#include<bits/stdc++.h>
#include<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>
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){
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 "";
}
  
int main(){
Puzzle in;
for(int i = 0; i < N2; i++){
cin >> in.f[i];
if(in.f[i] == 0){
in.f[i] = N2;
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<bits/stdc++.h>
#include<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>
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){
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 "";
}
  
int main(){
Puzzle in;
for(int i = 0; i < N2; i++){
cin >> in.f[i];
if(in.f[i] == 0){
in.f[i] = N2;
in.space = i;
}
}
string ans = bfs(in);
cout << ans.size() << endl;
  
return 0;
}

```