**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

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

int main() {
    int t;
    cin >> t;
    while(t--) {
        int n, x;
        cin >> n >> x;
        vector<int> v1(n);
        priority_queue<int> pq;
        for(int i = 0; i < n; i++) {
            cin >> v1[i];
            pq.push(v1[i]);
        }

        int cus = 0;
        vector<int> v;

        while(!pq.empty()) {

            int groupSize = min(x, (int)pq.size());
            for(int i = 0; i < groupSize; i++) {
                v.push_back(pq.top());
                pq.pop();
            }
            cus=cus+v[groupSize-1];

            for(int i = 0; i < groupSize - 1; i++) {
                int diff = v[i] - v[groupSize - 1];
                if(diff != 0) {
                    pq.push(diff);
                }
            }

            v.clear();
        }

        cout << cus << endl;
    }

    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 can maximize the failure probability. Here are the steps to consider:

1. **Edge Cases for `x`**: The value of `x` (the maximum number of cars a customer can buy) should be tested with values that are both small and large relative to the number of models `n`.

2. **Edge Cases for `n`**: The number of models `n` should be tested with both small and large values to see how the code handles different sizes of input.

3. **Edge Cases for `a_i`**: The number of cars for each model should be varied to include cases where some models have a large number of cars and others have a small number of cars.

4. **Special Cases**: Consider cases where all models have the same number of cars, or where there is a single model with a very large number of cars.

Given these considerations, a small-scale test input that can expose the bug might look like this:

```plaintext
3
1 1
10
2 1
1 1
3 3
1 2 3
```

### Explanation:
1. **Test Case 1**:
   - `n = 1`, `x = 1`
   - `a_1 = 10`
   - This tests the scenario where there is only one model of car and the customer can only buy one car at a time. The expected output is `10` because it takes 10 customers to sell all 10 cars.

2. **Test Case 2**:
   - `n = 2`, `x = 1`
   - `a_1 = 1`, `a_2 = 1`
   - This tests the scenario where there are two models of cars and the customer can only buy one car at a time. The expected output is `2` because it takes 2 customers to sell all 2 cars.

3. **Test Case 3**:
   - `n = 3`, `x = 3`
   - `a_1 = 1`, `a_2 = 2`, `a_3 = 3`
   - This tests the scenario where there are three models of cars and the customer can buy up to 3 cars at a time. The expected output is `2` because:
     - Customer 1 can buy 1 car from model 1, 1 car from model 2, and 1 car from model 3.
     - Customer 2 can buy the remaining 2 cars from model 3 and 1 car from model 2.

This test input is small and valid, and it should help expose the bug 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: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define int long long
#define cin(v) for(auto &i:v)cin>>i
#define cout(v) for(auto &i:v)cout<<i<<' '
#define all(v) v.begin(),v.end()
#define vc vector
typedef long double ld;
const ld esp=1e-6;
const int N=1e6+5;
signed main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    cout.tie(NULL);
    int t;
    cin>>t;
    while(t--){
        int n,x;cin>>n>>x;
        vector<int>v(n);
        cin(v);
        sort(v.rbegin(),v.rend());
        cout<<max(v[0],n)<<endl;
    }
    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 scenarios where the code's logic fails to correctly determine the minimum number of customers needed to sell all the cars. The buggy code incorrectly outputs `max(v[0], n)` as the result, which is not the correct approach.

### Key Points to Consider:
1. **Large Values of `a_i`**: The code might fail when there are large numbers of cars in a single model.
2. **Small `x` Values**: The code might fail when the maximum number of cars a customer can buy is small.
3. **Multiple Models with Similar Numbers of Cars**: The code might fail when there are multiple models with similar numbers of cars, making it difficult to distribute the cars efficiently.

### Challenging Test Input:
Let's create a test case with a large number of cars in one model and a small value of `x` to see if the code fails to correctly determine the minimum number of customers.

```plaintext
1
5 2
1000000 1 1 1 1
```

### Explanation:
- **Test Case 1**:
  - `n = 5`: There are 5 different models of cars.
  - `x = 2`: Karel can convince each customer to buy up to 2 cars.
  - `a_1 = 1000000`, `a_2 = 1`, `a_3 = 1`, `a_4 = 1`, `a_5 = 1`: There are 1,000,000 cars of the first model and 1 car each of the other models.

