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

using namespace std;

#define N 3
#define M 9

struct Puzz{
  int f[M];
  int space;
  string path;

  bool operator < (const Puzz &p) const {
    for(int i=0;i<M;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 dirc[4] = {'u','l','d','t'};

bool Tage(Puzz p){
  for(int i=0;i<M;i++)
  if(p.f[i] != (i+1)) return false;
  return true;
}

string afs(Puzz m){
  queue<Puzz> Q;
  map<Puzz, bool> V;
  Puzz u,v;
  m.path="";
  Q.push(m);
  V[m]=true;


  while(!Q.empty()){
    u=Q.front(); Q.pop();

    if(Tage(u)) return u.path;
    int mx=u.space/N;
    int my=u.space%N;

    for(int i=0;i<4;i++){
      int px=mx+dx[i];
      int py=my+dy[i];
      if(px<0 || py<0 || px>=N || py>=N) continue;
      v=u;
      swap(v.f[u.space],v.f[px*N+py]);
      v.space=px*N+py;

      if(!V[v]){
        V[v]=true;
        v.path+=dirc[i];
        Q.push(v);
      }
    }
  }
  return "unsolvable";
}

int main(){
  Puzz ln;

  for(int i=0;i<M;i++){
    cin >> ln.f[i];
    if(ln.f[i] ==0){
      ln.f[i]=M;
      ln.space =i;
    }
  }
  string answer = afs(ln);
  cout << answer.size() << endl;

  return 0;
}


### Prompt

In Cpp, your task is to solve 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 N 3
#define M 9

struct Puzz{
  int f[M];
  int space;
  string path;

  bool operator < (const Puzz &p) const {
    for(int i=0;i<M;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 dirc[4] = {'u','l','d','t'};

bool Tage(Puzz p){
  for(int i=0;i<M;i++)
  if(p.f[i] != (i+1)) return false;
  return true;
}

string afs(Puzz m){
  queue<Puzz> Q;
  map<Puzz, bool> V;
  Puzz u,v;
  m.path="";
  Q.push(m);
  V[m]=true;


  while(!Q.empty()){
    u=Q.front(); Q.pop();

    if(Tage(u)) return u.path;
    int mx=u.space/N;
    int my=u.space%N;

    for(int i=0;i<4;i++){
      int px=mx+dx[i];
      int py=my+dy[i];
      if(px<0 || py<0 || px>=N || py>=N) continue;
      v=u;
      swap(v.f[u.space],v.f[px*N+py]);
      v.space=px*N+py;

      if(!V[v]){
        V[v]=true;
        v.path+=dirc[i];
        Q.push(v);
      }
    }
  }
  return "unsolvable";
}

int main(){
  Puzz ln;

  for(int i=0;i<M;i++){
    cin >> ln.f[i];
    if(ln.f[i] ==0){
      ln.f[i]=M;
      ln.space =i;
    }
  }
  string answer = afs(ln);
  cout << answer.size() << endl;

  return 0;
}


```