#include <iostream>
#include <queue> 
#include <map>

using namespace std;

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'};
 
typedef struct Puzzle {
	int M[9];
	int space;
	string path;
	bool operator<(const Puzzle & p) const {
		for (int i=0;i<9;i++) {
			if (M[i] == p.M[i]) continue;
			return M[i] > p.M[i];
		}
		return false;
	}
} Puzzle;

bool check(Puzzle & p) {
	for (int i=0;i<8;i++) {
		if (p.M[i] != i+1) return false;
	}
	return true;
}

string bfs(Puzzle & p) { //广度优先搜索 
	queue<Puzzle> Q;
	map<Puzzle,bool> V; //标记该状态是否被访问
	Puzzle u,v;
	p.path = "";
	Q.push(p);
	V[p] = true;
	while (!Q.empty()) {
		u = Q.front();
		Q.pop();
		if (check(u)) return u.path;
		
		int sx = u.space / 3;
		int sy = u.space % 3;
		
		for (int r = 0;r<4;r++) {
			int tx = sx + dx[r];
			int ty = sy + dy[r];
			if (tx < 0 || ty < 0 || tx >= 3 || ty >= 3) continue;
			
			v = u;
			swap(v.M[u.space],v.M[tx * 3 + ty]);
			v.space = tx*3 + 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<9;i++) {
		cin >> in.M[i];
		if (in.M[i] == 0) {
			in.space = i;
		}
	}
	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 <queue> 
#include <map>

using namespace std;

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'};
 
typedef struct Puzzle {
	int M[9];
	int space;
	string path;
	bool operator<(const Puzzle & p) const {
		for (int i=0;i<9;i++) {
			if (M[i] == p.M[i]) continue;
			return M[i] > p.M[i];
		}
		return false;
	}
} Puzzle;

bool check(Puzzle & p) {
	for (int i=0;i<8;i++) {
		if (p.M[i] != i+1) return false;
	}
	return true;
}

string bfs(Puzzle & p) { //广度优先搜索 
	queue<Puzzle> Q;
	map<Puzzle,bool> V; //标记该状态是否被访问
	Puzzle u,v;
	p.path = "";
	Q.push(p);
	V[p] = true;
	while (!Q.empty()) {
		u = Q.front();
		Q.pop();
		if (check(u)) return u.path;
		
		int sx = u.space / 3;
		int sy = u.space % 3;
		
		for (int r = 0;r<4;r++) {
			int tx = sx + dx[r];
			int ty = sy + dy[r];
			if (tx < 0 || ty < 0 || tx >= 3 || ty >= 3) continue;
			
			v = u;
			swap(v.M[u.space],v.M[tx * 3 + ty]);
			v.space = tx*3 + 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<9;i++) {
		cin >> in.M[i];
		if (in.M[i] == 0) {
			in.space = i;
		}
	}
	string ans = bfs(in);
	cout << ans.size() << endl;
	return 0;
}

```