#include <iostream>
#include <algorithm>
#include <queue>
#include <map>
#define N  3
#define N2 9
using namespace std;
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 "unsolvable";
}

int main()
{
	Puzzle in;
	for (int i = 0; i < N2; ++i)
	{
		cin>>in.f[i];
		if (in.f[i]==0) {
			in.space = i;
		        in.f[i] = 9;
		}
	}
	string ans = bfs(in);
	cout<<ans.size()<<endl;
	return 0;
}
### Prompt

Construct a Cpp code solution to the problem outlined:

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 <algorithm>
#include <queue>
#include <map>
#define N  3
#define N2 9
using namespace std;
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 "unsolvable";
}

int main()
{
	Puzzle in;
	for (int i = 0; i < N2; ++i)
	{
		cin>>in.f[i];
		if (in.f[i]==0) {
			in.space = i;
		        in.f[i] = 9;
		}
	}
	string ans = bfs(in);
	cout<<ans.size()<<endl;
	return 0;
}
```