### Expected Output:
- The correct output should be `500003` because:
  - Karel needs 1 customer to sell the 2 cars of models 2 and 3.
  - Karel needs 1 customer to sell the 2 cars of models 4 and 5.
  - Karel needs 500000 customers to sell all 1,000,000 cars of model 1 (since each customer can only buy 2 cars).

### Buggy Code Output:
- The buggy code will output `max(1000000, 5)`, which is `1000000`.

This test case should expose the bug in the code, as the buggy code's output is clearly incorrect.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
for _ in range(int(input())):
    n, x = map(int, input().split())
    arr = list(map(int, input().split()))
    arr.sort()
    ans = 0
    while len(arr) > 0:
        ans += arr[-1]
        k = arr[-1]
        l = min(x, len(arr))
        for i in range(l):
            arr[-i-1] -= k
        left = []
        for _ in range(l):
            k = arr.pop()
            if k < 0:
                left.append(-k)
        while left != []:
            k = min(len(left), len(arr))
            for i in range(k):
                arr[-i-1] -= left[-i-1]
            left = []
            remove = []
            for i in range(k):
                if arr[-i-1] < 0:
                    left.append(arr[-i-1])
                if arr[-i-1] <=0:
                    remove.append(i)
            remove = remove[::-1]
            for i in remove:
                arr.pop(-i-1)
            left.sort()
        arr.sort()

    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
1
5 3
1 1 1 1 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import heapq

def solve():
    n, k = map(int, input().split())
    arr = list(map(int, input().split()))

    ans = max(arr)
    ans = max(n , ans )
    print(ans)

def main():
    t = int(input())
    for _ in range(t):
        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 generate a test input that exposes the bug in the given buggy code, we need to focus on the following points:

1. **Understanding the Bug**: The buggy code calculates the answer as the maximum of the largest number of cars of any single model (`max(arr)`) and the number of models (`n`). This approach is incorrect because it does not consider the actual constraints of the problem, such as the maximum number of cars a customer can buy (`x`).

2. **Edge Cases**:
   - **Single Model**: If there is only one model of cars, the number of customers needed is simply the number of cars divided by `x`, rounded up.
   - **Multiple Models with Few Cars**: If there are multiple models but each model has only a few cars, the number of customers needed is the total number of cars divided by `x`, rounded up.
   - **Large Number of Models with Few Cars**: If there are many models with a small number of cars, the number of customers needed is the total number of cars divided by `x`, rounded up.
   - **Mixed Scenarios**: A mix of models with different numbers of cars can help test the algorithm's ability to handle a variety of inputs.

3. **Test Input**:
   - We will create a test case with multiple models where the number of cars in each model is such that the total number of cars is not divisible by `x`, and the number of models is less than the maximum number of cars in any single model.

Here is a small-scale test input that should expose the bug:

```plaintext
3
5 2
1 2 3 4 5
3 1
10 20 30
7 4
1 1 1 1 1 1 1
```

### Explanation:
1. **Test Case 1**:
   - `n = 5`, `x = 2`
   - `a = [1, 2, 3, 4, 5]`
   - The total number of cars is 15. The minimum number of customers needed is 8 (since each customer can buy up to 2 cars, and 15 / 2 = 7.5, rounded up to 8).

2. **Test Case 2**:
   - `n = 3`, `x = 1`
   - `a = [10, 20, 30]`
   - The total number of cars is 60. The minimum number of customers needed is 60 (since each customer can buy only 1 car).

3. **Test Case 3**:
   - `n = 7`, `x = 4`
   - `a = [1, 1, 1, 1, 1, 1, 1]`
   - The total number of cars is 7. The minimum number of customers needed is 2 (since each customer can buy up to 4 cars, and 7 / 4 = 1.75, rounded up to 2).

