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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mma, mmi = math.max, math.min
local n, m = io.read("*n", "*n")
local edge = {}
local inf = n + 1
for i = 1, n do
  edge[i] = {}
end
for i = 1, m do
  local a, b = io.read("*n", "*n")
  table.insert(edge[a], b)
end

local len = {}
local function solve(z)
  for i = 1, n - 1 do
    len[i] = inf
  end
  len[n] = 0
  for idx = n - 1, 1, -1 do
    if #edge[idx] == 1 then
      len[idx] = 1 + len[edge[idx][1]]
    else
      local lensum, lenmax = 0, 0
      for i = 1, #edge[idx] do
        local v = len[edge[idx][i]]
        lensum = lensum + v
        lenmax = mma(lenmax, v)
      end
      if z == idx then
        len[idx] = 1 + (lensum - lenmax) / (#edge[idx] - 1)
      else
        len[idx] = 1 + lensum / #edge[idx]
      end
    end
  end
  return len[1]
end
local ret = solve(1)
for i = 2, n do
  ret = mmi(ret, solve(i))
end
print(string.format("%.9f", ret))