**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given a positive integer $k$ and a set $S$ of all integers from $l$ to $r$ (inclusive).

You can perform the following two-step operation any number of times (possibly zero):

  1. First, choose a number $x$ from the set $S$, such that there are at least $k$ multiples of $x$ in $S$ (including $x$ itself);    2. Then, remove $x$ from $S$ (note that nothing else is removed).

Find the maximum possible number of operations that can be performed.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of test cases follows.

The only line of each test case contains three integers $l$, $r$, and $k$ ($1\le l\le r\leq 10^9$, $1\leq k\le r-l+1$) — the minimum integer in $S$, the maximum integer in $S$, and the parameter $k$.

Output: For each test case, output a single integer — the maximum possible number of operations that can be performed.

Examples:
input:
8
3 9 2
4 9 1
7 9 2
2 10 2
154 220 2
147 294 2
998 24435 3
1 1000000000 2
output:
2
6
0
4
0
1
7148
500000000

Note: In the first test case, initially, $S = \\{3,4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 4$ for the first operation, since there are two multiples of $4$ in $S$: $4$ and $8$. $S$ becomes equal to $\\{3,5,6,7,8,9\\}$;    2. Choose $x = 3$ for the second operation, since there are three multiples of $3$ in $S$: $3$, $6$, and $9$. $S$ becomes equal to $\\{5,6,7,8,9\\}$.

In the second test case, initially, $S=\\{4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 5$, $S$ becomes equal to $\\{4,6,7,8,9\\}$;    2. Choose $x = 6$, $S$ becomes equal to $\\{4,7,8,9\\}$;    3. Choose $x = 4$, $S$ becomes equal to $\\{7,8,9\\}$;    4. Choose $x = 8$, $S$ becomes equal to $\\{7,9\\}$;    5. Choose $x = 7$, $S$ becomes equal to $\\{9\\}$;    6. Choose $x = 9$, $S$ becomes equal to $\\{\\}$.

In the third test case, initially, $S=\\{7,8,9\\}$. For each $x$ in $S$, no multiple of $x$ other than $x$ itself can be found in $S$. Since $k = 2$, you can perform no operations.

In the fourth test case, initially, $S=\\{2,3,4,5,6,7,8,9,10\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 2$, $S$ becomes equal to $\\{3,4,5,6,7,8,9,10\\}$;    2. Choose $x = 4$, $S$ becomes equal to $\\{3,5,6,7,8,9,10\\}$;    3. Choose $x = 3$, $S$ becomes equal to $\\{5,6,7,8,9,10\\}$;    4. Choose $x = 5$, $S$ becomes equal to $\\{6,7,8,9,10\\}$.

Buggy Code:
for _ in range(int(input())):
    l, r, k = map(int, input().split())
    ops = 0

    if k == 1:
        print(r - l + 1)

    else:
        x = r // 2
        while k*x > r:
            x -= 1
        ops = x - l + 1 if x >= l else 0
        # while k*l <= r:
        #     ops += 1
        #     l += 1

        print(ops)

**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 10 2
10 10 1
10 10 2
1 100 3
1 1000000000 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given a positive integer $k$ and a set $S$ of all integers from $l$ to $r$ (inclusive).

You can perform the following two-step operation any number of times (possibly zero):

  1. First, choose a number $x$ from the set $S$, such that there are at least $k$ multiples of $x$ in $S$ (including $x$ itself);    2. Then, remove $x$ from $S$ (note that nothing else is removed).

Find the maximum possible number of operations that can be performed.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of test cases follows.

The only line of each test case contains three integers $l$, $r$, and $k$ ($1\le l\le r\leq 10^9$, $1\leq k\le r-l+1$) — the minimum integer in $S$, the maximum integer in $S$, and the parameter $k$.

Output: For each test case, output a single integer — the maximum possible number of operations that can be performed.

Examples:
input:
8
3 9 2
4 9 1
7 9 2
2 10 2
154 220 2
147 294 2
998 24435 3
1 1000000000 2
output:
2
6
0
4
0
1
7148
500000000

Note: In the first test case, initially, $S = \\{3,4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 4$ for the first operation, since there are two multiples of $4$ in $S$: $4$ and $8$. $S$ becomes equal to $\\{3,5,6,7,8,9\\}$;    2. Choose $x = 3$ for the second operation, since there are three multiples of $3$ in $S$: $3$, $6$, and $9$. $S$ becomes equal to $\\{5,6,7,8,9\\}$.

In the second test case, initially, $S=\\{4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 5$, $S$ becomes equal to $\\{4,6,7,8,9\\}$;    2. Choose $x = 6$, $S$ becomes equal to $\\{4,7,8,9\\}$;    3. Choose $x = 4$, $S$ becomes equal to $\\{7,8,9\\}$;    4. Choose $x = 8$, $S$ becomes equal to $\\{7,9\\}$;    5. Choose $x = 7$, $S$ becomes equal to $\\{9\\}$;    6. Choose $x = 9$, $S$ becomes equal to $\\{\\}$.

In the third test case, initially, $S=\\{7,8,9\\}$. For each $x$ in $S$, no multiple of $x$ other than $x$ itself can be found in $S$. Since $k = 2$, you can perform no operations.

In the fourth test case, initially, $S=\\{2,3,4,5,6,7,8,9,10\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 2$, $S$ becomes equal to $\\{3,4,5,6,7,8,9,10\\}$;    2. Choose $x = 4$, $S$ becomes equal to $\\{3,5,6,7,8,9,10\\}$;    3. Choose $x = 3$, $S$ becomes equal to $\\{5,6,7,8,9,10\\}$;    4. Choose $x = 5$, $S$ becomes equal to $\\{6,7,8,9,10\\}$.

Buggy Code:
for _ in range(int(input())):
    l, r, k = map(int, input().split())
    ops = 0

    if l == 1 and k == 2:
        print(r // 2)
    else:
        while k*l <= r:
            ops += 1
            l += 1

        print(ops)

**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 10 2
1 10 3
1 10 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given a positive integer $k$ and a set $S$ of all integers from $l$ to $r$ (inclusive).

You can perform the following two-step operation any number of times (possibly zero):

  1. First, choose a number $x$ from the set $S$, such that there are at least $k$ multiples of $x$ in $S$ (including $x$ itself);    2. Then, remove $x$ from $S$ (note that nothing else is removed).

Find the maximum possible number of operations that can be performed.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of test cases follows.

The only line of each test case contains three integers $l$, $r$, and $k$ ($1\le l\le r\leq 10^9$, $1\leq k\le r-l+1$) — the minimum integer in $S$, the maximum integer in $S$, and the parameter $k$.

Output: For each test case, output a single integer — the maximum possible number of operations that can be performed.

Examples:
input:
8
3 9 2
4 9 1
7 9 2
2 10 2
154 220 2
147 294 2
998 24435 3
1 1000000000 2
output:
2
6
0
4
0
1
7148
500000000

Note: In the first test case, initially, $S = \\{3,4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 4$ for the first operation, since there are two multiples of $4$ in $S$: $4$ and $8$. $S$ becomes equal to $\\{3,5,6,7,8,9\\}$;    2. Choose $x = 3$ for the second operation, since there are three multiples of $3$ in $S$: $3$, $6$, and $9$. $S$ becomes equal to $\\{5,6,7,8,9\\}$.

In the second test case, initially, $S=\\{4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 5$, $S$ becomes equal to $\\{4,6,7,8,9\\}$;    2. Choose $x = 6$, $S$ becomes equal to $\\{4,7,8,9\\}$;    3. Choose $x = 4$, $S$ becomes equal to $\\{7,8,9\\}$;    4. Choose $x = 8$, $S$ becomes equal to $\\{7,9\\}$;    5. Choose $x = 7$, $S$ becomes equal to $\\{9\\}$;    6. Choose $x = 9$, $S$ becomes equal to $\\{\\}$.

In the third test case, initially, $S=\\{7,8,9\\}$. For each $x$ in $S$, no multiple of $x$ other than $x$ itself can be found in $S$. Since $k = 2$, you can perform no operations.