This test input should highlight the issues with the buggy code, which incorrectly calculates the number of customers needed.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
for i in range(int(input())):
  a,b = map(int,input().split())
  l = sorted(map(int,input().split()))
  l = l[::-1]
  c = 0
  v = sum(l)
  while v != 0:
    c+=1
    h = 0
    for i in range(a):
      if h == b:
        break
      if l[i] != 0:
        h+=1
        l[i]-=1
        v-=1;
  print(c)

**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
5 5
10 10 10 10 10
10 1
1 1 1 1 1 1 1 1 1 1
10 10
1 2 3 4 5 6 7 8 9 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import java.util.*;
public class B_Kar_Salesman{
  public static void main(String[] args) {
    Scanner s = new Scanner(System.in);
    int t=s.nextInt();
    int sum = 0;
    int cnt=0;
    for(int i=1; i<=t; i++){
          int n=s.nextInt();
          int x=s.nextInt();
          int[] a = new int[n];
          for(int j=0; j<n; j++){
            a[j] = s.nextInt();
          }
          sum = 0;
          cnt=0;
          for(int l=0; l<n; l++){   //sum lena hai
            sum=sum+a[l];
          }

          int max = a[0];
          for(int l=0; l<n; l++){  //max nikalo
            if(a[l]>max)max=a[l];
          }

          Arrays.sort(a);

          int spots = max * x;

          if(spots>=sum){
            System.out.println(max);

          }else {
            int left = sum - spots;
            int ad = left/x;
            System.out.println(max+ad+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:
To generate a challenging test input that exposes the bug in the given buggy code, let's analyze the code and identify the potential issues:

1. **Incorrect Calculation of `spots`**:
   - The code calculates `spots` as `max * x`, which is incorrect. The correct approach should consider the sum of the top `x` values in the array after sorting.

2. **Incorrect Calculation of `left`**:
   - The code calculates `left` as `sum - spots`, and then calculates `ad` as `left / x`. The final output is `max + ad + 1`, which is also incorrect. The correct approach should be to determine the number of customers needed to cover the remaining cars after the initial `spots`.

3. **Edge Cases**:
   - Cases where `x` is 1.
   - Cases where the number of cars of each model is very large.
   - Cases where the number of models is very large.
   - Cases where the number of cars of each model is the same.

Given these issues, a small-scale test input that can expose the bug is:

```plaintext
3
5 3
1 2 3 4 5
3 2
1 1 1
4 2
10 10 10 10
```

### Explanation:

1. **Test Case 1:**
   - `n = 5`, `x = 3`
   - `a = [1, 2, 3, 4, 5]`
   - The correct number of customers needed is 3 (one customer buys 3 cars from the top 3 models, and two more customers each buy 2 cars from the remaining models).

2. **Test Case 2:**
   - `n = 3`, `x = 2`
   - `a = [1, 1, 1]`
   - The correct number of customers needed is 2 (one customer buys 2 cars from any 2 models, and one more customer buys the remaining car).

3. **Test Case 3:**
   - `n = 4`, `x = 2`
   - `a = [10, 10, 10, 10]`
   - The correct number of customers needed is 10 (each customer buys 2 cars from any 2 models).

This test input should help expose the bugs in the given buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import java.util.*;
public class B_Kar_Salesman{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-->0)
        {
            int n = sc.nextInt();
            int k = sc.nextInt();
            int[] arr = new int[n];
            int max = 0;
            int min = Integer.MAX_VALUE;
            int sum = 0;

            for (int i = 0; i < n; i++) {
                arr[i] = sc.nextInt();
                if (arr[i] > max) {
                    max = arr[i];
                }
                if (arr[i] < min) {
                    min = arr[i];
                }
                sum+=arr[i];
            }

            if(max>=sum-max){
                System.out.println(max);
            }
            else{
                System.out.println(max+min);
            }
        }
    }
}

**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 3
1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
using ll=long long ;
#define all(x) (x).begin(),(x).end()
#define rall(v) v.rbegin(),v.rend()
#define IOS ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define sz(x) (int)(x.size())
#define db(...) ZZ(#__VA_ARGS__, __VA_ARGS__)
template <typename Arg1> void ZZ(const char* name, Arg1&& arg1){std::cerr << name << '=' << arg1 << endl;}
template <typename Arg1, typename... Args>void ZZ(const char* names, Arg1&& arg1, Args&&... args)

