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

using namespace std;
#define MAX 3
#define MAX2 9

struct Puzzle{
  int f[MAX2];
  int space;
  string path;
  
  bool operator < (const Puzzle &p) const{
    for(int i=0;i<MAX2;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<MAX2;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/MAX;
    int sy=u.space%MAX;
    for(int r=0;r<4;r++){
      int tx=sx+dx[r];
      int ty=sy+dy[r];
      if(tx<0 || ty<0 || tx>=MAX || ty>=MAX)continue;
      v=u;
      swap(v.f[u.space],v.f[tx*MAX+ty]);
      v.space=tx*MAX+ty;
      if(!V[v]){
	V[v]=true;
	v.path += dir[r];
	Q.push(v);
      }
    }
  }
  return "umsolvable";
}

int main() {
  Puzzle in;
  for(int i=0;i<MAX2;i++){
    cin >> in.f[i];
    if(in.f[i]==0){
      in.f[i]=MAX2;
      in.space=i;
    }
  }
  string ans=BFS(in);
  cout << ans.size() << endl;
  return 0;
}


### Prompt

Generate 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 MAX 3
#define MAX2 9

struct Puzzle{
  int f[MAX2];
  int space;
  string path;
  
  bool operator < (const Puzzle &p) const{
    for(int i=0;i<MAX2;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<MAX2;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/MAX;
    int sy=u.space%MAX;
    for(int r=0;r<4;r++){
      int tx=sx+dx[r];
      int ty=sy+dy[r];
      if(tx<0 || ty<0 || tx>=MAX || ty>=MAX)continue;
      v=u;
      swap(v.f[u.space],v.f[tx*MAX+ty]);
      v.space=tx*MAX+ty;
      if(!V[v]){
	V[v]=true;
	v.path += dir[r];
	Q.push(v);
      }
    }
  }
  return "umsolvable";
}

int main() {
  Puzzle in;
  for(int i=0;i<MAX2;i++){
    cin >> in.f[i];
    if(in.f[i]==0){
      in.f[i]=MAX2;
      in.space=i;
    }
  }
  string ans=BFS(in);
  cout << ans.size() << endl;
  return 0;
}


```