In the fourth test case, initially, $S=\\{2,3,4,5,6,7,8,9,10\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 2$, $S$ becomes equal to $\\{3,4,5,6,7,8,9,10\\}$;    2. Choose $x = 4$, $S$ becomes equal to $\\{3,5,6,7,8,9,10\\}$;    3. Choose $x = 3$, $S$ becomes equal to $\\{5,6,7,8,9,10\\}$;    4. Choose $x = 5$, $S$ becomes equal to $\\{6,7,8,9,10\\}$.

Buggy Code:
import java.util.*;
public class A_Set{
public static void main(String[] args){
Scanner sc = new Scanner(System.in);
int t = sc.nextInt();
while(t-->0){
long l = sc.nextLong();
long r = sc.nextLong();
long k = sc.nextLong();
long i = l;
for(i=l;i<=r;i++) if((i*k>r)) break;
System.out.println(i-l);
}
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 10 2
1 10 1
999 1000 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given a positive integer $k$ and a set $S$ of all integers from $l$ to $r$ (inclusive).

You can perform the following two-step operation any number of times (possibly zero):

  1. First, choose a number $x$ from the set $S$, such that there are at least $k$ multiples of $x$ in $S$ (including $x$ itself);    2. Then, remove $x$ from $S$ (note that nothing else is removed).

Find the maximum possible number of operations that can be performed.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $t$ ($1\le t\le 10^4$) — the number of test cases. The description of test cases follows.

The only line of each test case contains three integers $l$, $r$, and $k$ ($1\le l\le r\leq 10^9$, $1\leq k\le r-l+1$) — the minimum integer in $S$, the maximum integer in $S$, and the parameter $k$.

Output: For each test case, output a single integer — the maximum possible number of operations that can be performed.

Examples:
input:
8
3 9 2
4 9 1
7 9 2
2 10 2
154 220 2
147 294 2
998 24435 3
1 1000000000 2
output:
2
6
0
4
0
1
7148
500000000

Note: In the first test case, initially, $S = \\{3,4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 4$ for the first operation, since there are two multiples of $4$ in $S$: $4$ and $8$. $S$ becomes equal to $\\{3,5,6,7,8,9\\}$;    2. Choose $x = 3$ for the second operation, since there are three multiples of $3$ in $S$: $3$, $6$, and $9$. $S$ becomes equal to $\\{5,6,7,8,9\\}$.

In the second test case, initially, $S=\\{4,5,6,7,8,9\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 5$, $S$ becomes equal to $\\{4,6,7,8,9\\}$;    2. Choose $x = 6$, $S$ becomes equal to $\\{4,7,8,9\\}$;    3. Choose $x = 4$, $S$ becomes equal to $\\{7,8,9\\}$;    4. Choose $x = 8$, $S$ becomes equal to $\\{7,9\\}$;    5. Choose $x = 7$, $S$ becomes equal to $\\{9\\}$;    6. Choose $x = 9$, $S$ becomes equal to $\\{\\}$.

In the third test case, initially, $S=\\{7,8,9\\}$. For each $x$ in $S$, no multiple of $x$ other than $x$ itself can be found in $S$. Since $k = 2$, you can perform no operations.

In the fourth test case, initially, $S=\\{2,3,4,5,6,7,8,9,10\\}$. One possible optimal sequence of operations is:

  1. Choose $x = 2$, $S$ becomes equal to $\\{3,4,5,6,7,8,9,10\\}$;    2. Choose $x = 4$, $S$ becomes equal to $\\{3,5,6,7,8,9,10\\}$;    3. Choose $x = 3$, $S$ becomes equal to $\\{5,6,7,8,9,10\\}$;    4. Choose $x = 5$, $S$ becomes equal to $\\{6,7,8,9,10\\}$.

Buggy Code:
import java.util.*;

public class Vishwajeet {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();

        while (t-- > 0) {
            long l = sc.nextLong();
            long r = sc.nextLong();
            long k = sc.nextLong();

            long output = 0;

            long check = r / k;

            if (k == 1) {
                output = r - l + 1;
            } else {
                for (long i = l; i <= check; i++) {
                    if (r >= i * k) {
                        output++;
                    } else {
                        break;
                    }
                }
            }