{
const char* comma = strchr(names + 1, ',');
std::cerr.write(names, comma - names) << '=' << arg1;
ZZ(comma, args...);
}

const int mod=(int)(1e9)+7;
const int inf=(int)(2e9);

void testCases() {
       int n,x;
       cin>>n>>x;
       vector<int> nums(n);
       for(int &val:nums) cin>>val;
       priority_queue<int> pq;
       for(int i=0;i<n;i++){
           pq.push(nums[i]);
       }
       ll ans=0;
       while (!pq.empty())
       {
            vector<int> sale;
            int cnt=min(x,sz(pq));
            int i=0;
            while (i<cnt)
            {
                int ele=pq.top();
                pq.pop();
                sale.push_back(ele);
                i++;
            }
            i=0;
            while(i<cnt-1 && (sale[i]-sale[cnt-1])!=0){
                pq.push(sale[i]-sale[cnt-1]);
                i++;
            }
            ans+=sale[cnt-1];

       }

       cout<<ans<<'\n';

}

int32_t main()
{
IOS;
#ifndef ONLINE_JUDGE
   freopen("in.txt","r",stdin);
   freopen("out.txt","w",stdout);
#endif
// comment for single Test Case
int t=1;
 cin>>t;
 for(int T=1;T<=t;T++)
 {

   // cout << "Case #" << T << ": ";
   testCases();
}

return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, let's consider the logic and potential issues in the code. The code uses a priority queue to handle the largest numbers of cars of each model and tries to distribute them among customers. The main issue in the code is that it doesn't handle the distribution of cars correctly when the number of cars of different models is not uniform.

### Step-by-Step Analysis:
1. **Priority Queue Usage**: The code uses a priority queue to always process the models with the most cars first. This is a good approach but needs careful handling of the distribution.
2. **Sale Distribution**: The code tries to distribute the cars in a way that it subtracts the smallest number of cars from the larger ones and pushes the remaining back into the priority queue. This can lead to incorrect results if the distribution is not handled correctly.
3. **Edge Cases**: The code might fail when:
   - There are a large number of models with a small number of cars and a few models with a large number of cars.
   - The number of cars in each model is not evenly distributed.
   - The value of `x` is close to `n` or much smaller than `n`.

### Challenging Test Input:
To create a test input that maximizes the failure probability, we can use a scenario where the number of cars in each model is not evenly distributed and `x` is smaller than the number of models. This will force the code to handle the distribution more carefully.

```plaintext
1
5 2
10 9 8 7 6
```

### Explanation:
- **Number of Test Cases**: 1
- **Number of Models (n)**: 5
- **Maximum Cars per Customer (x)**: 2
- **Number of Cars per Model**: 10, 9, 8, 7, 6

### Expected Behavior:
- The code should handle the distribution of cars in such a way that it minimizes the number of customers needed.
- For the given input, the correct output should be 17, as Karel can distribute the cars as follows:
  - Customer 1: 2 cars (10 and 9)
  - Customer 2: 2 cars (8 and 7)
  - Customer 3: 2 cars (6 and 8)
  - Customer 4: 2 cars (7 and 6)
  - Customer 5: 2 cars (6 and 6)
  - Customer 6: 2 cars (6 and 6)
  - Customer 7: 2 cars (6 and 6)
  - Customer 8: 2 cars (6 and 6)
  - Customer 9: 2 cars (6 and 6)
  - Customer 10: 2 cars (6 and 6)
  - Customer 11: 2 cars (6 and 6)
  - Customer 12: 2 cars (6 and 6)
  - Customer 13: 2 cars (6 and 6)
  - Customer 14: 2 cars (6 and 6)
  - Customer 15: 2 cars (6 and 6)
  - Customer 16: 2 cars (6 and 6)
  - Customer 17: 2 cars (6 and 6)

This input should expose the bug in the code if it incorrectly handles the distribution of cars.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
/*                  Bismillahir Rahmanir Rahim.
                       I Like failure.
                       I got my mind full of unsaid things.
                                                         */
   #include<bits/stdc++.h>

   using namespace std;
   #define int long long int
   const int N = 2e5 + 9,
     mod = 1e9 + 7;
   void test() {
     int n;
     cin >> n;
     int q;
     cin >> q;
     multiset < int, greater < int >> st;
     map < int, int > mp;
     for (int i = 0; i < n; i++) {
       int x;
       cin >> x;
       st.insert(x);
       mp[x]++;
     }
     int ans = 0;

     while (!st.empty()) {
       auto it = --st.end();
       int en = * it;
       mp[en]--;
       if (mp[en] == 0) st.erase(en);
       int ct = 1;
       ans += en;
       std::vector < int > v;
       auto sit = st.begin();
       while (sit != st.end()) {
          if (ct >= q) break;
         ct++;
         int x = * sit;
         v.push_back(x);
         sit++;

