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

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);
string bfs(Puzzle s);

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

  return 0;
}


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";
}

	


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

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);
string bfs(Puzzle s);

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

  return 0;
}


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";
}

	


```