            System.out.println(output);
        }
    }
}

**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 10 1
1 10 2
1 10 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
#include <iostream>
#include <vector>
#include <algorithm>
#include <string>
#include <map>
#include <bitset>
#include <cmath>
#define ll int long long
#define makearr int arr[m]; for(int i=0;i<m;i++) cin>>arr[i];
#define For(i,j,n) for(int i=j;i<n;i++)
#define pb push_back
#define mp make_pair
#define pi pair<int,int>
#define pll pair<ll,ll>
#define ri(n) int n; cin>>n;
#define rll(n) ll n; cin>>n;
#define di(n) int n;
#define dll(n) ll n;
using namespace std;
bool pow2(int n){
    while(n%2==0){
        n/=2;
    }
    return (n==1);
}
int maxdiv(int n){
    int it=1;
    while(n%2==0){
        n/=2;it++;
    }int x=1;
    For(i,0,it){
        x*=2;
    }
    return --x;
}
int maxno(int n){
    int x=1;int it=1;
    while(x<=n){
        x*=2;it++;
    }
    if(x>n) {x/=2;it--;}
    return it;
}
int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    ri(t);For(ii,0,t){
        ri(n);ri(m);vector<int> finans(n,-1);
        makearr;reverse(arr,arr+m);vector<bool> chek(n+1,false);
        bool canmake=true;
        finans[0]=arr[0];
        For(i,2,n+1){
            if(chek[i]) continue;
            int it=1;
            for(int j=i;j<=n;j+=i){
                finans[j-1]=it++;
                chek[j]=true;
            }
        }
        For(i,1,n){
            if(finans[i]>=m){
                canmake=false;break;
            }
            finans[i]=arr[finans[i]];
        }
        if(canmake){
        For(i,0,n){
            cout<<finans[i]<<" ";
        }cout<<"\n";}
        else cout<<-1<<"\n";
    }
    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
1
4 2
2 4
```

**Explanation:**

1. **Test Case Setup:**
   - We have one test case (`t = 1`).
   - The length of the array `n` is 4.
   - The set `S` contains 2 elements: `{2, 4}`.

2. **Reasoning:**
   - The goal is to find the lexicographically largest array of length 4 where each element is from the set `{2, 4}` and the condition `a_{gcd(i, j)} ≠ gcd(a_i, a_j)` is satisfied for all pairs `1 ≤ i < j ≤ n`.

3. **Edge Case:**
   - This test case is designed to expose a potential bug in the code. The set `{2, 4}` has a specific property: the GCD of any two elements from the set is either 2 or 4.
   - For the array to satisfy the condition, the GCD of any two elements must not match the value at the GCD index. This is particularly challenging when the set is small and the elements have a common factor.

4. **Expected Output:**
   - The correct output for this test case should be `-1` because it is impossible to construct such an array. For example:
     - If `a = [4, 4, 4, 4]`, then `gcd(a_1, a_2) = 4` and `a_{gcd(1, 2)} = a_1 = 4`, which violates the condition.
     - If `a = [2, 4, 2, 4]`, then `gcd(a_1, a_2) = 2` and `a_{gcd(1, 2)} = a_1 = 2`, which also violates the condition.

This test case is designed to be small and manageable while still being challenging enough to expose the bug in the given code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
#include <iostream>
#include <algorithm>
#include <math.h>
#include <vector>
#include <set>
#include <queue>
#include <array>
#include <map>
#include <random>
#include <bitset>
#include <stack>

typedef long long ll;
typedef double ld;
using namespace std;

inline void solve_();

#define ff first
#define ss second
#define len(v) (int)v.size()

const int N = 1e5 + 7;

int mind[N];

void prec() {
    for (int i = 0; i < N; ++i)
        mind[i] = i;
    for (int i = 2; i < N; ++i) {
        if (mind[i] == i) {
            for (int j = i; j < N; j += i) {
                mind[j] = min(mind[j], i);
            }
        }
    }
}

void solve() {
    int n, m;
    cin >> n >> m;

    vector<int> s(m);
    for (int i = 0; i < m; ++i) cin >> s[i];

    vector<int> ans(n, -1);

    vector<int> used(m, 0);

    for (int i = 1; i <= n; ++i) {
        int v = i;
        if (i == 1) {
            ans[i - 1] = m - 1;
        }
        else if (mind[i] == i) {
            ans[i - 1] = m - 2;
            if (m <= 1) {
                cout << "-1\n";
                return;
            }
        }
        else {
            used[m - 1] = i;

            while (v > 1) {
                used[ans[mind[v] - 1]] = i;
                v /= mind[v];
            }
            for (int j = m - 1; j >= 0; --j) {
                if (used[j] != i) {
                    ans[i - 1] = j;
                    break;
                }
            }
            if (ans[i - 1] == -1) {
                cout << "-1\n";
                return;
            }
        }
    }
    for (int i = 0; i < n; ++i) {
        cout << s[ans[i]] << ' ';
    }
    cout << '\n';
}

signed main() {
    prec();
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);
    cout.precision(20);

    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 generate a challenging test input that exposes the bug in the given buggy code, we need to consider edge cases and scenarios where the code might fail. The primary issue in the code seems to be related to the handling of the lexicographically largest array and the constraints on the GCD conditions.

Let's break down the potential issues:
1. **Single Element Set**: If the set \( S \) contains only one element, the code should return \(-1\) because it's impossible to satisfy the GCD condition.
2. **Multiple Elements with Specific GCD Conditions**: If the set \( S \) contains multiple elements, the code might fail to correctly handle the GCD conditions, especially when the elements have specific relationships.

Given these considerations, let's create a test input that includes:
- A single test case with \( n \) and \( m \) such that the set \( S \) contains multiple elements, but the GCD conditions are tricky to satisfy.
- A test case where \( S \) contains only one element to ensure the code correctly returns \(-1\).

Here is the test input:

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

### Explanation:
1. **Test Case 1**:
   - \( n = 4 \)
   - \( m = 2 \)
   - Set \( S = \{1, 2\} \)
   - This case is designed to test the code's ability to handle multiple elements and the GCD conditions. The GCD conditions for this set are:
     - \( \gcd(1, 2) = 1 \)
     - \( \gcd(1, 3) = 1 \)
     - \( \gcd(1, 4) = 1 \)
     - \( \gcd(2, 3) = 1 \)
     - \( \gcd(2, 4) = 2 \)
     - \( \gcd(3, 4) = 1 \)
   - The code should find a lexicographically largest array that satisfies these conditions.

2. **Test Case 2**:
   - \( n = 3 \)
   - \( m = 1 \)
   - Set \( S = \{3\} \)
   - This case is designed to test the code's ability to handle a single element set. The GCD condition for this set is:
     - \( \gcd(1, 2) = 1 \)
     - \( \gcd(1, 3) = 1 \)
     - \( \gcd(2, 3) = 1 \)
   - The code should correctly return \(-1\) because it's impossible to satisfy the GCD condition with a single element.

This test input should help expose any bugs in the code, particularly in handling multiple elements and the GCD conditions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
#pragma GCC optimize("Ofast")
#pragma GCC optimize("O3")

#include <iostream>
#include<string>
#include<math.h>
#include<algorithm>
#include<vector>
#include<map>
#include<deque>
#include<queue>
#include <numeric>
#include <iomanip>
#include<set>
#include <unordered_map>
#include<stack>

#define ali ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define ll long long
//#define LONG_MAX 1e14
#define mod (ll)(1e9 + 7)
using namespace std;

/*vector<ll>primes;
bool notPrime[200005];

void sieve() {

	for (ll i = 2; i * i < 200005; i++) {
		if (!notPrime[i]) {
			for (ll j = i * i; j < 200005; j += i) {

				notPrime[j] = true;
			}
		}
	}
	for (ll i = 2; i < 200005; i++) {
		if (!notPrime[i]) {
			primes.push_back(i);
		}

	}
}

vector<ll> getFactors(ll n) {
	ll tmp = n;
	vector<ll>v;

	for (ll i : primes) {
		if (n == 1) {
			break;
		}
		if (n % i == 0 && tmp / i >= 3) {
			v.push_back(i);
		}
		while (n % i == 0) {
			n /= i;
		}
	}
	if (n != 1 && tmp / n >= 3) {
		v.push_back(n);
	}
	return v;
}

vector<pair<ll, ll>> getFactors(ll n, ll power) {
	ll tmp = n;
	vector<pair<ll, ll>>tt;
	bool first = true;
	for (int i = 0; primes[i] * primes[i] <= n; i++) {
		if (tmp == 1) {
			break;
		}
		while (tmp % primes[i] == 0) {
			if (first) {
				tt.push_back({ primes[i], 1 });
				first = false;
			}
			else {
				tt[tt.size() - 1].second++;
			}
			tmp /= primes[i];
		}
		if (!first) {
			tt[tt.size() - 1].second *= power;
			first = true;
		}
	}
	if (tmp != 1) {
		tt.push_back({ tmp, power });
	}
	return tt;
}*/

ll fastPow(ll base, ll power) {
	if (power == 1) {
		return base;
	}
	if (power == 0) {
		return 1;
	}
	ll out = fastPow(base, power / 2);
	//out = ((out % mod) * (out % mod)) % mod;
	out *= out;

	if (power % 2 != 0) {
		//out = ((out % mod) * (base % mod)) % mod;
		out *= base;
	}
	return out;

}

ll fastPow(ll base, ll power, ll modd) {
	if (power == 1) {
		return base;
	}
	ll out = fastPow(base, power / 2, modd);
	//cout << out << " " << ((out % modd) * (out % modd)) << '\n';
	if (out % modd != 0) {

		out = ((out % modd) * (out % modd));
	}
	else {
		out *= out;
	}
	//cout << out << '\n';
	if (out % modd != 0) {
		out %= modd;
	}
	//cout << out << '\n';
	//out *= out;

	if (power % 2 != 0) {
		if ((out % modd) * (base % modd) == 0) {
			out *= base;
		}
		else {
			out = ((out % modd) * (base % modd));
		}
		if (out % modd != 0) {
			out %= modd;
		}
		//out *= base;
	}
	return out;

}

ll gcd(ll a, ll b)
{

	if (a == 0)
		return b;
	if (b == 0)
		return a;
	return gcd(b % a, a);
}

ll fact(ll n1, ll end, ll n2) {
	ll f1 = 1, f2 = 1;
	ll i, j, gc;

	for (i = n1, j = 2; i <= end && j <= n2; i++, j++) {
		f1 *= i;
		f2 *= j;
		gc = gcd(f1, f2);
		f1 /= gc;
		f2 /= gc;

	}
	for (; i <= end; i++) {
		f1 *= i;
		gc = gcd(f1, f2);
		f1 /= gc;
		f2 /= gc;
	}
	for (; j <= n2; j++) {
		f2 *= j;
		gc = gcd(f1, f2);
		f1 /= gc;
		f2 /= gc;
	}

	return f1;
}

ll c(ll n, ll r) {

	if (r > n - r) {
		return fact(r + 1, n, n - r);
	}
	else {
		return fact(n - r + 1, n, r);
	}
}

int main() {
	ali
	int t = 1;
	cin >> t;
	while(t--) {
		bool can = true;
		ll n, m, tmp;
		cin >> n >> m;
		vector<ll>arr, v;
		map<ll, ll>mp;
		for(int i = 0; i < m; i++){
			cin >> tmp;
			arr.push_back(tmp);
		}
		v.push_back(tmp);
		arr.pop_back();
		if(arr.empty() && n > 1){
			cout << -1 << '\n';
			continue;
		}
		ll pointer;
		for(int i = 2; i <= n; i++){
			pointer = m - 2;
			for(int j = 2; j * j <= i; j++){
				if (i % j == 0){
					if(mp[j] == arr[pointer]){
						pointer--;
					}
					else if (mp[i / j] == arr[pointer]){
						pointer--;
					}
					if(pointer < 0){
						can = false;
						break;
					}
				}
			}
			if(!can){
				break;
			}
			v.push_back(arr[pointer]);
			mp[i] = arr[pointer];
		}
		if(can){
			for(int i = 0; i < n; i++){
				cout << v[i] << " ";
			}
		}
		else{
			cout << -1;
		}
		cout << '\n';
	}

	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 provided buggy code, we need to consider the constraints and the conditions of the problem. The buggy code seems to have issues with handling certain edge cases and specific combinations of `n` and `m`.

### Step-by-Step Analysis:
1. **Understanding the Bug**:
   - The code appears to have issues with handling cases where the set `S` has very few elements, especially when `m` is small.
   - The code also seems to have issues with the condition `a_{gcd(i, j)} ≠ gcd(a_i, a_j)`.

2. **Edge Cases**:
   - **Small `m` and `n`**: When `m` is very small, the code might struggle to find a valid array.
   - **Single Element Set**: When `m = 1`, the code should always return `-1` if `n > 1` because it's impossible to satisfy the condition.
   - **Specific Values in `S`**: Certain values in `S` might cause the code to fail, especially when the values are close to each other.

3. **Test Input**:
   - We will create a test case with a small `m` and `n` to maximize the probability of exposing the bug.
   - We will use a set `S` with values that are close to each other to test the gcd condition.

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

### Explanation:
- **Test Case 1**:
  - `n = 4`: The array length is 4.
  - `m = 2`: The set `S` has 2 elements.
  - `S = {2, 3}`: The set contains the elements 2 and 3.

### Why This Test Input?
- **Small `m` and `n`**: This makes it more likely to expose issues in the code's handling of small sets.
- **Close Values in `S`**: The values 2 and 3 are close, which can make it difficult to satisfy the gcd condition.
- **Expected Output**: The correct output for this test case should be `-1` because it's impossible to create an array of length 4 with elements from `{2, 3}` that satisfies the condition `a_{gcd(i, j)} ≠ gcd(a_i, a_j)`.

### Expected Behavior:
- The buggy code should ideally return `-1` for this test case. If it fails to do so, it indicates a bug in the logic.

This test input is designed to be small and complete, ensuring it fits within the output length limit while effectively exposing the bug in the provided code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
from math import gcd
def works(a):
    n = len(a)
    for i in range(n):
        for j in range(i+1, n):
            k = gcd(i+1, j+1) - 1
            if a[k] == gcd(a[i], a[j]):
                return False
    return True

t = int(input())
for _ in range(t):
    n, m = map(int, input().split())
    a = list(map(int, input().split()))
    if m == 1 and n > 1:
        print(-1)
        continue
    ans = [a[-1]] * n
    step = 2
    while step < n:
        for i in range(step-1, n, step):
            j = a.index(ans[i])
            ans[i] = a[j-1]
        step += 1
    if not works(ans):
        print(-1)
    else:
        print(*ans)

**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
2
4 2
1 2
3 2
1 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
from math import gcd

def solve():
    n, m = map(int, input().split(" "))
    a = input().split(" ")

    a.sort(reverse=True)

    if n == 1:
        print(a[0])
        return

    ans = [0] + [1] * (n - 1)

    for i in range(1, n + 1):
        if i * 2 <= n:
            if ans[i - 1] + 1 >= m:
                print(-1)
                return
            ans[i * 2 - 1] = ans[i - 1] + 1

    for i in range(0, n):
        print(a[ans[i]], end = " ")
    print()

def main():
    t = 1
    t = int(input())
    while t:
        t -= 1
        solve()

if __name__ == '__main__':
    main()

**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 buggy code, we need to focus on edge cases that are likely to cause issues. The main problem in the code is that it does not correctly handle the lexicographically largest array and the GCD conditions. Specifically, the code does not ensure that the GCD condition is met for all pairs, and it incorrectly assigns values to the array.

### Steps to Generate the Test Input:
1. **Single Element Set**: Test cases where the set \( S \) contains only one element. This should force the code to return -1 if \( n > 1 \) because the GCD condition cannot be satisfied.
2. **Small \( n \) and \( m \)**: Test cases with small values of \( n \) and \( m \) to ensure the code handles small inputs correctly.
3. **Large \( n \) and \( m \)**: Test cases with larger values of \( n \) and \( m \) to ensure the code can handle larger inputs and the GCD condition is met.
4. **Mixed Values in \( S \)**: Test cases with a mix of values in \( S \) to ensure the lexicographically largest array is generated correctly.

### Chosen Test Input:
Let's create a test input that includes a mix of these edge cases:

```plaintext
4
3 2
1 2
4 3
2 3 4
6 3
3 4 6
10 4
2 4 6 8
```

### Explanation:
1. **Test Case 1**:
   - \( n = 3 \), \( m = 2 \)
   - \( S = \{1, 2\} \)
   - The only possible arrays are permutations of \([1, 1, 1]\), \([1, 1, 2]\), \([1, 2, 1]\), \([1, 2, 2]\), \([2, 1, 1]\), \([2, 1, 2]\), \([2, 2, 1]\), \([2, 2, 2]\). None of these satisfy the GCD condition for all pairs, so the output should be \(-1\).

2. **Test Case 2**:
   - \( n = 4 \), \( m = 3 \)
   - \( S = \{2, 3, 4\} \)
   - The lexicographically largest array that satisfies the GCD condition is \([4, 3, 3, 2]\).

3. **Test Case 3**:
   - \( n = 6 \), \( m = 3 \)
   - \( S = \{3, 4, 6\} \)
   - The lexicographically largest array that satisfies the GCD condition is \([6, 4, 4, 3, 4, 3]\).

4. **Test Case 4**:
   - \( n = 10 \), \( m = 4 \)
   - \( S = \{2, 4, 6, 8\} \)
   - The lexicographically largest array that satisfies the GCD condition is \([8, 6, 6, 4, 6, 4, 4, 2, 4, 2]\).

This test input should expose the bug in the given buggy code by covering a variety of edge cases and ensuring that the GCD condition is met for all pairs.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
def get_primes_until(z):
	if z == 1:
		return []
	else:
		primes = [2]
		k = 3
		while k <= z:
			prime = True
			for i in range(len(primes)):
				if k%primes[i] == 0:
					prime = False
					break
			if prime:
				primes.append(k)
			k += 2
		return primes

for _ in range(int(input())):
	n, m = map(int, input().split())
	a = [int(x) for x in input().split()]
	#if first term is highest possible, everything else must be smaller
	#primes!!

	vals = [[1]]
	vals.append(get_primes_until(n))
	while len(vals[-1]) > 0:
		vals.append([])
		for i in range(len(vals[-2])):
			if vals[-2][i]*2 <= n:
				vals[-1].append(vals[-2][i]*2)
	ans = [0]*n
	yes = True
	for i in range(len(vals)):
		for j in range(len(vals[i])):
			if m-i-1 >= 0:
				ans[vals[i][j]-1] = a[m-i-1]
			else:
				yes = False
	if yes:
		print(' '.join([str(num) for num in ans]))
	else:
		print(-1)

**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
1
5 2
2 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

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

public class Main {
    public static void main(String[] args) {
 		FastScanner fs = new FastScanner();
        PrintWriter out = new PrintWriter(System.out);

        int testcases = 1;
        testcases = fs.nextInt();

        for(int testcase = 1; testcase <= testcases; testcase++) {
            new Solution(fs, out).solve(testcase);
        }

        out.close();
    }
}

class Solution {

    public final int mod = 1000000007;

	void fillAns(int ans[], int prefix, int number) {
		ans[2 * prefix] = number;
		for(int i = 3 * prefix; i < ans.length; i += 2 * prefix) {
			ans[i] = number;
		}
	}

    void solve(int testCaseNumber) {
        //Let the magic happen
		int n, m;
		n = fs.nextInt();
		m = fs.nextInt();

		List<Integer> A = fs.nextIntList(m);

		int ans[] = new int[n + 1];

		ans[1] = A.get(m - 1);

		int prefix = 1;
		int index = m - 2;
		while(prefix * 2 <= n && index >= 0) {
			fillAns(ans, prefix, A.get(index));
			prefix *= 2;
			index--;
		}

		if(prefix * 2 <= n) {
			out.println(-1);
		} else {
			for(int i = 1; i <= n; ++i) {
				out.print(ans[i] + " ");
			}
			out.println();
		}
    }

    Solution(FastScanner fs, PrintWriter out) {
        this.fs = fs;
        this.out = out;
    }
    private final FastScanner fs;
    private final PrintWriter out;
}

class Pair<U,V> {
    private U first;
    private V second;

    Pair(U first, V second) {
        this.first = first;
        this.second = second;
    }
    U first() {
        return this.first;
    }
    V second() {
        return this.second;
    }
    static <S,T> Pair<S,T> of(S s, T t) {
        return new Pair<S,T>(s, t);
    }
	@Override
	public String toString() {
		return "{\n" +
				"\tfirst: " + first.toString() + ",\n" +
			   	"\tsecond: " + second.toString() + "\n" +
				"}";
	}
}

class FastScanner {
   final private int BUFFER_SIZE = 1 << 16;
   private DataInputStream din;
   private byte[] buffer;
   private int bufferPointer, bytesRead;

   public FastScanner() {
       din = new DataInputStream(System.in);
       buffer = new byte[BUFFER_SIZE];
       bufferPointer = bytesRead = 0;
   }

   public String next(int n) {
       byte[] buf = new byte[n]; // line length
       int cnt = 0, c;
       while ((c = read()) != -1) {
           if (c <= 32) {
               if (cnt != 0) {
                   break;
               }
               else {
                   continue;
               }
           }
           buf[cnt++] = (byte)c;
       }
       return new String(buf, 0, cnt);
   }

   public int nextInt() {
       int ret = 0;
       byte c = read();
       while (c <= ' ') {
           c = read();
       }
       boolean neg = (c == '-');
       if (neg)
           c = read();
       do {
           ret = ret * 10 + c - '0';
       } while ((c = read()) >= '0' && c <= '9');

       if (neg)
           return -ret;
       return ret;
   }

   public ArrayList<Integer> nextIntList(int n) {
	ArrayList<Integer> arrList = new ArrayList<>(n);
   	for(int i = 0; i < n; ++i) {
   		arrList.add(nextInt());
   	}
   	return arrList;
   }

   public long nextLong() {
       long ret = 0;
       byte c = read();
       while (c <= ' ')
           c = read();
       boolean neg = (c == '-');
       if (neg)
           c = read();
       do {
           ret = ret * 10 + c - '0';
       } while ((c = read()) >= '0' && c <= '9');
       if (neg)
           return -ret;
       return ret;
   }

   public double nextDouble() {
       double ret = 0, div = 1;
       byte c = read();
       while (c <= ' ')
           c = read();
       boolean neg = (c == '-');
       if (neg)
           c = read();

       do {
           ret = ret * 10 + c - '0';
       } while ((c = read()) >= '0' && c <= '9');

       if (c == '.') {
           while ((c = read()) >= '0' && c <= '9') {
               ret += (c - '0') / (div *= 10);
           }
       }

       if (neg)
           return -ret;
       return ret;
   }

   private byte read() {
       if (bufferPointer == bytesRead)
           fillBuffer();
       return buffer[bufferPointer++];
   }

   private void fillBuffer() {
       try {
           bytesRead = din.read(buffer, bufferPointer = 0,
                               BUFFER_SIZE);
           if (bytesRead == -1)
               buffer[0] = -1;
       } catch (IOException e) {
           System.err.println(e.getMessage());
       }
   }

   public void close() {
       if (din == null)
           return;
       try {
	       din.close();
       } catch (IOException e) {
           System.err.println(e.getMessage());
       }
   }
}

class Helper {
    static final Random random = new Random();
    static final int mod = 1_000_000_007;

    static void ruffleSort(int[] a) {
        int n = a.length;// shuffle, then sort
        for (int i = 0; i < n; i++) {
            int oi = random.nextInt(n), temp = a[oi];
            a[oi] = a[i];
            a[i] = temp;
        }
        Arrays.sort(a);
    }

    static long add(long a, long b) {
        return (a + b) % mod;
    }

    static long sub(long a, long b) {
        return ((a - b) % mod + mod) % mod;
    }

    static long mul(long a, long b) {
        return (a * b) % mod;
    }

    static long exp(long base, long exp) {
        if (exp == 0)
            return 1;
        long half = exp(base, exp / 2);
        if (exp % 2 == 0)
            return mul(half, half);
        return mul(half, mul(half, base));
    }

	static long gcd(long a, long b) {
		if(b == 0) {
			return a;
		}
		return gcd(b % a, a);
	}

    static long[] factorials = new long[2_000_001];
    static long[] invFactorials = new long[2_000_001];

    static void precompFacts() {
        factorials[0] = invFactorials[0] = 1;
        for (int i = 1; i < factorials.length; i++)
            factorials[i] = mul(factorials[i - 1], i);
        invFactorials[factorials.length - 1] = exp(factorials[factorials.length - 1], mod - 2);
        for (int i = invFactorials.length - 2; i >= 0; i--)
            invFactorials[i] = mul(invFactorials[i + 1], i + 1);
    }

    static long nCk(int n, int k) {
        return mul(factorials[n], mul(invFactorials[k], invFactorials[n - k]));
    }
}

**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 challenging test input that exposes the bug in the given code, let's analyze the problem and the code to identify potential edge cases and scenarios where the code might fail.

### Analysis of the Problem and Code

1. **Problem Requirements:**
   - We need to construct the lexicographically largest array \( a \) such that \( a_i \in S \) for each \( 1 \le i \le n \) and \( a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j) \) for all pairs \( 1 \le i < j \le n \).

2. **Code Logic:**
   - The code initializes the array \( ans \) and sets \( ans[1] \) to the largest element in \( S \).
   - It then fills the array using the `fillAns` method, which places the next largest element in specific positions based on the prefix.

3. **Potential Bugs:**
   - The code might fail if it does not correctly handle the condition \( a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j) \).
   - The code might also fail if it does not correctly handle the case where no valid array can be constructed.

### Challenging Test Input

To expose the bug, we need to create a test case where the condition \( a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j) \) is difficult to satisfy. This can happen in several scenarios:
- When the set \( S \) contains only a few elements.
- When the set \( S \) contains elements that are multiples of each other, making it hard to satisfy the GCD condition.

### Test Input

Let's create a test input where \( S \) contains elements that are multiples of each other and see if the code can handle it correctly.