       }
       for (auto i: v) {
         mp[i]--;
         if (mp[i] == 0) st.erase(i);
         if (i > en) {
           st.insert(i - en);
           mp[i - en]++;
         }
       }
     }
     cout << ans << endl;

   }
   signed main() {
     ios_base::sync_with_stdio(0);
     cin.tie(0);
     int tt = 1, cs = 1;
     cin >> tt;
     while (tt--) {
       // cout<<"Case "<<cs<<": ";
       test();
       cs++;

     }
     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 following:

1. **Edge Cases**: Test cases with extreme values, such as the maximum number of models and the maximum number of cars per model.
2. **Special Cases**: Test cases where the number of cars per model is very close to each other or where the number of cars per model is highly varied.
3. **Boundary Conditions**: Test cases where the number of cars per model is just enough to be divisible by the maximum number of cars a customer can buy.

Let's construct a test case that will expose the bug:

- **Number of test cases**: 1 (to keep it simple)
- **Number of models**: 5 (a moderate number to ensure the test case is not too large)
- **Maximum cars a customer can buy**: 3 (a moderate number to ensure the test case is not too large)
- **Number of cars per model**: 1, 2, 3, 4, 5 (to create a varied distribution)

This test case should be challenging because it requires the algorithm to handle a mix of small and large numbers of cars per model, and it should be able to distribute the cars efficiently.

Here is the test input:

```plaintext
1
5 3
1 2 3 4 5
```

This test case will help expose any issues in the logic for distributing the cars among customers and ensuring that the minimum number of customers is used.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
n = int(input())
for i in range(n):
    mod_max = [int(x) for x in input().split()]
    mod, max = mod_max[0], mod_max[1]
    cars = sorted([int(x) for x in input().split()])[::-1]
    cnt = 0
    while 0 in cars:
        cars.remove(0)
    while True:
        j=0
        while j<max and j<len(cars):
            cars[j]-=1
            j+=1
        cnt+=1
        while 0 in cars:
            cars.remove(0)
        if len(cars)==0:
            break
        elif len(cars)==1:
            cnt+=cars[0]
            break
    print(cnt)

**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 1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import sys
from heapq import *
from bisect import bisect_left, bisect_right
from os import path
from math import sqrt, gcd, factorial, log
from io import BytesIO, IOBase
from collections import defaultdict, Counter, deque

BUFSIZE = 8192

class FastIO(IOBase):
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)

class IOWrapper(IOBase):
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")

# sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
# input = lambda: sys.stdin.readline().rstrip("\r\n")
# sys.setrecursionlimit(4000)

# if sys.version_info[0] < 3:
#     sys.stdin, sys.stdout = FastIO(sys.stdin), FastIO(sys.stdout)
# else:
#     sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)

input = lambda: sys.stdin.readline().rstrip("\r\n")

mod = int(1e9) + 7
MOD = 998244353

def lcm(a,b):
    return (a*b)//(gcd(a%b,b)) if a*b != 0 else a+b

def mex(l):
    n = max(l)
    m = (n+1)*[0]
    for i in l :
        m[i] = 1
    for i in range(n+1) :
        if m[i] == 0 :
            return i
    return n+1

def exp(a,b):
    ans = 1
    while b :
        if b&1 :
            ans *= a
            ans %= mod
        a *= a
        b >>= 1
    return ans

def SieveOfEratosthenes(n):
    a = [True for i in range(n+1)]
    p = 2
    while (p * p <= n):
        if (a[p] == True):
            for i in range(p * p, n+1, p):
                a[i] = False
        p += 1
    return a

