**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Uninteresting Number
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a number $$$n$$$ with a length of no more than $$$10^5$$$.

You can perform the following operation any number of times: choose one of its digits, square it, and replace the original digit with the result. The result must be a digit (that is, if you choose the digit $$$x$$$, then the value of $$$x^2$$$ must be less than $$$10$$$).

Is it possible to obtain a number that is divisible by $$$9$$$ through these operations?

Input: The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

The only line of each test case contains the number $$$n$$$, without leading zeros. The length of the number does not exceed $$$10^5$$$.

It is guaranteed that the sum of the lengths of the numbers across all test cases does not exceed $$$10^5$$$.

Output: For each test case, output "YES" if it is possible to obtain a number divisible by $$$9$$$ using the described operations, and "NO" otherwise.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

Examples:
input:
9
123
322
333333333333
9997
5472778912773
1234567890
23
33
52254522632
output:
NO
YES
YES
NO
NO
YES
NO
YES
YES

Note: In the first example, from the integer $$$123$$$, it is possible to obtain only $$$123$$$, $$$143$$$, $$$129$$$, and $$$149$$$, none of which are divisible by $$$9$$$.

In the second example, you need to replace the second digit with its square; then $$$n$$$ will equal $$$342 = 38 \cdot 9$$$.

In the third example, the integer is already divisible by $$$9$$$.

Buggy Code:
import java.util.*;
import java.io.*;
import java.util.StringTokenizer;

public class uinintrestingnumber {
	static PrintWriter out;
	static int t;
	static BufferedReader in;
	static StringTokenizer st;

	public static void main(String[] args) throws IOException {

	in = new BufferedReader(new InputStreamReader(System.in));
	out  = new PrintWriter(System.out);

    //in = new BufferedReader(new FileReader("haybales.in"));
	//out  = new PrintWriter("haybales.out");

	st = new StringTokenizer(in.readLine());
    int t = Integer.parseInt(st.nextToken());

	for (int i = 0 ; i < t ; i ++)
		{
			solve();
		}

		in.close();
		out.close();
	}

	static void solve() throws IOException // hashmap way faster than treemap
	{

        st = new StringTokenizer(in.readLine());
        String n = st.nextToken();

        long start = 0;

        int three = 0;
        int two = 0;
        long digitSum = 0;
        long sum = 0;
        long diff = 0;
        long threeSum = 0;
        long twoSum = 0;
        long threeSumN = 0;

        String number = new String("");

        for(int i = 0; i < n.length() ; i ++)
        {
            number = n.substring(i,i+1);
            digitSum += Integer.parseInt(number);

            if(Integer.parseInt(number) == 3)
                three++;
            else if(Integer.parseInt(number) == 2)
                two++;
        }

      //  System.out.println(three + " "+ two);

        if(digitSum%9 == 0)
        {
            System.out.println("Yes");
        }
        else if(digitSum%2 == 0) // even case
        {
           // System.out.println(n);
            // find next biggest even number bigger than the sum of its digits

            start = (digitSum+9-1) / 9;

            long nextEven = start * 9;

            while (nextEven%2 != 0) // find next larger odd # thats divisible by 9
            {
                start++;
                nextEven = start * 9;
            }

          //  System.out.println(nextEven);

            diff = nextEven - digitSum;

            sum = three*6 + two *2;

            if(sum >= diff)
            {
                threeSum = (diff/6)*6;
                twoSum = diff-threeSum;

                threeSumN = (threeSum/6) - three;

             //   System.out.println(threeSumN);

                if(threeSumN > 0 && twoSum/2 >= (two-(3*threeSumN)))
                {
                    three += threeSumN;
                    two -= 3*threeSumN;
                }

                if((threeSum/6) <= three && (twoSum/2) <= two)
                {
                    System.out.println("Yes");
                }
                else
                    System.out.println("No");
            }
            else
            {
                System.out.println("No");
            }
        }
        else
        {
         //    System.out.println(n);
            // find next biggest odd number bigger than the sum of its digits

            start = (digitSum+9-1) / 9;

            long nextOdd = start * 9;
         //   start = (digitSum+9-1 / 9);

            while (nextOdd%2 == 0) // find next larger odd # thats divisible by 9
            {
                start++;
                nextOdd = start * 9;
            }

           // System.out.println(nextOdd);

            diff = nextOdd - digitSum;

            sum = three*6 + two *2;

          if(sum >= diff)
            {
                threeSum = (diff/6)*6;
                twoSum = diff-threeSum;

                threeSumN = (threeSum/6) - three;

              //  System.out.println(threeSumN);

                if(threeSumN > 0 && twoSum/2 >= (two-(3*threeSumN)))
                {
                    three += threeSumN;
                    two -= 3*threeSumN;
                }

                if((threeSum/6) <= three && (twoSum/2) <= two)
                {
                    System.out.println("Yes");
                }
                else
                    System.out.println("No");
            }
            else
            {
                System.out.println("No");
            }

        }

    }

    public static ArrayList<Integer> psum(ArrayList<Integer> frequencies) {
		ArrayList<Integer> prefixSums = new ArrayList<>();
        prefixSums.add(0);

        int sum = 0;

		 for(int i = 0; i < frequencies.size(); i ++) {
            sum += frequencies.get(i);
            prefixSums.add(sum);
        }

        return prefixSums;
	}

     static int largestSum (ArrayList<Integer> prefixSum, int k, int n)
    {
        int lo = -1, hi = prefixSum.size() , mid = (lo + hi) / 2;
        while (lo < mid && mid < hi)
        {
            if (prefixSum.get(mid) <= k && mid < n ) lo = mid;
            else hi = mid;
            mid = (lo + hi) / 2;
        }
            return hi;
    }

}

class Edge {
	public int l, r;

	public Edge(int l, int r) {
		this.l = l;
		this.r = r;
	}

    public Integer getL() { return l; }
    public Integer getR() { return r; }
}

class Computer {

    // Attributes
    int price;
    String type;

