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 Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import Data.Maybe
import Data.List
import Control.Monad
import qualified Data.ByteString.Char8 as BS
--import qualified Data.Vector.Unboxed as VU
{-
ave :: [Double] -> Maybe Double
ave [] = Nothing
ave x = Just $ (sum x) / (fromIntegral $ length x)

f :: [Int] -> [[Int]] -> Maybe [(Int,Double)]
f [] a = Just [(0,-1)]
f (x:xs) a = do 
    ys <- f xs a
    z <- ave $ map (\z -> snd $ fromJust $ find (\(n,e) -> n == (last z)) ys) $ filter (\z -> (head z) == x) a
    return $ (x,z+1):ys
    -- return $ (x,((ave $ map (\z -> snd $ fromJust $ find (\(n,e) -> n == (last z)) ys) $ filter (\z -> (head z) == x) a) + 1):ys)


g :: [a] -> [[a]]
g (x:xs) = xs:(map (\y -> x:y) (g xs))
g [] = [] 
-- [1,2,3,4] |-> [[2,3,4],[1,3,4],[1,2,4],[1,2,3]]

main :: IO ()
main = do
    [n,m] <- (map (fst . fromJust . BS.readInt)) . BS.words <$> BS.getLine
    a <- replicateM m ((map (fst . fromJust . BS.readInt)) . BS.words <$> BS.getLine)
    let b = map (\x -> [n,0]:x) (a:(g a))
    putStrLn $ show $ minimum (map (snd . head) $ catMaybes $ map (f [1..n]) b)
-}
main :: IO ()
main = do
    [n,m] <- (map (fst . fromJust . BS.readInt)) . BS.words <$> BS.getLine
    a <- replicateM m ((map (fst . fromJust . BS.readInt)) . BS.words <$> BS.getLine)
    let ((b,c,d):_) = f [1..n] ([n,0]:a)
    putStrLn $ show $ min c d

f :: [Int] -> [[Int]] -> [(Int,Double,Double)]
f [] a = [(0,-1,-1)]
f (x:xs) a = (x, (w / l) + 1, d + 1):ys where
    ys = f xs a
    zs = map (\(_:(b:_)) -> b ) $ filter (\z -> (head z) == x) a
    ws = filter (\(b,c,d) -> elem b zs) ys
    l = fromIntegral $ length ws
    w = sum (map (\(b,c,d) -> c) ws)
    b = (w + (minimum $ map (\(b,c,d) -> d - c) ws)) / l
    c = if l == 1 then -1 else (w - (maximum $ map (\(b,c,d) -> c) ws)) / (l - 1)
    d = if c == -1 then b else min b c