# *****************    Be careful with the use of 1 and 0 while using log    *****************

def solve(test):
    # n = int(input())
    n,k = map(int,input().split())
    a = list(map(int,input().split()))
    # b = list(map(int,input().split()))
    # s = input()
    # t = input()
    a.sort()
    if a[-1] >= sum(a) - a[-1] :
        print(a[-1])
        return
    # print(a)
    cur = -1
    ans = 0
    i = 0
    while i+k-1 < n :
        a[i+k-1] -= a[i]
        i += (k-1)
    ans = 0
    # print(a)
    for i in range(0,n,k):
        ans += a[i]
    print(ans)

def main():
    if path.exists("D:/Academics/Competetive_Programming/Programs/Python/input.txt"):
        sys.stdin = open("D:/Academics/Competetive_Programming/Programs/Python/input.txt", 'r')
        sys.stdout = open("D:/Academics/Competetive_Programming/Programs/Python/output.txt", 'w')
    test = 1
    for test in range(int(input())):
        solve(test)
    # solve(test)

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:
```plaintext
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import java.util.*;
public class B_Kar_Salesman{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-->0)
        {
            int n = sc.nextInt();
            int k = sc.nextInt();
            int[] arr = new int[n];
            int max = 0;
            int min = Integer.MAX_VALUE;
            int sum = 0;

            for (int i = 0; i < n; i++) {
                arr[i] = sc.nextInt();
                if (arr[i] > max) {
                    max = arr[i];
                }
                if (arr[i] < min) {
                    min = arr[i];
                }
                sum+=arr[i];
            }

            if(max>=sum-max){
                System.out.println(max);
            }
            else{
                System.out.println(max+k-min);
            }
        }
    }
}

**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 1 1 1 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

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

// Add your class name here
/**
 * B_Kar_Salesman
 */
public class B_Kar_Salesman {

  public static void solve() throws IOException {
    int n = in();
    int x = in();
    long[] a = new long[n];
    for (int i = 0; i < n; i++) {
      a[i] = inl();
    }

    Arrays.sort(a);

    // if (x >= n) {
    // out.println(a[n - 1]);
    // return;
    // }
    int i = n - 1;
    long[] d = new long[x];
    int j = x - 1;
    while (j >= 0) {
      d[j] = a[i];
      j--;
      i--;
    }
    long cut = 0;
    boolean flag = true;
    while (flag) {
      Arrays.sort(d);
      long min = Math.max(0, d[0]);
      for (int k = 0; k < x; k++) {
        if (i >= 0 && d[k] == min) {
          d[k] = a[i];
          i--;
        } else {
          d[k] = d[k] - min;
        }

      }

      cut += (min == -1 ? 0 : min);

      if (i < 0) {
        flag = false;
      }
    }
    Arrays.sort(d);
    if (d[x - 1] != -1) {
      cut += d[x - 1];
    }
    out.println(cut);

  }

  static boolean MULTI_CASE = true;

  public static void main(String[] args) throws IOException {
    if (MULTI_CASE) {
      int T = sc.nextInt();
      for (int i = 0; i < T; ++i) {
        solve();
      }
    } else {
      solve();
    }
    out.close();
  }

  static int in() throws IOException {
    return sc.nextInt();
  }

  static long inl() throws IOException {
    return sc.nextLong();
  }

  static void py() {
    out.println("YES");
  }

  static void pn() {
    out.println("NO");
  }

  static InputReader sc = new InputReader();
  static PrintWriter out = new PrintWriter(new OutputStreamWriter(System.out));

  static class InputReader {
    private StringTokenizer st;
    private BufferedReader bf;

    public InputReader() {
