#include <bits/stdc++.h>
#define MAX 3
#define MAX2 9

using namespace std;

struct Puzzle{
  int f[MAX2];
  int space;
  string path;

  bool operator < (const Puzzle &p) const {
  int i;
    for(i = 0; i < MAX2; i++){
      if( f[i] == p.f[i] ){
        continue;
      }
      return f[i] > p.f[i];
    }
    return false;
  }
};

string bfs(Puzzle);
bool isTarget(Puzzle);

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

bool isTarget(Puzzle p){
  int i;
  for(i = 0; i < MAX2; i++){
    if( p.f[i] != (i+1) ){
      return false;
    }
  }
  return true;
}//end of isTarget func

int main(){
  Puzzle in;
  int i;
  for(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;
}//end of main func

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

  int r,tx,ty;
  int sx,sy;
  int tmp;

  while( !Q.empty() ){
    u = Q.front();
    Q.pop();
    if( isTarget(u) ){
      return u.path;
    }
    sx = u.space / MAX;
    sy = u.space % MAX;
    for(r = 0; r < 4; r++){
      tx = sx + dx[r];
      ty = sy + dy[r];
      if( tx < 0 || ty < 0 || tx >= MAX || ty >= MAX){
        continue;
      }
      v = u;
      /*swap*/
      tmp = v.f[u.space];
      v.f[u.space] = v.f[tx * MAX + ty];
      v.f[tx * MAX + ty] = tmp;

      v.space = tx * MAX + ty;
      if( !V[v] ){
        V[v] = true;
        v.path += dir[r];
        Q.push(v);
      }
    }
  }
  return "unsolvable";
}
### 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 <bits/stdc++.h>
#define MAX 3
#define MAX2 9

using namespace std;

struct Puzzle{
  int f[MAX2];
  int space;
  string path;

  bool operator < (const Puzzle &p) const {
  int i;
    for(i = 0; i < MAX2; i++){
      if( f[i] == p.f[i] ){
        continue;
      }
      return f[i] > p.f[i];
    }
    return false;
  }
};

string bfs(Puzzle);
bool isTarget(Puzzle);

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

bool isTarget(Puzzle p){
  int i;
  for(i = 0; i < MAX2; i++){
    if( p.f[i] != (i+1) ){
      return false;
    }
  }
  return true;
}//end of isTarget func

int main(){
  Puzzle in;
  int i;
  for(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;
}//end of main func

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

  int r,tx,ty;
  int sx,sy;
  int tmp;

  while( !Q.empty() ){
    u = Q.front();
    Q.pop();
    if( isTarget(u) ){
      return u.path;
    }
    sx = u.space / MAX;
    sy = u.space % MAX;
    for(r = 0; r < 4; r++){
      tx = sx + dx[r];
      ty = sy + dy[r];
      if( tx < 0 || ty < 0 || tx >= MAX || ty >= MAX){
        continue;
      }
      v = u;
      /*swap*/
      tmp = v.f[u.space];
      v.f[u.space] = v.f[tx * MAX + ty];
      v.f[tx * MAX + ty] = tmp;

      v.space = tx * MAX + ty;
      if( !V[v] ){
        V[v] = true;
        v.path += dir[r];
        Q.push(v);
      }
    }
  }
  return "unsolvable";
}
```