```plaintext
3
6 3
2 4 6
4 2
2 4
3 3
1 2 3
```

### Explanation

1. **Test Case 1:**
   - \( n = 6 \), \( m = 3 \), \( S = \{2, 4, 6\} \)
   - This test case is challenging because the elements in \( S \) are multiples of each other, making it hard to satisfy the GCD condition.

2. **Test Case 2:**
   - \( n = 4 \), \( m = 2 \), \( S = \{2, 4\} \)
   - This test case is also challenging because the elements in \( S \) are multiples of each other.

3. **Test Case 3:**
   - \( n = 3 \), \( m = 3 \), \( S = \{1, 2, 3\} \)
   - This test case is challenging because the elements in \( S \) are relatively prime to each other, but the small size of \( n \) and \( m \) can still make it difficult to satisfy the GCD condition.

This test input should help expose any issues in the code's logic for handling the GCD condition and constructing the lexicographically largest array.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
import java.util.*;

public class Main{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();

        outer:
        for (int tc = 0; tc < t; tc++) {
            int n = sc.nextInt();
            int m = sc.nextInt();
            int[] S = new int[m];
            for (int i = 0; i < m; i++) {
                S[i] = sc.nextInt();
            }
            Arrays.sort(S);
            reverse(S);

            int[] a = new int[n + 1];
            Arrays.fill(a, -1);

            for (int i = 1; i <= n; i++) {
                for (int x : S) {
                    boolean valid = true;
                    for (int j = 1; j < i; j++) {
                        if (x == gcd(a[gcd(i, j)], a[j])) {
                            valid = false;
                            break;
                        }
                    }

                    if (valid) {
                        a[i] = x;
                        break;
                    }
                }

                if (a[i] == -1) {
                    System.out.println(-1);
                    continue outer;
                }
            }

            for (int i = 1; i <= n; i++) {
                System.out.print(a[i] + (i == n ? "\n" : " "));
            }
        }
    }

    public static void reverse(int[] array) {
        int left = 0;
        int right = array.length - 1;
        while (left < right) {
            int temp = array[left];
            array[left] = array[right];
            array[right] = temp;
            left++;
            right--;
        }
    }

    public static int gcd(int a, int b) {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }
}

// 15hZs9LRrNMgY36FODp

**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
2
4 2
1 2
3 2
1 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

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

#define Morshedy_22 ios_base::sync_with_stdio(false);cin.tie(nullptr);
#define all(v) v.begin(),v.end()
#define rall(v) v.rbegin(),v.rend()
#define FILL(a, b) memset(a, b, sizeof(a))
#define popCnt(x) (__builtin_popcountll(x))
#define sz(x) (int)(x).size()
#define el '\n'
#define int long long

const int N = 1e5+5, mod = 1e9+7;
const int d4x[4]={-1, 0, 1, 0}, d4y[4]={0, 1, 0, -1};
const int d8x[8]={-1, -1, 0, 1, 1, 1, 0, -1}, d8y[8]={0, 1, 1, 1, 0, -1, -1, -1};

void dbg_out() { cerr << endl; }
template <typename Head, typename... Tail>
void dbg_out(Head H, Tail... T) {
    cerr << ' ' << H;
    dbg_out(T...);
}
#define dbg(...) cerr << "(" << #__VA_ARGS__ << "):", dbg_out(__VA_ARGS__)

/*
---> وتظل تسعى جاهدا في همةٍ   ..  والله يعطي من يشاءُ إذا شكر <---
*/

void testCase()
{
    int n, m;
    cin >> n >> m;

    vector<int> v(m);
    for(auto &i : v)
        cin >> i;

    sort(rall(v));

    vector<int> ans(n+1);
    vector<int> cnt(n+1, 1);

    cnt[1] = 0;
    int flag = 0;
    for(int i=2; i*i<=n; i++)
    {
        for(int j = i*i; j <= n; j+=i)
            cnt[j]++;
    }

    for(int i=1; i<=n; i++)
    {
        if(cnt[i] >= m)
            return cout << -1 << el, void();

        ans[i] = v[cnt[i]];
    }

    for(int i=1; i<=n; i++)
        cout << ans[i] << " ";
    cout << el;

}

int32_t main()
{
    Morshedy_22

    int tc = 1;
    cin >> tc;
    while(tc--)
        testCase();

    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 provided buggy code, we need to focus on edge cases and scenarios that maximize the failure probability. Let's break down the requirements and constraints:

1. **Edge Case with Small `n` and `m`:**
   - Test cases with small values of `n` and `m` can help identify issues in the logic that handles small inputs.

2. **Edge Case with Large `n` and Small `m`:**
   - Test cases with a large `n` and a small `m` can stress the algorithm and reveal issues in handling large arrays with limited choices.

3. **Edge Case with `n` and `m` Equal:**
   - Test cases where `n` and `m` are equal can help identify issues in handling the maximum number of choices.

4. **Edge Case with No Solution:**
   - Test cases where no valid solution exists can help identify issues in correctly identifying and handling such scenarios.

Given these considerations, let's generate a small-scale test input that is complete and valid:

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

### Explanation of the Test Cases:

1. **Test Case 1:**
   - `n = 3`, `m = 2`, `S = {1, 2}`
   - This test case has a small `n` and `m`. The only possible arrays are `[2, 2, 2]`, `[2, 2, 1]`, `[2, 1, 2]`, and `[1, 2, 2]`. The lexicographically largest array that satisfies the condition is `[2, 1, 2]`.

2. **Test Case 2:**
   - `n = 5`, `m = 3`, `S = {2, 3, 4}`
   - This test case has a larger `n` and a small `m`. The algorithm should be able to find a valid array, such as `[4, 3, 4, 2, 4]`.

3. **Test Case 3:**
   - `n = 7`, `m = 4`, `S = {1, 2, 3, 4}`
   - This test case has a larger `n` and a larger `m`. The algorithm should be able to find a valid array, such as `[4, 3, 4, 2, 4, 3, 4]`.

4. **Test Case 4:**
   - `n = 8`, `m = 5`, `S = {1, 2, 3, 4, 5}`
   - This test case has a larger `n` and a larger `m`. The algorithm should be able to find a valid array, such as `[5, 4, 5, 3, 5, 4, 5, 2]`.

These test cases are designed to cover a range of scenarios and should help expose any 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: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

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

typedef long long ll;
typedef unsigned long long ull;
typedef pair<int,int> pi;
typedef pair<long long,long> pl;

#define all(s) s.begin(),s.end()
#define F first
#define S second
#define sz(a) int(a.size())

const ll mod = 1e9+7;
const ll INF = 1e18;
const int N = 1e5+200;
const int inf = 1e9+200;

ll binpow(ll a, ll b, ll m) {
    a %= m;
    ll res = 1;
    while (b > 0) {
        if (b & 1)
            res = res * a % m;
        a = a * a % m;
        b >>= 1;
    }
    return res;
}

map<int,int> was;

void solve() {
    int n,m;
    cin>>n>>m;
    int s[m+3];
    for(int i=1;i<=m;++i) {
        cin>>s[i];
    }
    if(m<floor(log2(n))+1) {
        cout<<"-1\n";
        return;
    }
    s[0]=0;
    sort(s+1,s+1+m);
    vector<int> ans(n+1);
    ans[1]=s[m];
    ll i=1;
    was[s[m]]=1;
    int l=0;
    for(int k=m-1;k>=1;--k) {
        if(i*2>n) {
            l=k;
            break;
        }
        was[s[k]]=i*2;
        i*=2;
    }
    s[0]=0;
    for(int i=m;i>max(l-1,0);--i) {
        for(int j=was[s[i]]*2;j<=n;j+=was[s[i]]) {
            if(was[s[i]]==0) {
                break;
            }
            ans[j]=s[i-1];
        }
    }/*
    for(auto [u,x]:was) {
        cout<<u<<" "<<x<<'\n';
    }*/
    for(int i=1;i<=n;++i) {
        if(i!=was[ans[i]]&&i%was[ans[i]]==0) {
            cout<<"error\n";
            continue;
        }
        cout<<ans[i]<<" ";
    }
    was.clear();
    cout<<'\n';
}

int main() {
    ios_base::sync_with_stdio(0) , 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:
```plaintext
3
4 2
2 3
5 2
2 4
6 3
2 3 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
t = int(input())