    // Constructor
    public Computer(int price, String type)
    {
        // This keyword refers to current instance itself
        this.price = price;
        this.type = type;
    }

    // Method of Student class
    // To print student details in main()
    public String toString()
    {

        // Returning attributes of Student
      return this.price + " " + this.type;
    }

	public Integer getPrice() { return price; }

    public String getType() { return type; }

  //  public Character getOperator() { return operator; }

//	public Integer getChange() { return change; }

}

// Class 2
// Helper class implementing Comparator interface
class Sortbynum implements Comparator<Computer> {

    // Method
    // Sorting in ascending order of roll number
    public int compare(Computer a, Computer b)
    {
        return a.price - b.price;
    }
}

// Class 3
// Helper class implementing Comparator interface
/*
class Sortbyname implements Comparator<Student> {

    // Method
    // Sorting in ascending order of name
    public int compare(Student a, Student b)
    {

        return a.name.compareTo(b.name);
    }
}
*/

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the provided code, we need to consider edge cases that might cause the algorithm to fail. The key aspects to consider are:

1. **Numbers with digits that can be squared to form valid digits (0-9)**: Digits 0, 1, 2, 3.
2. **Numbers that are already divisible by 9**.
3. **Numbers that cannot be made divisible by 9 through the operations**.
4. **Numbers with a mix of digits that can and cannot be squared**.

The code seems to have issues with correctly handling the conditions for making a number divisible by 9. Specifically, it might fail when the sum of digits is close to a multiple of 9 but requires specific digit manipulations to reach that multiple.

Let's consider a small-scale test input that can expose these issues:

### Test Input
```plaintext
5
123
322
333
222
111
```

### Explanation
1. **123**: This number cannot be made divisible by 9 through the operations.
2. **322**: This number can be made divisible by 9 by changing the second digit (2) to 4, making the number 342.
3. **333**: This number is already divisible by 9.
4. **222**: This number cannot be made divisible by 9 through the operations.
5. **111**: This number cannot be made divisible by 9 through the operations.

This test input includes a mix of numbers that are already divisible by 9, numbers that can be made divisible by 9, and numbers that cannot be made divisible by 9. This should help in exposing any logical errors in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Uninteresting Number
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a number $$$n$$$ with a length of no more than $$$10^5$$$.

You can perform the following operation any number of times: choose one of its digits, square it, and replace the original digit with the result. The result must be a digit (that is, if you choose the digit $$$x$$$, then the value of $$$x^2$$$ must be less than $$$10$$$).

Is it possible to obtain a number that is divisible by $$$9$$$ through these operations?

Input: The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

The only line of each test case contains the number $$$n$$$, without leading zeros. The length of the number does not exceed $$$10^5$$$.

It is guaranteed that the sum of the lengths of the numbers across all test cases does not exceed $$$10^5$$$.

Output: For each test case, output "YES" if it is possible to obtain a number divisible by $$$9$$$ using the described operations, and "NO" otherwise.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

Examples:
input:
9
123
322
333333333333
9997
5472778912773
1234567890
23
33
52254522632
output:
NO
YES
YES
NO
NO
YES
NO
YES
YES

Note: In the first example, from the integer $$$123$$$, it is possible to obtain only $$$123$$$, $$$143$$$, $$$129$$$, and $$$149$$$, none of which are divisible by $$$9$$$.

In the second example, you need to replace the second digit with its square; then $$$n$$$ will equal $$$342 = 38 \cdot 9$$$.

In the third example, the integer is already divisible by $$$9$$$.

Buggy Code:
import java.io.*;
import java.util.*;

public class UninterestingNumber {
    public static void main(String[] args) throws IOException {
        BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
        StringTokenizer tokenizer = new StringTokenizer(in.readLine());

        int t = Integer.parseInt(tokenizer.nextToken());
        ArrayList<String> ans = new ArrayList<>();

        loop:
        for (int i = 0; i < t; i++) {
            tokenizer = new StringTokenizer(in.readLine());
            String n = tokenizer.nextToken();
            int num = simplify(n);

            if (num == 0) {
                ans.add("YES");
                continue;
            }

            int num2 = numTwos(n);
            int num3 = numThrees(n);

            int a = 0;
            int b = 0;

            if (num == 6 && num3 >= 2) {
                ans.add("YES");
                continue;
            }

            while (a < num2 || b < num3) {
                if (a < num2) {
                    num += 2;
                    a++;
                    if (num % 9 == 0) {
                        ans.add("YES");
                        continue loop;
                    }
                    a--;
                    num -= 2;
                }
                if (b < num3) {
                    num += 6;
                    b++;
                    if (num % 9 == 0) {
                        ans.add("YES");
                        continue loop;
                    }
                    b--;
                    num -= 6;
                }
                if (a < num2) {
                    a++;
                    num += 2;
                } else if (b < num3) {
                    b++;
                    num += 6;
                }
            }

            ans.add("NO");
        }

        for (String s : ans) {
            System.out.println(s);
        }
    }

    public static int numTwos(String x) {
        int ret = 0;
        for (char a : x.toCharArray()) {
            if ((a - '0') == 2) ret++;
        }
        return ret;
    }

    public static int numThrees(String x) {
        int ret = 0;
        for (char a : x.toCharArray()) {
            if ((a - '0') == 3) ret++;
        }
        return ret;
    }

    public static int simplify(String x) {
        int ret = 0;
        for (char a : x.toCharArray()) {
            ret += (a - '0');
        }
        return ret % 9;
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
222
333
2333
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
#include <iostream>
using namespace std;
inline void pass(){return;}
int t,n,m,a[15][15];

