Problem D
Hopscotch 50

There’s a new art installation in town, and it inspires you…to play a childish game. The art installation consists of a floor with an $n\! \times \! n$ matrix of square tiles. Each tile holds a single number from $1$ to $k$. You want to play hopscotch on it. You want to start on some tile numbered $1$, then hop to some tile numbered $2$, then $3$, and so on, until you reach some tile numbered $k$. You are a good hopper, so you can hop any required distance. You visit exactly one tile of each number from $1$ to $k$.

What’s the shortest possible total distance over a complete game of Hopscotch? Use the Manhattan distance: the distance between the tile at $(x_1,y_1)$ and the tile at $(x_2,y_2)$ is $|x_1-x_2| + |y_1-y_2|$.

Input

The first line of input contains two space-separated integers $n$ ($1 \le n \le 50$) and $k$ ($1 \le k \le n^2$), where the art installation consists of an $n\! \times \! n$ matrix with tiles having numbers from $1$ to $k$.

Each of the next $n$ lines contains $n$ space-separated integers $x$ ($1 \le x \le k$). This is the art installation.

Output

Output a single integer, which is the total length of the shortest path starting from some $1$ tile and ending at some $k$ tile, or $-1$ if it isn’t possible.

10 5
5 1 3 4 2 4 2 1 2 1
4 5 3 4 1 5 3 1 1 4
4 2 4 1 5 4 5 2 4 1
5 2 1 5 5 3 5 2 3 2
5 5 2 3 2 3 1 5 5 5
3 4 2 4 2 2 4 4 2 3
1 5 1 1 2 5 4 1 5 3
2 2 4 1 2 5 1 4 3 5
5 3 2 1 4 3 5 2 3 1
3 4 2 5 2 5 3 4 4 2

5

10 5
5 1 5 4 1 2 2 4 5 2
4 2 1 4 1 1 1 5 2 5
2 2 4 4 4 2 4 5 5 4
2 4 4 5 5 5 2 5 5 2
2 2 4 4 4 5 4 2 4 4
5 2 5 5 4 1 2 4 4 4
4 2 1 2 4 4 1 2 4 5
1 2 1 1 2 4 4 1 4 5
2 1 2 5 5 4 5 2 1 1
1 1 2 4 5 5 5 5 5 5

-1