Problem:
Score : 200 points

Problem Statement

#nck {
width: 30px;
height: auto;
}

We have a grid of H rows and W columns. Initially, there is a stone in the top left cell. Shik is trying to move the stone to the bottom right cell. In each step, he can move the stone one cell to its left, up, right, or down (if such cell exists). It is possible that the stone visits a cell multiple times (including the bottom right and the top left cell).

You are given a matrix of characters a_{ij} (1 \leq i \leq H, 1 \leq j \leq W). After Shik completes all moving actions, a_{ij} is # if the stone had ever located at the i-th row and the j-th column during the process of moving. Otherwise, a_{ij} is .. Please determine whether it is possible that Shik only uses right and down moves in all steps.

Constraints

2 \leq H, W \leq 8

a_{i,j} is either # or ..

There exists a valid sequence of moves for Shik to generate the map a.

Input

The input is given from Standard Input in the following format:

H W
a_{11}a_{12}...a_{1W}
:
a_{H1}a_{H2}...a_{HW}

Output

If it is possible that Shik only uses right and down moves, print Possible. Otherwise, print Impossible.

Sample Input 1

4 5
##...
.##..
..##.
...##

Sample Output 1

Possible

The matrix can be generated by a 7-move sequence: right, down, right, down, right, down, and right.

Sample Input 2

5 3
###
..#
###
#..
###

Sample Output 2

Impossible

Sample Input 3

4 5
##...
.###.
.###.
...##

Sample Output 3

Impossible

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

import Data.Array
type Pos = (Int,Int)
type Field  = Array Pos Bool
type Size  = [Int]
type Range  = (Pos,Pos)
readsize :: String -> Range
readsize = (\[h,w] -> fieldrange h w ) . map read . words . head . lines
readfield :: Range -> String -> Field
readfield range  = listArray range . map onPath . concat . tail . lines
fieldrange h w = ((0,0),(h-1, w-1)) 
onPath :: Char -> Bool
onPath x
    | x == '#' = True
    | x == '.' = False
main = do
    str <- getContents
    let frange = readsize str
        field = readfield frange str
    putStrLn $ getAns $ and $ map (hasCorrectPath field) $ path field

right (x,y) = (x,y+1)
left  (x,y) = (x,y-1)
down  (x,y) = (x+1,y)
up    (x,y) = (x-1,y)
has :: (Pos->Pos) -> Field -> Pos -> Bool
has f field pos = field `el`  f  pos
hasCorrectPath :: Field -> Pos -> Bool
hasCorrectPath field pos = (has up field pos `nand` has left field pos) && (has down field pos `nand` has right field pos)
path = map (\(i,x)->i) . filter (\(i,x) -> x) . assocs

getAns True = "Possible"
getAns False = "Impossible"

nand a b = not $ a && b

el :: Field -> Pos -> Bool
el a i
    | inRange (bounds a) i  = a ! i
    | otherwise             = False