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:
def solve():
    import sys
    input = sys.stdin.readline

    N, M = map(int, input().split())
    adjL = [[] for _ in range(N)]
    adjLRev = [[] for _ in range(N)]
    outdegs = [0] * N
    for _ in range(M):
        s, t = map(int, input().split())
        s, t = s-1, t-1
        adjL[s].append(t)
        adjLRev[t].append(s)
        outdegs[s] += 1

    probs = [0] * N
    expes = [0] * N
    probs[0] = 1
    for v in range(N):
        prob, expe, outdeg = probs[v], expes[v], outdegs[v]
        if prob != 0:
            num = expe / prob
            for v2 in adjL[v]:
                probs[v2] += prob / outdeg
                expes[v2] += prob * (num+1) / outdeg

    probRevs = [0] * N
    expeRevs = [0] * N
    probRevs[-1] = 1
    for v in reversed(range(N)):
        prob, expe = probRevs[v], expeRevs[v]
        if prob != 0:
            num = expe / prob
            for v0 in adjLRev[v]:
                probRevs[v0] += prob / outdegs[v0]
                expeRevs[v0] += prob * (num+1) / outdegs[v0]

    ans = expes[-1]
    for vRem in range(N-1):
        if outdegs[vRem] == 1: continue
        values = []
        prob, expe, outdeg = probs[vRem], expes[vRem], outdegs[vRem]
        for v2 in adjL[vRem]:
            value = expe*probRevs[v2] + expeRevs[v2]*prob + prob*probRevs[v2]
            values.append(value)
        sumV = sum(values)
        ans2 = expes[-1] - sumV / outdeg
        for value in values:
            ans = min(ans, ans2 + (sumV-value) / (outdeg-1))

    print(ans)


solve()