void solve(){
    cin >> n >> m;
    for(int i=1;i<=n;i++){
        for(int j=1;j<=m;j++) cin>>a[i][j];
    }
    if(n==1&&m==1) cout<<"-1"<<endl;
    else if(n==1){
        cout << a[1][m];
        for(int i=1;i<m;i++) cout<<" "<<a[1][i];
        cout << endl;
    }
    else if(m==1){
        cout << a[n][1];
        for(int i=1;i<n;i++) cout<<endl<<a[1][i];
        cout << endl;
    }
    else{
        for(int i=1;i<=n;i++){
            cout << a[i][m];
            for(int j=1;j<m;j++) cout<<" "<<a[i][j];
            cout << endl;
        }
    }
}

int main(){
    cin >> t;
    for(int ran=0;ran<t;ran++) solve();
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
2 2
1 2
3 4
3 3
1 2 3
4 5 6
7 8 9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
#include <iostream>
#include <vector>
#include <algorithm>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <list>
#include <string>
#include <limits>
#include<cmath>
#include <numeric>
using namespace std;
typedef long long ll;
typedef vector<long long> vll;
typedef vector<double> vd;
typedef vector<pair<ll,ll> > vpll;
typedef vector<vector<ll> > vvll;
#define fo(i,n) for(long long i = 0; i < n; i++)
#define foa(i,k,n) for(long long i = k; i < n; i++)
#define fob(i,k,n) for(long long i = k; i >= n; i--)
#define sortuniq(v) {sort(v.begin(), v.end()); v.erase(unique(v.begin(), v.end()), v.end());}
#define sz(x) int((x).size())
#define pb push_back;
#define F first;
#define S second;
#define uniq(v) {v.erase(unique(v.begin(), v.end()), v.end());}
#define mod 1000000007
int primec(int n)
{
for (int i = 2; i*i <= n; i++)
 if (n % i == 0)
return 0;
if(n==1){return 0;}
return 1;
}
ll bin_expo(ll n,ll m,ll p ){
if(p==0)return 1;
if(p==1)return n%m;
if(p%2==0){ll a=bin_expo(n,m,p/2);
return ((a%m)*(a%m))%m;
}
else {
    ll a=bin_expo(n,m,p-1);
    return ((n%m)*(a%m))%m;
}
}
ll ncr(ll n,ll r,ll m){ /* m ahould be prime */
    if(r>n)return 0;
    if(n-r<r)r=n-r;
    ll num=1;
    ll den=1;
    for(ll i=n-r+1;i<=n;i++){
    num*=i;
    num%=m;
    }
    for(ll i=1;i<=r;i++){
    den*=i;
    den%=m;
    }
    den=bin_expo(den,m,m-2);
    return (num*den)%m;
}
void solve(){
    ll n,m;cin>>n>>m; vvll a(n,vll(m));vvll b(n,vll(m));
    fo(i,n){fo(j,m){cin>>a[i][j];}}
    if(n==1 && m==1){cout<<-1<<endl;}
    else{
        reverse(a.begin(),a.end());
        if(n%2==1){
            reverse(a[n/2].begin(),a[n/2].end());
            if(m%2==1){
                ll t=a[n/2][m-1];
                a[n/2][m-1]=a[n/2][m/2];
                a[n/2][m/2]=t;
            }
        }
        fo(i,n){fo(j,m){cout<<a[i][j]<<' ';}cout<<endl;}
    }
}
int main(){
    ios_base::sync_with_stdio(0); cin.tie(0);
    int tc;
    cin>>tc;
    while(tc--){solve();}
    return 0;}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To create a test input that exposes the bug in the given code, we need to focus on edge cases and scenarios that are likely to cause the code to fail. The code provided has a specific approach to generating the matrix `b` by reversing the rows of matrix `a` and then performing some additional operations to ensure no elements are in the same position. Here are the steps to think through:

1. **Single Element Matrix**:
   - This is already handled correctly in the code with a special case. No need to test this.

2. **Matrix with Odd Rows and Odd Columns**:
   - The code reverses the matrix and then swaps the middle element if both dimensions are odd. This might not always result in a valid matrix `b`.

3. **Matrix with Even Rows and Even Columns**:
   - The code simply reverses the matrix. This should generally work, but it's worth testing.

4. **Matrix with Odd Rows and Even Columns**:
   - The code reverses the matrix and then reverses the middle row. This might not always result in a valid matrix `b`.

5. **Matrix with Even Rows and Odd Columns**:
   - The code reverses the matrix and then reverses the middle column. This might not always result in a valid matrix `b`.

Let's create a test case for a matrix with odd rows and odd columns to see if the code's approach fails.

### Test Case:
```plaintext
1
3 3
1 2 3
4 5 6
7 8 9
```

### Explanation:
- The matrix `a` is:
  ```
  1 2 3
  4 5 6
  7 8 9
  ```
- The code will reverse the matrix to get:
  ```
  7 8 9
  4 5 6
  1 2 3
  ```
- Then it will reverse the middle row (which is the second row):
  ```
  7 8 9
  6 5 4
  1 2 3
  ```
- Finally, it will swap the middle element (5) with the last element of the middle row (4):
  ```
  7 8 9
  6 4 5
  1 2 3
  ```

### Expected Output:
- The expected output should be a valid matrix `b` where no element is in the same position as in matrix `a`. However, the code's approach might not always guarantee this.

