#include "bits/stdc++.h"
using namespace std;
#define N 3
#define N2 9

struct Puzzle{
	int f[N2];
	int 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;
	}
};

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

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

Please formulate 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 "bits/stdc++.h"
using namespace std;
#define N 3
#define N2 9

struct Puzzle{
	int f[N2];
	int 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;
	}
};

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

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