      bf = new BufferedReader(new InputStreamReader(System.in));
      st = null;
    }

    public String next() throws IOException {
      while (st == null || !st.hasMoreTokens()) {
        st = new StringTokenizer(bf.readLine());
      }
      return st.nextToken();
    }

    public String nextLine() throws IOException {
      return bf.readLine();
    }

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

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

    public double nextDouble() throws IOException {
      return Double.parseDouble(next());
    }
  }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios that can highlight issues in the algorithm. Here are the steps to think through this:

1. **Edge Cases for `n` and `x`:**
   - When `x` is very large compared to `n`, the code should handle it correctly.
   - When `x` is very small compared to `n`, the code should still work efficiently.

2. **Large Number of Cars in One Model:**
   - If one model has a significantly larger number of cars compared to others, the code should handle this efficiently.

3. **Multiple Test Cases:**
   - The test input should include multiple test cases to ensure the code handles all scenarios correctly.

4. **Special Cases:**
   - When all models have the same number of cars.
   - When the number of cars in each model is a multiple of `x`.

Based on these considerations, here is a small-scale test input that is complete and valid:

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

This test input includes:
- A case where `x` is smaller than `n` and there are varying numbers of cars in each model.
- A case where `x` is equal to `n`.
- A case where one model has a significantly larger number of cars.
- A case with a larger `n` and `x` where the number of cars in each model is a mix of small and large values.

This input should help expose any issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

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

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);

    int t;
    cin >> t;

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

        vector<int> v(n);
        int total = 0;
        for (int i = 0; i < n; i++) {
            cin >> v[i];
            total += v[i];
        }

        sort(v.begin(), v.end(), greater<int>());
        int c = max(v[0], (total + x - 1) / x);
        cout << c << "\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 scenarios where the code's logic fails. The buggy code calculates the number of customers required using a formula that may not always be correct, especially when the distribution of cars across different models is uneven.

### Step-by-Step Analysis:
1. **Understanding the Bug**:
   - The buggy code sorts the number of cars in descending order and then uses the formula `c = max(v[0], (total + x - 1) / x)` to determine the number of customers.
   - This formula might not correctly account for the distribution of cars across different models, especially when the number of cars in one model is significantly larger than the others.

2. **Edge Cases**:
   - **Highly Skewed Distribution**: When one model has a very large number of cars compared to the others.
   - **Equal Distribution**: When the number of cars in each model is almost the same.
   - **Small Number of Models**: When `n` is small and `x` is large.
   - **Large Number of Models**: When `n` is large and `x` is small.

3. **Challenging Scenario**:
   - **Highly Skewed Distribution**: This is likely to expose the bug because the formula might overestimate the number of customers needed.

### Challenging Test Input:
Let's create a test case where one model has a very large number of cars, and the other models have a small number of cars. This will force the code to handle a highly skewed distribution.

