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 PYTHON3):

Solution:
import sys
from collections import defaultdict

sys.setrecursionlimit(10 ** 6)
input = sys.stdin.readline

def main():
    n, m = map(int, input().split())
    to = defaultdict(list)
    ot = defaultdict(list)
    for _ in range(m):
        s, t = map(int, input().split())
        s, t = s - 1, t - 1
        to[s].append(t)
        ot[t].append(s)
    # print(to)
    # 各頂点からゴールまでの期待値
    ev = [0] * n
    for u in range(n - 2, -1, -1):
        s = 0
        for ku in to[u]:
            s += ev[ku]
        ev[u] = s / len(to[u]) + 1
    # print(ev)
    # 各辺を通る確率
    pe = [[0] * n for _ in range(n)]
    # 各頂点を通る確率
    pn = [0] * n
    # 各頂点を通る確率を計算
    for u in range(n):
        p = 0
        for ou in ot[u]:
            p += pe[ou][u]
        if u == 0: p = 1
        pn[u] = p
        sz = len(to[u])
        for ku in to[u]:
            pe[u][ku] = p / sz
    # print(pn)
    # 変化量の最大値
    dx = 0
    for u in range(n - 1):
        sz = len(to[u])
        if sz == 1: continue
        mx = 0
        for ku in to[u]:
            mx = max(ev[ku], mx)
        nev = ((ev[u] - 1) * sz - mx) / (sz - 1) + 1
        de = pn[u] * (ev[u] - nev)
        dx = max(dx, de)
        # print(u, mx, de, nev)
    print(ev[0] - dx)

main()