 1 + i, ts.length - 1 - i);
    }
    return true;
  }
  
  public static void shuffle(int[] array) {
    Random random = new Random();
    int n = array.length;
    for (int i = 0; i < n; i++) {
      int randomIndex = i + random.nextInt(n - i);
      
      int temp = array[i];
      array[i] = array[randomIndex];
      array[randomIndex] = temp;
    }
  }
  
  public static void shuffle(long[] array) {
    Random random = new Random();
    int n = array.length;
    for (int i = 0; i < n; i++) {
      int randomIndex = i + random.nextInt(n - i);
      
      long temp = array[i];
      array[i] = array[randomIndex];
      array[randomIndex] = temp;
    }
  }
  
  public static void sort(int[] array) {
    shuffle(array);
    Arrays.sort(array);
  }
  
  public static void sort(long[] array) {
    shuffle(array);
    Arrays.sort(array);
  }
}

class UnionFind {
  int[] parents;
  int[] ranks;

  UnionFind(int n) {
    parents = new int[n];
    ranks = new int[n];

    for (int i = 0; i < n; i++) {
      parents[i] = i;
    }
  }

  public int getRoot(int index) {
    if (parents[index] == index) {
      return index;
    } else {
      parents[index] = getRoot(parents[index]);
      return parents[index];
    }
  }

  public boolean sameGroup(int a, int b) {
    return getRoot(a) == getRoot(b);
  }

  public void merge(int a, int b) {
    int rootA = getRoot(a);
    int rootB = getRoot(b);
    if (rootA == rootB) {
      return;
    }
    if (ranks[rootA] < ranks[rootB]) {
      parents[rootA] = rootB;
    } else if (ranks[rootB] < ranks[rootA]) {
      parents[rootB] = rootA;
    } else {
      parents[rootA] = rootB;
      ranks[rootB]++;
    }
  }
}