#include <iostream>
#include <cmath>
#include <string>
#include <queue>
#include <map>
using namespace std;

#define N 3
#define N2 9

//?§???¨??????
static const int dx[4]={0,-1,0,1};
static const int dy[4]={1,0,-1,0};
static const char dir[4]={'u','l','d','r'};

//Puzzle{????????????????????????}
struct Puzzle
{ 
	int f[N2], space;
	string path;
	
	bool operator < (const Puzzle &p) const
	{
		for(int i=0;i<N2;i++)
		{
			if(f[i]==p.f[i])continue;
			return f[i]<p.f[i];
		}
		return false;
	}
};

bool isTarget(Puzzle p)
{
	for(int i=0;i<N2;i++)
		if(p.f[i]!=(i+1))return false;
	return true;
}

string bfs(Puzzle s)
{
	queue<Puzzle>Q;
	map<Puzzle,bool> V;
	Puzzle u,v;
	s.path="";
	Q.push(s);
	V[s]=true;
	
	while(!Q.empty())
	{
		u=Q.front();Q.pop();
		if(isTarget(u))return u.path;
		int sx=u.space/N;
		int sy=u.space%N;
		for(int r=0;r<4;r++)
		{
			int tx=sx+dx[r];
			int ty=sy+dy[r];
			if(tx<0||ty<0||tx>=N||ty>=N)continue;
			v=u;
			swap(v.f[u.space],v.f[tx*N+ty]);
			v.space=tx*N+ty;
			if(!V[v])
			{
				V[v]=true;
				v.path+=dir[r];
				Q.push(v);
			}
		}
	}
	return "unsolveable";
}

int main()
{
	Puzzle in;
	
	for(int i=0;i<N2;i++)
	{
		cin>>in.f[i];
		if(in.f[i]==0)
		{
			in.f[i]=N2;
			in.space=i;
		}
	}
	string ans=bfs(in);
	cout<<ans.size()<<endl;
	//cout<<ans;
}
### Prompt

Your task is to create a CPP solution to the following problem:

The goal of the 8 puzzle problem is to complete pieces on $3 \times 3$ cells where one of the cells is empty space.

In this problem, the space is represented by 0 and pieces are represented by integers from 1 to 8 as shown below.


1 3 0
4 2 5
7 8 6


You can move a piece toward the empty space at one step. Your goal is to make the pieces the following configuration in the shortest move (fewest steps).


1 2 3
4 5 6
7 8 0


Write a program which reads an initial state of the puzzle and prints the fewest steps to solve the puzzle.

Constraints

* There is a solution.

Input

The $3 \times 3$ integers denoting the pieces or space are given.

Output

Print the fewest steps in a line.

Example

Input

1 3 0
4 2 5
7 8 6


Output

4

### Response

```cpp
#include <iostream>
#include <cmath>
#include <string>
#include <queue>
#include <map>
using namespace std;

#define N 3
#define N2 9

//?§???¨??????
static const int dx[4]={0,-1,0,1};
static const int dy[4]={1,0,-1,0};
static const char dir[4]={'u','l','d','r'};

//Puzzle{????????????????????????}
struct Puzzle
{ 
	int f[N2], space;
	string path;
	
	bool operator < (const Puzzle &p) const
	{
		for(int i=0;i<N2;i++)
		{
			if(f[i]==p.f[i])continue;
			return f[i]<p.f[i];
		}
		return false;
	}
};

bool isTarget(Puzzle p)
{
	for(int i=0;i<N2;i++)
		if(p.f[i]!=(i+1))return false;
	return true;
}

string bfs(Puzzle s)
{
	queue<Puzzle>Q;
	map<Puzzle,bool> V;
	Puzzle u,v;
	s.path="";
	Q.push(s);
	V[s]=true;
	
	while(!Q.empty())
	{
		u=Q.front();Q.pop();
		if(isTarget(u))return u.path;
		int sx=u.space/N;
		int sy=u.space%N;
		for(int r=0;r<4;r++)
		{
			int tx=sx+dx[r];
			int ty=sy+dy[r];
			if(tx<0||ty<0||tx>=N||ty>=N)continue;
			v=u;
			swap(v.f[u.space],v.f[tx*N+ty]);
			v.space=tx*N+ty;
			if(!V[v])
			{
				V[v]=true;
				v.path+=dir[r];
				Q.push(v);
			}
		}
	}
	return "unsolveable";
}

int main()
{
	Puzzle in;
	
	for(int i=0;i<N2;i++)
	{
		cin>>in.f[i];
		if(in.f[i]==0)
		{
			in.f[i]=N2;
			in.space=i;
		}
	}
	string ans=bfs(in);
	cout<<ans.size()<<endl;
	//cout<<ans;
}
```