for _ in range(t):
    n, m = map(int, input().split())

    primes = [1 for i in range(n + 3)]
    for i in range(2, n):
        if primes[i] == 1:
            for j in range(i+i, len(primes), i):
                primes[j] = 0
    primes[1] = 0

    a = list(map(int, input().split()))

    if n == 1 and m == 1:
        print(a[0])
        continue
    elif n > 1 and m == 1:
        print(-1)
        continue
    a = list(reversed(a))
    ans = [0 for i in range(n+1)]

    ans[1] = a[0]
    curs = [0 for i in range(n+1)]
    curs[1]=0
    for i in range(2, n+1):
        if primes[i] == 1:
            ans[i] = a[1]
            continue
        if primes[i] == 0:
            flag = False
            cur = 1
            for j in range(2, i):
                if i % j == 0:
                    cur = max(cur, curs[i//j])
            cur += 1
            if cur >= m:
                ans = [-1]
                break

            ans[i] = a[cur]
    if ans == [-1]:
        print(-1)
        continue
    for i in ans[1:]:
        print(i, end=" ")
    print()

**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
2
4 2
2 4
3 3
1 2 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
I = lambda : list(map(int, input().split(' ')))
R = lambda : (int(input()))

import sys
import math
from collections import deque, defaultdict
from bisect import bisect_left, bisect_right

def get_primes(n):

    a = []
    b = [True for i in range(n+1)]
    for i in range(2,int(n**0.5)+1):
        if b[i] == False: continue
        if b[i] == True:
            for j in range(2*i, n+1, i): b[j] = False

    for i in range(2,n+1):
        if b[i] == True: a.append(i)

    return a

for kp in range(R()):

    n, m = I()
    a = I()
    a.sort()
    primes = get_primes(n)[::1]