Let's see if the code produces a valid matrix `b` for this test case. If it fails, it will expose the bug.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
#include<iostream>
using namespace std;
#include<vector>
#include <algorithm>

int main() {
    int t;
    cin >> t;

    while (t--) {
        int n, m;
        cin >> n >> m;

        vector<vector<int>> a(n, vector<int>(m));
        vector<int> all_elements;

        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                cin >> a[i][j];
                all_elements.push_back(a[i][j]);
            }
        }

        sort(all_elements.begin(), all_elements.end());

        vector<vector<int>> b(n, vector<int>(m));
        int k = 0;

        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                b[i][j] = all_elements[k];
                ++k;
            }
        }

        bool possible = true;

        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                if (a[i][j] == b[i][j]) {
                    if (j + 1 < m && a[i][j + 1] != b[i][j] && a[i][j] != b[i][j + 1]) {
                        swap(b[i][j], b[i][j + 1]);
                    } else if (i + 1 < n && a[i + 1][j] != b[i][j] && a[i][j] != b[i + 1][j]) {
                        swap(b[i][j], b[i + 1][j]);
                    } else {
                        possible = false;
                    }
                }
            }
        }

        if (!possible) {
            cout << -1 << endl;
        } else {
            for (int i = 0; i < n; ++i) {
                for (int j = 0; j < m; ++j) {
                    cout << b[i][j] << " ";
                }
                cout << endl;
            }
        }
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
t=input()
for i in range(int(t)):
    n,m= map(int, input().split())
    a=[]
    for i in range(n):
        a.append(input().split())
    if n+m<=2:
        print(-1)
        continue
    if m==1:
        up=0
        down=n-1
        while up<down:
            a[up][0],a[down][0]=a[down][0], a[up][0]
            up+=1
            down-=1
        if n%2==1:
            a[n//2][0],a[0][0]=a[0][0],a[n//2][0]

    else:
        for i in range(n):
            left=0
            right=m-1
            while left<right:
                a[i][left],a[i][right]=a[i][right], a[i][left]
                left+=1
                right-=1
            if n%2==1:
                a[i][m//2],a[i][0]=a[i][0],a[i][m//2]
    for i in range(n):
        print(' '.join(a[i]))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
2 2
1 2
3 4
2 2
1 2
4 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
def new_value(j: int, k: int, current: int, available_numbers: set):

    ordered_numbers = iter(sorted(available_numbers))
    for num in ordered_numbers:
        if num != grid_a[j][k]:
            available_numbers.remove(num)
            return num

t = int(input())
for i in range(t):

    row, col = map(int, input().split())
    grid_a = []

    for j in range(row):
        line = list(map(int, input().split()))
        grid_a.append(line)

    if row == 1 and col == 1:
        print("-1")
        continue

    grid_b = [row[:] for row in grid_a]
    # Initialize the set of available numbers
    numbers = set(range(1, row * col + 1))

    current = 0
    for j in range(row):
        for k in range(col):
            current = new_value(j, k, current, numbers)
            grid_b[j][k] = current

    for row in grid_b:
        print(" ".join(map(str, row)))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios where the code might fail. The main issues with the buggy code are:

1. **Incorrect handling of available numbers**: The `new_value` function does not correctly ensure that the selected number is different from the current number in the same position.
2. **Incorrect initialization and usage of the `numbers` set**: The set of available numbers is not properly managed, leading to potential issues with duplicate or missing numbers.

Given these issues, a small-scale test input that can expose the bug is one where the matrix size is small but the numbers are arranged in a way that makes it difficult to find a valid `b` matrix.

Let's consider a 2x2 matrix where the numbers are arranged in a way that makes it challenging to find a valid `b` matrix:

```plaintext
1
2 2
1 2
3 4
```

### Explanation:
- **Test Case 1:**
  - **Matrix Size**: 2x2
  - **Matrix `a`**:
    ```
    1 2
    3 4
    ```
  - **Expected Output**:
    - A valid matrix `b` should be:
      ```
      3 4
      1 2
      ```
    - The buggy code might fail to correctly manage the available numbers and could potentially return an invalid matrix or get stuck in an infinite loop.

This test case is small and complete, making it easy to verify the output and check for the bug. It also ensures that the sum of `n * m` over all test cases does not exceed the limit.

```plaintext
1
2 2
1 2
3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
t = int(input())
for _ in range(t):
    n, m = map(int, input().split())
    lst = []
    for i in range(n):
        line = list(map(int, input().split()))
        lst.append(line)
    if n == 1 and m == 1:
        print(-1)
        continue
    if m == 1:
        for i in range(n-1, -1, -1):
            print(lst[i][0])
        continue
    for i in range(n):
        x = lst[i][0]
        for j in range(1, m):
            print(lst[i][j], end=" ")
        print(x)

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code might fail. The buggy code has the following issues:

1. It handles the case where `n == 1` and `m == 1` correctly by printing `-1`.
2. It handles the case where `m == 1` by printing the elements in reverse order, which is incorrect for this problem.
3. For other cases, it shifts the elements in each row by one position to the left and prints the first element at the end, which might not always satisfy the condition that no element in the same position is the same.

To expose these issues, we can use the following test cases:

1. A matrix with `n == 1` and `m == 1` to ensure it prints `-1`.
2. A matrix with `m == 1` and `n > 1` to ensure it prints the elements in reverse order, which is incorrect.
3. A matrix with `n > 1` and `m > 1` where the shifting logic might not work as intended.

Here is a small-scale test input that covers these scenarios:

```plaintext
3
1 1
1
2 1
1
2
3 3
1 2 3
4 5 6
7 8 9
```

Explanation:
1. The first test case is a single element matrix, which should output `-1`.
2. The second test case is a matrix with `m == 1` and `n == 2`, which should not be printed in reverse order.
3. The third test case is a 3x3 matrix where the shifting logic might not work as intended, especially if the elements are in a specific order that makes it difficult to shift them without violating the condition.

This test input should effectively expose the bugs in the provided buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
import java.util.*;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;

public class Solution {
    public static void main(String[] args) {
        FastScanner fs = new FastScanner();
        PrintWriter out = new PrintWriter(System.out);
        int t = fs.nextInt();
        while (t-- != 0) {
            int n = fs.nextInt();
            int m = fs.nextInt();
            int[][] a = new int[n][m];
            HashSet<Integer> set = new HashSet<>();
            for(int i = 1 ; i<=n*m ; i++){
                set.add(i);
            }
            for(int i = 0 ; i<n ; i++){
                for(int j = 0 ; j<m ; j++){
                    a[i][j] = fs.nextInt();
                }
            }
            if (n==m && n==1) {
                out.println(-1);
            }
            else {
                for(int i = 0 ; i<n ; i++){
                    for(int j = 0 ; j<m ; j++){
                        for(int e:set){
                            if (e!=a[i][j]) {
                                a[i][j] = e;
                                set.remove(e);
                                break;
                            }
                        }
                    }
                }
                for(int i = 0 ; i<n ; i++){
                    for(int j = 0 ; j<m ; j++){
                        out.print(a[i][j]+" ");
                    }
                    out.println();
                }
            }
        }
        out.close();
    }
}

class Pair {
    int x, y;

    public Pair(int x, int y) {
        this.x = x;
        this.y = y;
    }
}

class FastScanner {
    BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
    StringTokenizer st = new StringTokenizer("");

    String next() {
        while (!st.hasMoreTokens())
            try {
                st = new StringTokenizer(br.readLine());
            } catch (IOException e) {
            }
        return st.nextToken();
    }

    int nextInt() {
        return Integer.parseInt(next());
    }

    long nextLong() {
        return Long.parseLong(next());
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
2 2
1 2
3 4
3 3
1 2 3
4 5 6
7 8 9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
import java.util.*;
public class Classroom {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
         int t = sc.nextInt();
         while(t-->0){
             int n = sc.nextInt();
             int m = sc.nextInt();
             HashMap<Integer,Integer> map = new HashMap<>();
             int arr[][] = new int[n][m];

             for(int i = 0; i < n;i++){
                for(int j = 0; j < m;j++){
                    int x = sc.nextInt();
                    arr[i][j] = x;
                    map.put(x,map.getOrDefault(x,0)+1);
                }
             }

             if(map.size() == 1){
                System.out.println(-1);
             }else{
                boolean ans = false;
                for(int i = 0; i < n;i++){
                    for(int j = 0; j < m;j++){
                        for(Integer k :map.keySet()){
                            if(k != arr[i][j]){
                                arr[i][j] = k;
                                ans = true;
                                map.put(k,map.get(k)-1);
                                if(map.get(k) == 0){
                                    map.remove(k);
                                }
                                break;
                            }
                        }
                        if(ans == false){
                           break;
                        }
                    }
                    if(ans == false){
                        break;
                    }
                }

                if(ans == false){
                    System.out.println(-1);
                }else{
                  for(int i = 0; i < n;i++){
                    for(int j = 0; j < m;j++){
                        System.out.print(arr[i][j] + " ");
                    }
                    System.out.println();
                  }
                }
             }
         }
         sc.close();
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
2 2
1 2
3 4
3 3
1 2 3
4 5 6
7 8 9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
/*
```CPP{.line-numbers}

```
*/
#include <bits/stdc++.h>
using namespace std;
#define umap unordered_map
#define cy cout << "YES" << endl
#define cn cout << "NO" << endl
#define ll long long
#define rep(i, l, r) for (int i = l; i <= r; i++)
#define rep_(i, l, r) for (int i = l; i >= r; i--)
#define db(a) cout << #a << "=" << a << endl;
#define pb(x) push_back(x)
const int inf = 0x3f3f3f3f;
const int N = 10 + 5;

int n, m;
int a[N][N], b[N][N];
void solve() {
	cin >> n >> m;
	map<int, int> mp;
	rep(i, 1, n) rep(j, 1, m) cin >> a[i][j];

	if (n * m == 1) {
		cout << -1 << endl;
		return;
	}

	rep(i, 1, n) {
		rep(j, 1, m) {
			int x = rand() % (n * m) + 1;
			while (mp[x] || x == a[i][j]) x = rand() % (n * m) + 1;
			b[i][j] = x;
			cout << b[i][j] << ' ';
			mp[x] = 1;
		}
		cout << endl;
	}
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	cout.tie(0);
	int T = 1;
	cin >> T;

	while (T--) solve();
	return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that are likely to cause issues. The main issue in the buggy code is the use of `rand()` to generate random numbers without ensuring that the generated numbers are unique and different from the corresponding elements in the input matrix `a`.

### Step-by-Step Analysis:
1. **Small Matrix with All Elements Unique**:
   - The matrix should be small to ensure it fits within the constraints.
   - All elements should be unique and within the range `[1, n * m]`.

2. **Ensuring the Random Generation Fails**:
   - The random generation in the buggy code might fail if the matrix `a` is such that it is very difficult to find a valid `b` matrix using the given method.
   - A good example is a matrix where the elements are already in a specific order that makes it hard for the random generation to find a valid `b` matrix.

3. **Example with a 2x2 Matrix**:
   - Consider a 2x2 matrix where the elements are `[1, 2, 3, 4]`.
   - The code should generate a matrix `b` where no element in `b` is the same as the corresponding element in `a`.

### Challenging Test Input:
```plaintext
1
2 2
1 2
3 4
```

### Explanation:
- **Test Case 1**:
  - Matrix `a` is:
    ```
    1 2
    3 4
    ```
  - The code needs to generate a matrix `b` where:
    - `b[1][1] != 1`
    - `b[1][2] != 2`
    - `b[2][1] != 3`
    - `b[2][2] != 4`
  - The random generation in the buggy code might fail to find a valid `b` matrix because it relies on random numbers and does not ensure that the generated numbers are unique and different from the corresponding elements in `a`.

This test input should expose the bug in the random number generation and the lack of a proper check to ensure all elements in `b` are different from the corresponding elements in `a`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define int long long int

/* UTILS */
#define mod 1000000007
#define pi 3.1415926535897932384626433832795
#define vi vector<int>
#define vvi vector<vector<int>>
#define vvvi vector<vector<vector<int>>>
int gcd(int a,int b) { if (b==0) return a; return gcd(b, a%b); }
int lcm(int a,int b) { return a/gcd(a,b)*b; }
bool prime(int a) { if (a==1) return 0; for (int i=2;i<=round(sqrt(a));++i) if (a%i==0) return 0; return 1; }
void yes() { cout<<"YES\n"; }
void no() { cout<<"NO\n"; }

class DisjointSet{
    public:
    vector<int> parent,size;
    DisjointSet(int n){
        parent.resize(n+1);
        size.resize(n+1,1);
        for (int i=1;i<=n;i++) parent[i]=i;
    }
    int findPar(int u){
        if (u==parent[u]){
            return u;
        }
        return parent[u]=findPar(parent[u]);
    }
    void Union(int u,int v){
        int ulp_u=findPar(u);
        int ulp_v=findPar(v);
        if (ulp_u==ulp_v) return ;
        if (size[ulp_u]<size[ulp_v]){
            size[ulp_v]+=size[ulp_u];
            parent[ulp_u]=ulp_v;
        }
        else{
            size[ulp_u]+=size[ulp_v];
            parent[ulp_v]=ulp_u;
        }
    }
};

void solve() {
    int n,m;
    cin >> n >> m;
    vvi v(n, vi(m));
   set<int> st;
    for (int i=0;i<n;i++){
        for (int j=0;j<m;j++){
            int c;
            cin >> c;
            v[i][j]=c;
            st.insert(c);
        }
    }
    if (n==1 and m==1){
        cout << -1 << endl;
        return;
    }

    vvi res=v;
    for (int i=0;i<n;i++){
        for (int j=0;j<m;j++){
            for (auto it = st.begin(); it != st.end(); ++it) {
                if (*it == v[i][j]) continue;
                res[i][j] = *it;
                st.erase(it);
                break;
            }
        }
    }
    for (int i=0;i<n;i++){
        for (int j=0;j<m;j++) cout << res[i][j] << " ";
        cout << endl;
    }
}

int32_t main() {
      ios_base::sync_with_stdio(false);
      cin.tie(NULL);
    int t=1;
    cin >> t;
    while (t--){
        solve();
}
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the given code, we need to focus on edge cases and scenarios that maximize the failure probability. The code's primary issue is in the way it constructs the matrix `res`. It simply iterates over the set `st` and assigns the first available number that is not equal to `v[i][j]` to `res[i][j]`. This approach does not guarantee that the resulting matrix `res` will meet the requirement that `a[i][j] != b[i][j]` for all `i` and `j`.

Here are the steps to generate a challenging test input:

1. **Edge Case 1: Single Element Matrix**
   - This is already handled correctly in the code, so we can skip this.

2. **Edge Case 2: Small Matrices with Specific Patterns**
   - Matrices where the numbers are arranged in a specific pattern that makes it difficult to find a valid `b[i][j]` for all `i` and `j`.

3. **Edge Case 3: Matrices with Repeated Patterns**
   - Matrices where the numbers are arranged in a way that makes it difficult to find a valid `b[i][j]` for all `i` and `j`.

Let's create a small-scale test input that will expose the bug:

```plaintext
3
2 2
1 2
3 4
2 2
4 3
2 1
2 2
1 3
2 4
```

### Explanation:
1. **First Test Case:**
   - Matrix:
     ```
     1 2
     3 4
     ```
   - The code should be able to find a valid matrix `b` for this case.

2. **Second Test Case:**
   - Matrix:
     ```
     4 3
     2 1
     ```
   - The code should be able to find a valid matrix `b` for this case.

3. **Third Test Case:**
   - Matrix:
     ```
     1 3
     2 4
     ```
   - This case is designed to be more challenging. The code might struggle to find a valid matrix `b` where all elements are different from the corresponding elements in matrix `a`.

This test input is small and complete, and it should expose the bug in the given code by challenging its ability to construct a valid matrix `b` for all test cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
def solve():
    n, m = map(int, input().split())
    lst = []
    for i in range(n):
        l = [int(x) for x in input().split()]
        lst.append(l)

    elements = [lst[i][j] for i in range(n) for j in range(m)]
    elements.sort()

    b = [[0] * m for _ in range(n)]
    idx = 0

    for i in range(n):
        for j in range(m):
            b[i][j] = elements[idx]
            idx += 1

    for i in range(n):
        for j in range(m):
            if lst[i][j] == b[i][j]:
                if j + 1 < m:
                    b[i][j], b[i][j + 1] = b[i][j + 1], b[i][j]
                elif i + 1 < n:
                    b[i][j], b[i + 1][j] = b[i + 1][j], b[i][j]
                else:
                    print(-1)
                    return

    for row in b:
        print(" ".join(map(str, row)))

t = int(input())
while t > 0:
    t -= 1
    solve()

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
2 2
1 2
3 4
2 2
1 2
2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
t = int(input())

for _ in range(t):
    temp = list(map(int, input().split()))
    n = temp[0]
    m = temp[1]
    if n == 1 and m == 1:
        nothing = input()
        print(-1)
        continue
    else:
        lst = []
        for i in range(n):
            temp = input()
            row = f"{temp[1:]} {temp[0]}"
            row = list(map(int, row.split()))
            lst.append(row)
            lst.reverse()
    for row in lst:
        print(" ".join(map(str, row)))

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To expose the bug in the given code, we need to focus on edge cases where the code might fail. The code has a specific issue in how it handles the input and the matrix transformation. Specifically, it reverses the entire list of rows, which is incorrect and can lead to incorrect output.

Let's consider the following steps to generate a challenging test input:

1. **Single Element Matrix (1x1):**
   - This case is already handled correctly in the code, so we don't need to focus on it.

2. **2x1 Matrix:**
   - This case should be handled correctly by the code, but it's worth checking to ensure the transformation is correct.

3. **2x2 Matrix:**
   - This is a more interesting case because the code reverses the entire list of rows. This can lead to incorrect output if the transformation is not done correctly.

4. **3x3 Matrix:**
   - This case is more complex and can expose issues with the row reversal logic.

5. **Matrix with Repeated Elements:**
   - This is not possible according to the problem constraints, but we should ensure the input is valid.

Given these considerations, let's generate a test input that includes a 2x2 matrix and a 3x3 matrix, which are more likely to expose the bug in the code.

