#include<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>
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 itarget(Puzzle x){
  for(int i=0;i<N2;i++)
    if(x.f[i]!=(i+1))return false;
    return true;
  }

  string bfs(Puzzle a){
    queue<Puzzle>Q;
    map<Puzzle, bool>V;
    Puzzle u, v;
    a.path="";
    Q.push(a);
    V[a]=true;

    while(!Q.empty()){
      u=Q.front(); Q.pop();
      if(itarget(u))return u.path;
      int xx=u.space/N;
      int xy=u.space%N;

      for(int k=0;k<4;k++){
	int yx=xx+dx[k];
	int yy=xy+dy[k];
	if(yx<0 || yy<0 || yx >=N || yy>=N)continue;
	v=u;
	swap(v.f[u.space],v.f[yx*N+yy]);
	v.space=yx*N+yy;
	if(!V[v])
	  {
	    V[v]=true;
	    v.path+=dir[k];
	    Q.push(v);
	  }
      }
    }
    return "unsolvable";
  }
  

int main(){
  Puzzle n;
  for(int i=0;i<N2;i++){
    cin>>n.f[i];
    if(n.f[i]==0){
      n.f[i]=N2;
      n.space=i;
    }
  }
  string ans=bfs(n);
  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<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>
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 itarget(Puzzle x){
  for(int i=0;i<N2;i++)
    if(x.f[i]!=(i+1))return false;
    return true;
  }

  string bfs(Puzzle a){
    queue<Puzzle>Q;
    map<Puzzle, bool>V;
    Puzzle u, v;
    a.path="";
    Q.push(a);
    V[a]=true;

    while(!Q.empty()){
      u=Q.front(); Q.pop();
      if(itarget(u))return u.path;
      int xx=u.space/N;
      int xy=u.space%N;

      for(int k=0;k<4;k++){
	int yx=xx+dx[k];
	int yy=xy+dy[k];
	if(yx<0 || yy<0 || yx >=N || yy>=N)continue;
	v=u;
	swap(v.f[u.space],v.f[yx*N+yy]);
	v.space=yx*N+yy;
	if(!V[v])
	  {
	    V[v]=true;
	    v.path+=dir[k];
	    Q.push(v);
	  }
      }
    }
    return "unsolvable";
  }
  

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


```