    # initialization
    indices = set(list(range(2,n+1)))
    res = [-1 for i in range(n)]
    res[0] = a.pop()
    factor = 1
    flag = 1
    pr = 0

    # try :

    while len(indices) > 0 and flag != -1:

        l = []
        curr_num = None

        for i in primes:

            if (factor * i) > n: break

            elif (factor * i) in indices :

                if curr_num == None:
                    if len(a) > 0 : curr_num = a.pop()
                    else: flag = -1 ;  break

                res[ (factor*i) - 1 ] = curr_num ; l.append(factor*i)

        for i in l: indices.remove(i)

        # print(indices, factor, primes)
        factor += 1
        # pr += 1

    if flag != -1: print(*res)
    else: print(-1)

    # except Exception as e:
    #     print(n)

    # two_power = 1
    # # iteration
    # while len(indices) > 0:
    #
    #     l = []
    #     curr_ind = pow(2,two_power)
    #     curr_num = a.pop()
    #
    #     res[curr_ind - 1] = curr_num ; indices.remove(curr_ind)
    #
    #     for i in indices:
    #         if i % curr_ind != 0:
    #             res[i-1] = curr_num
    #             l.append(i)
    #
    #     for i in l: indices.remove(i)
    #     two_power += 1
    #
    # print(*res)

**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
2
3 3
1 2 3
4 2
1 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
import java.util.*;

public class C {
	static boolean isPowerofTwo(int n)
    {
        if (n == 0)
            return false;
        if ((n & (~(n - 1))) == n)
            return true;
        return false;
    }
	static int highestPowerof2(int x)
    {

        x |= x >> 1;
        x |= x >> 2;
        x |= x >> 4;
        x |= x >> 8;
        x |= x >> 16;

        return x ^ (x >> 1);

    }
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int t1 = sc.nextInt();
		StringBuilder sb = new StringBuilder();

		while (t1-- > 0) {
			int n = sc.nextInt();
			int m = sc.nextInt();
			PriorityQueue<Integer> queue = new PriorityQueue<Integer> (Collections.reverseOrder());
			for(int i=0;i<m;i++) {
				queue.add(sc.nextInt());
			}
			int arr[] = new int[n+1];
			boolean res = true;
			for(int i=1;i<=n;i++) {
				if(isPowerofTwo(i)) {
					if(queue.size()==0) {
						res = false;
						break;
					}
					else {
						arr[i] = queue.remove();
					}
				}
			}
			if(!res)
				sb.append("-1\n");
			else {
				for(int i=3;i<=n;i+=2) {
					arr[i] = arr[2];
				}
				if(n>4) {
					int temp = 4;
					for(int i= temp; i<=highestPowerof2(n);i=i*2) {
						for(int j=i;j<=n;j+=2) {
							if(j%i!=0 && arr[j]==0) {
								arr[j] = arr[i];
							}
						}
					}
				}
				for(int i=1;i<=n;i++) {
					sb.append(arr[i]+" ");
				}
				sb.append("\n");
			}
		}

		System.out.print(sb.toString());
		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
2
8 4
1 2 3 4
6 3
1 2 4
```

### Explanation:

1. **Test Case 1:**
   - **Input:** `8 4`
   - **Set S:** `{1, 2, 3, 4}`
   - **Challenge:** This test case includes a set with multiple elements and a relatively large `n`. The challenge here is to ensure that the algorithm correctly handles the constraints for all pairs `(i, j)` and finds a lexicographically largest array that satisfies the conditions. The set includes powers of 2 and other numbers, which can be tricky for the algorithm to handle.

2. **Test Case 2:**
   - **Input:** `6 3`
   - **Set S:** `{1, 2, 4}`
   - **Challenge:** This test case is designed to test the algorithm's ability to handle a set that includes only powers of 2. The challenge is to ensure that the algorithm correctly fills the array while avoiding the condition where `a_gcd(i, j) == gcd(a_i, a_j)`. This can be particularly difficult because powers of 2 have specific properties that can lead to easy violations of the condition.

### Expected Output:
For the given test cases, the expected output should be:
```plaintext
4 3 3 2 3 2 2 1
-1
```

### Reasoning:
- **Test Case 1:** The lexicographically largest array that satisfies the conditions is `[4, 3, 3, 2, 3, 2, 2, 1]`.
- **Test Case 2:** No such array exists because the set `{1, 2, 4}` and the constraints make it impossible to satisfy the condition for all pairs `(i, j)`.

These test cases are designed to expose potential bugs in the algorithm, especially in handling powers of 2 and ensuring the lexicographical order.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
import java.util.*;

public class C {
	static boolean isPowerofTwo(int n)
	{
		if (n == 0)
			return false;
		if ((n & (~(n - 1))) == n)
			return true;
		return false;
	}
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int t1 = sc.nextInt();
		StringBuilder sb = new StringBuilder();

		while (t1-- > 0) {
			int n = sc.nextInt();
			int m = sc.nextInt();
			PriorityQueue<Integer> queue = new PriorityQueue<Integer> (Collections.reverseOrder());
			for(int i=0;i<m;i++) {
				queue.add(sc.nextInt());
			}
			int arr[] = new int[n+1];
			boolean res = true;
			for(int i=1;i<=n;i++) {
				if(isPowerofTwo(i)) {
					if(queue.size()==0) {
						res = false;
						break;
					}
					else {
						arr[i] = queue.remove();
					}
				}
			}
			if(!res)
				sb.append("-1\n");
			else {
				for(int i=3;i<=n;i+=2) {
					arr[i] = arr[2];
				}
				for(int i=4;i<=n;i=i*2) {
					for(int j=i;j<=n;j+=2) {
						if(j%i!=0 && arr[j]==0) {
							arr[j] = arr[i];
						}
					}
				}
				for(int i=1;i<=n;i++) {
					sb.append(arr[i]+" ");
				}
				sb.append("\n");
			}
		}

		System.out.print(sb.toString());
		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
1
5 3
1 2 3
```

### Explanation:
This test case is designed to expose the bug in the buggy code by focusing on a small but critical edge case.

1. **Test Case Size**: The test case is small, with \( n = 5 \) and \( m = 3 \), making it easy to verify manually and ensuring that the output is within the length limit.

2. **Set \( S \)**: The set \( S \) contains the integers \( \{1, 2, 3\} \). This set is chosen to test the algorithm's ability to handle a simple but non-trivial set of integers.

3. **GCD Conditions**: The test case is designed to check if the algorithm can correctly handle the GCD conditions. Specifically:
   - For \( i = 1 \) and \( j = 2 \), \( \gcd(a_1, a_2) \) should not equal \( a_{\gcd(1, 2)} = a_1 \).
   - For \( i = 1 \) and \( j = 3 \), \( \gcd(a_1, a_3) \) should not equal \( a_{\gcd(1, 3)} = a_1 \).
   - For \( i = 1 \) and \( j = 4 \), \( \gcd(a_1, a_4) \) should not equal \( a_{\gcd(1, 4)} = a_1 \).
   - For \( i = 1 \) and \( j = 5 \), \( \gcd(a_1, a_5) \) should not equal \( a_{\gcd(1, 5)} = a_1 \).
   - For \( i = 2 \) and \( j = 4 \), \( \gcd(a_2, a_4) \) should not equal \( a_{\gcd(2, 4)} = a_2 \).

4. **Buggy Code Analysis**:
   - The buggy code incorrectly handles the GCD conditions. Specifically, it only checks if \( i \) is a power of two and assigns the largest available value from the set \( S \) to such indices. This approach is insufficient to ensure the GCD condition for all pairs \( (i, j) \).

5. **Expected Output**:
   - The correct output for this test case should be a lexicographically largest array that satisfies the GCD conditions, or \(-1\) if no such array exists. The buggy code is likely to fail this test case because it does not properly handle the GCD conditions for all pairs.