```plaintext
2
2 2
1 2
3 4
3 3
1 2 3
4 5 6
7 8 9
```

This input includes:
- A 2x2 matrix: The code should not reverse the entire list of rows, but it does.
- A 3x3 matrix: The code should not reverse the entire list of rows, but it does.

This test input should help expose the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
//package competitiveprogramming;
//import competitiveprogramming.sum;
//
//import javax.swing.*;
//import java.beans.Introspector;
//import javax.swing.*;
import java.nio.channels.InterruptedByTimeoutException;
import java.util.*;

public class Main {
   static Scanner sc = new Scanner(System.in);
    static long mod  = (long)Math.pow(10,9)+7;

    public static void main(String[] args) {
        int t = sc.nextInt();
        out:while(t-->0){
            int n= sc.nextInt();
            int m = sc.nextInt();
            Set<Integer> st = new HashSet<>();
            int arr[][] = new int[n][m];
            for(int i=0; i<n; i++){
                for(int j=0; j<m; j++){
                    arr[i][j] = sc.nextInt();
                    st.add(arr[i][j]);
                }
            }
            if(st.size()!=n*m){
              System.out.println(-1);
              continue out;
            }
            if(n*m==1){
              System.out.println(-1);
              continue out;
            }
            if(n%2==1 && m%2==1){
                int a=0;
                int b=0;
                int x=n*m+1;
                for(int i=0; i<n; i++){
                    for(int j=0; j<m; j++){
                        arr[i][j] = x-arr[i][j];
                        if(x==2*arr[i][j]){
                            a=i;
                            b=j;
                        }
                    }
                }
                int temp = 0;
                if(a!=n-1 && b!=m-1){
                    temp = arr[n-1][m-1];
                    arr[n-1][m-1] = arr[a][b];
                    arr[a][b] =temp;
                }
                else{
                    temp = arr[0][0];
                    arr[0][0] = arr[a][b];
                    arr[a][b] =temp;
                }
            }
            else{
                int x=n*m+1;
                for(int i=0; i<n; i++){
                    for(int j=0; j<m; j++){
                        arr[i][j] = x-arr[i][j];
                    }
                }
            }
            for(int i=0; i<n; i++){
                for(int j=0; j<m; j++){
                    System.out.print(arr[i][j]+" ");
                }
                System.out.println();
            }

        }

    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
3
1 1
1
3 3
1 2 3
4 5 6
7 8 9
2 2
1 2
3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.StringTokenizer;
public class DiverseGame {
    static class FastReader {
        BufferedReader br;
        StringTokenizer st;
        public FastReader() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }
        String next() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }
        int nextInt() {
            return Integer.parseInt(next());
        }
        long nextLong() {
            return Long.parseLong(next());
        }
        double nextDouble() {
            return Double.parseDouble(next());
        }
        String nextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }
    }

    public static class Pair implements Comparable<Pair>{
        int first;
        int second;
        public Pair(int first, int second) {
            this.first = first;
            this.second = second;
        }
        @Override
        public boolean equals(Object obj) {
            if (this == obj) return true;
            if (obj == null || getClass() != obj.getClass()) return false;
            Pair pair = (Pair) obj;
            if (first != pair.first) return false;
            return second == pair.second;
        }
        @Override
        public int compareTo(Pair that){
            return Integer.compare(this.second, that.second);
        }
        @Override
        public int hashCode() {
            int result = first;
            result = 31 * result + second;
            return result;
        }
    }

    public static void main(String args[]){
        FastReader sc = new FastReader();
        int t = sc.nextInt();
        while(t-- != 0){
            int n = sc.nextInt();
            int m = sc.nextInt();
            int arr[][] = new int[n][m];
            for(int i=0; i<n; i++)
                for(int j=0; j<m; j++)
                    arr[i][j] = sc.nextInt();

            HashMap<Integer, Integer> hm = new HashMap<>();
            for(int i=0; i<n; i++)
                for(int j=0; j<m; j++)
                    hm.put(arr[i][j], hm.getOrDefault(arr[i][j], 0)+1);

            Pair pair[] = new Pair[hm.size()];
            int j = 0;
            for(var i : hm.keySet()){
                pair[j++] = new Pair(i, hm.get(i));
            }

            Arrays.sort(pair);

            ArrayList<Integer> list = new ArrayList<>();
            for(int i=1; i<=n*m; i++)
                list.add(i);

            if(pair[pair.length-1].second > (n*m)/2){
                System.out.println(-1);
            }
            else{
                for(int i=0; i<n; i++){
                    for(j=0; j<m; j++){
                        if(list.get(0) == arr[i][j]){
                            System.out.print(list.get(1)+" ");
                            list.remove(1);
                        }
                        else{
                            System.out.print(list.get(0)+" ");
                            list.remove(0);
                        }
                    }
                    System.out.println();
                }
            }
        }
    }
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that can maximize the failure probability. The key issues in the buggy code are:

1. The code uses a `HashMap` to count the occurrences of each number, which is unnecessary since the problem guarantees that all numbers in the matrix are distinct.
2. The code sorts the numbers based on their frequency, which is also unnecessary and can lead to incorrect behavior.
3. The code does not correctly handle the case where the matrix is a 1x1 matrix, which is impossible to satisfy the condition.

Let's consider the following scenarios:

1. **1x1 Matrix**: This is the simplest case where it is impossible to construct a matrix `b` that satisfies the condition.
2. **2x1 Matrix**: This is a slightly more complex case where the numbers can be swapped to satisfy the condition.
3. **2x2 Matrix**: This case requires more careful handling to ensure that no number in the same position in both matrices is the same.

Here is a small-scale test input that includes these scenarios:

```plaintext
3
1 1
1
2 1
2
1
2 2
1 2
3 4
```

This test input includes:
- A 1x1 matrix, which should output `-1`.
- A 2x1 matrix, which should output a valid matrix with swapped values.
- A 2x2 matrix, which should output a valid matrix with no numbers in the same position.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Diverse Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: Petr, watching Sergey's stream, came up with a matrix $$$a$$$, consisting of $$$n$$$ rows and $$$m$$$ columns (the number in the $$$i$$$-th row and $$$j$$$-th column is denoted as $$$a_{i, j}$$$), which contains all integers from $$$1$$$ to $$$n \cdot m$$$. But he didn't like the arrangement of the numbers, and now he wants to come up with a new matrix $$$b$$$, consisting of $$$n$$$ rows and $$$m$$$ columns, which will also contain all integers from $$$1$$$ to $$$n \cdot m$$$, such that for any $$$1 \leq i \leq n, 1 \leq j \leq m$$$ it holds that $$$a_{i, j} \ne b_{i, j}$$$.

You are given the matrix $$$a$$$, construct any matrix $$$b$$$ that meets Petr's requirements, or determine that it is impossible.

Hurry up! Otherwise, he will donate all his money to the stream in search of an answer to his question.

Input: Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^3$$$) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 10$$$) — the number of rows and columns of matrix $$$a$$$.

The next $$$n$$$ lines contain $$$m$$$ integers each, describing matrix $$$a$$$. The $$$i$$$-th of these lines contains the elements of matrix $$$a_{i, 1}, a_{i, 2}, \ldots, a_{i, m}$$$.

It is guaranteed that all numbers in matrix $$$a$$$ are distinct and $$$1 \leq a_{i, j} \leq n \cdot m$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$5 \cdot 10^4$$$.

Output: For each test case, output $$$n \cdot m$$$ integers — any suitable matrix $$$b$$$, or $$$-1$$$ if such a matrix does not exist.

Examples:
input:
5
1 1
1
2 1
2
1
1 5
2 4 5 3 1
2 4
1 2 3 4
5 6 7 8
3 3
4 2 1
9 8 3
6 7 5
output:
-1
1
2
4 5 3 1 2
6 7 8 5
2 3 4 1
8 3 9
7 5 6
2 1 4

Note: In the first test case, there is only one element in the matrix, so matrix $$$b$$$ is the only matrix and it does not fit.

In the second test case $$$a_{1, 1} = 2 \neq 1 = b_{1, 1}$$$, $$$a_{2, 1} = 1 \neq 2 = b_{2, 1}$$$.

Buggy Code:
#include <bits/stdc++.h>
#define JS ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define ll long long
using namespace std;

void flipAndPush(vector<vector<ll>>& arr, ll n, ll m) {
    for (int i = 0; i < n; ++i) {
        ll first = arr[i][0];

        for (int j = 1; j < m; ++j) {
            arr[i][j - 1] = arr[i][j];
        }

        arr[i][m - 1] = first;
    }

    ll top = 0, bottom = n - 1;
    while (top < bottom) {
        for (int j = 0; j < m; ++j) {
            ll temp = arr[top][j];
            arr[top][j] = arr[bottom][j];
            arr[bottom][j] = temp;
        }
        ++top;
        --bottom;
    }
}

void Solve(){
    ll t;
    cin >> t;
    while(t--){
        ll n, m;
        cin >> n >> m;

        vector<vector<ll>> arr(n, vector<ll>(m));
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                cin >> arr[i][j];
            }
        }

        if(n==1 && m==1){
            cout << "-1" << endl;
        } else {
            bool allEqual = true;
            ll firstElement = arr[0][0];
            for (int i = 0; i < n; ++i) {
                for (int j = 0; j < m; ++j) {
                    if(arr[i][j] != firstElement){
                        allEqual = false;
                        break;
                    }
                }
                if(!allEqual) break;
            }

            if(allEqual){
                cout << "-1" << endl;
            } else {
                flipAndPush(arr, n, m);
                for (int i = 0; i < n; ++i) {
                    for (int j = 0; j < m; ++j) {
                        cout << arr[i][j] << " ";
                    }
                    cout << endl;
                }
            }
        }
    }
}

int main() {
    JS
    Solve();
    return 0;
}

**Instructions:**
- Focus on edge cases or scenarios that maximize the failure probability in faulty solutions.
- Due to the output length limit, you should generate a **small-scale** test input that is **complete and valid**.
- Output the test input directly, not code to generate it.

**Output format:**
```plaintext
{test input}
```

Think step by step.

Response:
```plaintext
5
1 1
1
2 2
1 2
3 4
2 3
1 2 3
4 5 6
3 3
1 2 3
4 5 6
7 8 9
2 4
1 2 3 4
5 6 7 8
```

### Explanation:
1. **Test Case 1 (1x1 Matrix):**
   - This is the simplest case where it's impossible to create a different matrix.
   - Input: `1 1 1`
   - Expected Output: `-1`

2. **Test Case 2 (2x2 Matrix):**
   - This is a simple 2x2 matrix where all elements are different.
   - Input: `2 2 1 2 3 4`
   - Expected Output: A matrix where no element is the same as in the original matrix.

3. **Test Case 3 (2x3 Matrix):**
   - This is a 2x3 matrix with a sequence of numbers.
   - Input: `2 3 1 2 3 4 5 6`
   - Expected Output: A matrix where no element is the same as in the original matrix.

4. **Test Case 4 (3x3 Matrix):**
   - This is a 3x3 matrix with a sequence of numbers.
   - Input: `3 3 1 2 3 4 5 6 7 8 9`
   - Expected Output: A matrix where no element is the same as in the original matrix.

5. **Test Case 5 (2x4 Matrix):**
   - This is a 2x4 matrix with a sequence of numbers.
   - Input: `2 4 1 2 3 4 5 6 7 8`
   - Expected Output: A matrix where no element is the same as in the original matrix.

These test cases cover a range of matrix sizes and ensure that the code is tested for both the simplest and more complex scenarios.