#include<iostream>
#include<cmath>
#include<string>
#include<queue>
#include<map>

using namespace std;
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'};

class puzzle{
public :
  int f[9],space;
  string path;

  bool operator < (const puzzle &p) const{
    for(int i=0;i<9;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<9;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/3;
    int sy=u.space%3;

    for(int r=0;r<4;r++){
      int tx=sx+dx[r];
      int ty=sy+dy[r];
      if(tx<0||ty<0||tx>=3||ty>=3)continue;
      v=u;
      swap(v.f[u.space],v.f[tx*3+ty]);
      v.space = tx*3+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<9;i++){
    cin >> in.f[i];
    if(in.f[i] ==0){
      in.f[i] = 9;
      in.space=i;
    }
  }
  string ans =bfs(in);
  cout << ans.size()<<endl;

  return 0;

}

### Prompt

Please create a solution in Cpp 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;
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'};

class puzzle{
public :
  int f[9],space;
  string path;

  bool operator < (const puzzle &p) const{
    for(int i=0;i<9;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<9;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/3;
    int sy=u.space%3;

    for(int r=0;r<4;r++){
      int tx=sx+dx[r];
      int ty=sy+dy[r];
      if(tx<0||ty<0||tx>=3||ty>=3)continue;
      v=u;
      swap(v.f[u.space],v.f[tx*3+ty]);
      v.space = tx*3+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<9;i++){
    cin >> in.f[i];
    if(in.f[i] ==0){
      in.f[i] = 9;
      in.space=i;
    }
  }
  string ans =bfs(in);
  cout << ans.size()<<endl;

  return 0;

}

```