This test case is designed to be challenging and to expose the limitations of the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Shohag Loves GCD
time_limit_ms: 2000
memory_limit_mb: 256
Description: Shohag has an integer $$$n$$$ and a set $$$S$$$ of $$$m$$$ unique integers. Help him find the lexicographically largest$$$^{\text{∗}}$$$ integer array $$$a_1, a_2, \ldots, a_n$$$ such that $$$a_i \in S$$$ for each $$$1 \le i \le n$$$ and $$$a_{\operatorname{gcd}(i, j)} \neq \operatorname{gcd}(a_i, a_j)$$$$$$^{\text{†}}$$$ is satisfied over all pairs $$$1 \le i \lt j \le n$$$, or state that no such array exists.

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le m \le n \le 10^5$$$).

The second line contains $$$m$$$ unique integers in increasing order, representing the elements of the set $$$S$$$ ($$$1 \le x \le n$$$ for each $$$x \in S$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.

Output: For each test case, if there is no solution print $$$-1$$$, otherwise print $$$n$$$ integers — the lexicographically largest integer array that satisfies the conditions.

Examples:
input:
3
6 3
3 4 6
1 1
1
2 1
2
output:
6 4 4 3 4 3
1
-1

Note: In the first test case, every element in the array belongs to the given set $$$S = \{3, 4, 6\}$$$, and all pairs of indices of the array satisfy the necessary conditions. In particular, for pair $$$(2, 3)$$$, $$$a_{\operatorname{gcd}(2, 3)} = a_1 = 6$$$ and $$$\operatorname{gcd}(a_2, a_3) = \operatorname{gcd}(4, 4) = 4$$$, so they are not equal. There are other arrays that satisfy the conditions as well but this one is the lexicographically largest among them.

In the third test case, there is no solution possible because we are only allowed to use $$$a = [2, 2]$$$ but for this array, for pair $$$(1, 2)$$$, $$$a_{\operatorname{gcd}(1, 2)} = a_1 = 2$$$ and $$$\operatorname{gcd}(a_1, a_2) = \operatorname{gcd}(2, 2) = 2$$$, so they are equal which is not allowed!

Buggy Code:
#include <bits/stdc++.h>
#define all(c) c.begin(), c.end()
#define sz(c) ll(c.size())
#define ll long long
#define tr(a, it) for (auto it = a.begin(); it != a.end(); it++)
#define present(c, x) (c.find(x) != c.end())
#define cpresent(c, x) (find(all(c), x) != c.end())
#define vi vector<int>
#define vll vector<ll>
#define vvll vector<vll>
#define vpii vector<pair<int, int>>
#define vpll vector<pair<ll, ll>>
#define pll pair<ll, ll>
#define inf 1e9
#define MOD 1000000007
#define MAXN 10000
#define madar 1e18
#define moda 998244353
#define loop(i, s, e) for (ll i = s; i < e; i++)
#define revloop(i, s, e) for (ll i = s; i >= e; i--)
#define newl cout << endl
#define pb push_back
#define f first
#define s second

using namespace std;

////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

// DEBUG GG
#define debug(x)     \
    cout << #x << " "; \
    _print(x);         \
    cout << endl;

void _print(int x) { cout << x; }
void _print(char x) { cout << x; }
void _print(string s) { cout << s; }
void _print(uint64_t s) { cout << s; }
void _print(ll x) { cout << x; }
void _print(bool a) { cout << a; }

template <class T, class V>
void _print(pair<T, V> p);
template <class T>
void _print(vector<T> v);
template <class T>
void _print(set<T> v);
template <class T, class V>
void _print(map<T, V> v);
template <class T>
void _print(multiset<T> v);

template <class T, class V>
void _print(pair<T, V> p) {
    cout << "{";
    _print(p.first);
    cout << ",";
    _print(p.second);
    cout << "}";
}
template <class T>
void _print(vector<T> v) {
    cout << "[ ";
    for (T i : v) {
        _print(i);
        cout << " ";
    }
    cout << "]";
}
template <class T>
void _print(set<T> st) {
    cout << "[ ";
    for (T i : st) {
        _print(i);
        cout << " ";
    }
    cout << "]";
}
template <class T>
void _print(multiset<T> v) {
    cout << "[ ";
    for (T i : v) {
        _print(i);
        cout << " ";
    }
    cout << "]";
}
template <class T, class V>
void _print(map<T, V> v) {
    cout << "[ ";
    for (auto i : v) {
        _print(i);
        cout << " ";
    }
    cout << "]";
}

/////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

ll gcd(ll a, ll b){
    if(!(a%b)) return b;
    return gcd(b, a%b);
}

ll lcm(ll a, ll b) {
    return (a*b)/gcd(a,b);
}

ll MSB(ll n)
{
    ll ans = 0;
    while (n) {
        ans++;
        n /= 2;
    }
    return ans;
}

ll power(ll a, ll b, ll mod) { // to calculate (a^b)%mod
    ll p = 1;
    while (b > 0) {
        if (b & 1)
            p = (p * a) % mod;
        a = (a * a) % mod;
        b = b >> 1;
    }
    return p;
}

ll inv_mod(ll n, ll mod) {
    return power(n, mod - 2, mod);
}

///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// MAX, MIN SEGMENT TREE

vector<pair<ll,ll> > t(4*MAXN); // t is the segment tree, stores max, min in range

void build(vll a, ll v, ll tl, ll tr) {
    if (tl == tr) {
        t[v].first = a[tl];
        t[v].second = a[tl];
    } else {
        ll tm = (tl + tr) / 2;
        build(a, v*2, tl, tm);
        build(a, v*2+1, tm+1, tr);
        t[v].first = max(t[v*2].first , t[v*2+1].first);
        t[v].second = min(t[v*2].second , t[v*2+1].second);
    }
}

ll get_max(ll v, ll tl, ll tr, ll l, ll r) {
    if (l > r)
        return 0;
    if (l == tl && r == tr) {
        return t[v].first;
    }
    ll tm = (tl + tr) / 2;
    return max( get_max(v*2, tl, tm, l, min(r, tm)) , get_max(v*2+1, tm+1, tr, max(l, tm+1), r) ) ;
}

ll get_min(ll v, ll tl, ll tr, ll l, ll r) {
    if (l > r)
        return 0;
    if (l == tl && r == tr) {
        return t[v].second;
    }
    ll tm = (tl + tr) / 2;
    return min( get_min(v*2, tl, tm, l, min(r, tm)) , get_min(v*2+1, tm+1, tr, max(l, tm+1), r) ) ;
}

///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
// DIJKSTRA

void dijkstra(ll s, vector<ll> & d, vector<vector<pair<ll, ll>>> & adj) {
    ll n = adj.size();
    d.assign(n, LLONG_MAX);

    d[s] = 0;
    priority_queue<pll, vector<pll>, greater<pll>> q;
    q.push({0, s});
    while (!q.empty()) {
        ll v = q.top().second;
        ll d_v = q.top().first;
        q.pop();
        if (d_v != d[v])
            continue;

        for (auto edge : adj[v]) {
            ll to = edge.first;
            ll len = edge.second;

            if (d[v] + len < d[to]) {
                d[to] = d[v] + len;
                q.push({d[to], to});
            }
        }
    }
}
///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

void solve(int tc) {
    ll n,m; cin >> n >> m;
    vll a(m);
    loop(i,0,m) cin >> a[i];
    sort(all(a));
    // 6 4 4 3 4 3 4 3 4 3
    // 6 4
    if(m>=3 || (m==2 && n<=2) || (m==1 && n==1)){
        vll b(n);
        if(n==1) cout << a[0] << endl;
        else if(n==2){
            cout << a[m-1] << " " << a[m-2] << endl;
        }else{
            b[0] = a[m-1]; b[1] = b[2] = a[m-2];
            for(ll i=3; i<n; i++){
                if(i%2)
                    b[i] = a[m-3];
                else
                    b[i] = a[m-2];
            }
            for(auto i: b) cout << i << " ";
            newl;
        }
    }else{
        cout << -1 << endl;
    }

    // if(m==2 && n>3) false;

    // if(m==1 && n>1) false;

}

int main() {
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t;
    cin >> t;
    int tc = 0;
    while (t--) {
        tc++;
        solve(tc);
    }
    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
6 3
3 4 6
1 1
1
2 1
2
```