#include<bits/stdc++.h>
using namespace std;

int Y[4]={-1,0,1,0};
int X[4]={0,1,0,-1};

bool in(int a,int b){
  if(a<0 || b<0 || 3<=a || 3<=b)return false;
  return true;
}

int main()
{
  int a;
  string start="",goal="123456780";

  for(int i=0;i<9;i++){
    cin>>a;
    start+=(a+'0');
  }
  
  queue<string> q;
  map<string,int> mp;
  mp[start]=0;
  q.push(start);
  while(!q.empty()){
    string u=q.front();
    q.pop();
    int s=-1,cost=mp[u];
    if(u==goal){
      cout<<mp[u]<<endl;
      break;
    }

    for(int i=0;i<u.length();i++){
      if(u[i]=='0')s=i;
    }

    int y=s/3,x=s%3;

    for(int i=0;i<4;i++){
      int ny=y+Y[i],nx=x+X[i];
      if(!in(ny,nx))continue;
      int index=ny*3+nx;
      string t=u;
      swap(t[s],t[index]);
      if(mp.find(t)==mp.end()){
	mp[t]=cost+1;
	q.push(t);
      }
    }
  }
  return 0;
}
### Prompt

Your challenge is to write 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>
using namespace std;

int Y[4]={-1,0,1,0};
int X[4]={0,1,0,-1};

bool in(int a,int b){
  if(a<0 || b<0 || 3<=a || 3<=b)return false;
  return true;
}

int main()
{
  int a;
  string start="",goal="123456780";

  for(int i=0;i<9;i++){
    cin>>a;
    start+=(a+'0');
  }
  
  queue<string> q;
  map<string,int> mp;
  mp[start]=0;
  q.push(start);
  while(!q.empty()){
    string u=q.front();
    q.pop();
    int s=-1,cost=mp[u];
    if(u==goal){
      cout<<mp[u]<<endl;
      break;
    }

    for(int i=0;i<u.length();i++){
      if(u[i]=='0')s=i;
    }

    int y=s/3,x=s%3;

    for(int i=0;i<4;i++){
      int ny=y+Y[i],nx=x+X[i];
      if(!in(ny,nx))continue;
      int index=ny*3+nx;
      string t=u;
      swap(t[s],t[index]);
      if(mp.find(t)==mp.end()){
	mp[t]=cost+1;
	q.push(t);
      }
    }
  }
  return 0;
}
```