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

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;
	}
};

bool isTarget(Puzzle p) {
	for (int i = 0; i < N2; ++i) {
		if (p.f[i] != i + 1)
			return false;
	}
	return true;
}

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

string bfs(Puzzle p) {
	queue<Puzzle> Q;
	p.path = "";
	Q.push(p);
	map<Puzzle, bool> M;
	M[p] = true;
	Puzzle u, v;
	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 (!M[v]) {
				M[v] = true;
				v.path += dir[r];
				Q.push(v);
			}
		}
	}
	return "unresolved";
}

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

	return 0;
}
### Prompt

Please provide a CPP coded solution to the problem described below:

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

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;
	}
};

bool isTarget(Puzzle p) {
	for (int i = 0; i < N2; ++i) {
		if (p.f[i] != i + 1)
			return false;
	}
	return true;
}

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

string bfs(Puzzle p) {
	queue<Puzzle> Q;
	p.path = "";
	Q.push(p);
	map<Puzzle, bool> M;
	M[p] = true;
	Puzzle u, v;
	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 (!M[v]) {
				M[v] = true;
				v.path += dir[r];
				Q.push(v);
			}
		}
	}
	return "unresolved";
}

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

	return 0;
}
```