```plaintext
1
5 3
1 1 1 1 1000000
```

### Explanation:
- **Number of Test Cases**: 1
- **Number of Models (n)**: 5
- **Maximum Cars per Customer (x)**: 3
- **Number of Cars in Each Model**: 1, 1, 1, 1, 1000000

### Expected Output:
- The correct output for this test case should be 333334, because:
  - Karel can sell 3 cars per customer.
  - To sell 1000004 cars, Karel needs \( \lceil \frac{1000004}{3} \rceil = 333335 \) customers.

### Why This Input is Challenging:
- The buggy code will likely overestimate the number of customers because it uses a formula that doesn't correctly handle the highly skewed distribution.
- The correct solution should consider the distribution and ensure that the maximum number of cars per customer is used efficiently.

This test input should expose 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: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
/*

⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⣀⣠⡴⠖⠋⠀⠀⠀⠀⢿⣿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⣠⡆⠰⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⢠⣤⡈⠀⠀⠀⠀⢻⡇⢀⣷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⢸⡿⠗⠀⠀⠀⠀⠈⠛⠛⡉⠛⠁⠀⠀⠀⣧⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠴⠞⠉⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠸⣿⣧⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢻⣿⠷⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡿⢻⡇⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⣿⣧
⢠⡆⠀⠀⠀⠀⣾⠃⣾⡇⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⢿⣿⡀⠀⠀⢸⣿
⢸⡇⠀⠀⠀⢠⡿⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠘⣿⣇⠀⠀⠘⣿⡄
⢸⣧⠀⠀⠀⢸⡇⠀⣿⡀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⢹⡟⠀⠀⠀⣿⡇
⠈⣿⡄⠀⢀⣿⠇⠀⣿⣧⡀⠀⠀⠀⣠⣿⠋⣿⣿⣦⣀⣀⣠⣾⡇⠀⠀⠀⢿⡇
⠀⢻⣿⣶⣾⡿⠀⠀⠹⣿⣿⣶⣶⣿⣿⠋⠀⠈⠻⣿⣿⣿⡿⠟⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

*/
/*
    "BISMILLAHIR RAHMANIR RAHIM"
    ("IN THE NAME OF ALLAH, THE MOST GRACIOUS AND THE MOST MERCIFUL")
    CodeMonsterShu
    Department of Computer Science and Engineering
    Sheikh Hasina University, Netrokona, Bangladesh.
*/

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

void solve()
{
    int n, x;
    cin >> n >> x;
    int a[n];
    for (int i = 0; i < n; i++)
    {
        cin >> a[i];
    }
    int ans = 0;

    while (true)
    {
        int bad[n] = {0}; // To mark indices that are already selected
        int selected = 0;

        for (int j = 0; j < x; j++)
        {
            int max_pos = -1, max_value = 0;

            for (int i = 0; i < n; i++)
            {
                if (!bad[i] && a[i] > max_value)
                {
                    max_value = a[i];
                    max_pos = i;
                }
            }

            if (max_value == 0)
                break;
            bad[max_pos] = 1;
            a[max_pos]--;
            selected++;
        }

        if (selected == 0)
            break;
        ans++;
    }
    cout << ans << endl;
}

signed main()
{
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);

    int t;
    cin >> t;
    while (t--)
    {
        solve();
    }
    return 0;
}

/*===============================================
  :::::::::::::Author :CodeMonsterShu:::::::::::::
  ===============================================*/

**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
4
3 1
1 1 1
3 2
2 2 2
5 3
3 3 3 3 3
7 4
4 4 4 4 4 4 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import sys
#import random,os
from math import ceil,floor,gcd,lcm,sqrt,isqrt #, log, factorial, comb, perm,
#log10, log2, log, sin, asin, tan, atan, radians
#from collections import defaultdict as dd
#from collections import OrderedDict as Od  #ordered set
#from collections import Counter
#from collections import deque as dq #deque  e.g. myqueue=dq(list)
#from bisect import * #bisect_left, bisect(a,value)
from heapq import * #heappop(Q),heappush,heapify,Q[0] is smallest
#from itertools import permutations(p,r)#combinations(p,r)
#combinations(p,r) gives r-length tuples #combinations_with_replacement
#sys.setrecursionlimit(100000) #default is 1000
# input = BytesIO(os.read(0, os.fstat(0).st_size)).readline
input = lambda: sys.stdin.readline().rstrip("\r\n")
sint = lambda: int(input())
mint = lambda: map(int, input().split())
lint = lambda: list(map(int, input().split()))
###############################################
#dij = [(0, 1), (-1, 0), (0, -1), (1, 0)] #4 directions
#dij = [(0, 1), (-1, 0), (0, -1), (1, 0), (1, 1), (1, -1), (-1, 1), (-1, -1)] #8 d
#big = float('inf')
#mod=10**9+7
#T = 1

for _ in range(sint()):
    a,b=mint()
    l=lint()
    # l.sort()
    # print(l)
    for i in range(len(l)):
        l[i]*=-1
    heapify(l)
    c=0
    while len(l)>0:
        x=[]
        for i in range(b):
            x.append(heappop(l))
            if len(l)==0:
                break
        for i in x:
            if abs(i)-1>0:
                heappush(l,-1*(abs(i)-1))
        c+=1
    print(c)
    # 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:
To create a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that can maximize the failure probability. Here are the steps to consider:

1. **Maximum Number of Test Cases**: Ensure the test input includes the maximum number of test cases allowed, which is \(10^4\).
2. **Edge Cases for \(n\) and \(x\)**: Test the minimum and maximum values for \(n\) and \(x\).
3. **Large Values for \(a_i\)**: Use large values for \(a_i\) to ensure the algorithm handles large numbers correctly.
4. **Mixed Values for \(a_i\)**: Include a mix of small and large values for \(a_i\) to test the algorithm's ability to handle varied inputs.
5. **Specific Patterns**: Create patterns in \(a_i\) that can highlight the bug, such as having a large number of cars for a few models and a small number for others.

Given the constraints, a small-scale test input that is complete and valid can be:

```plaintext
1
5 3
1 2 3 4 5
```

This test case includes:
- \(n = 5\) (a moderate number of car models).
- \(x = 3\) (a moderate number of cars Karel can sell to one customer).
- \(a_i\) values are 1, 2, 3, 4, and 5, which provide a mix of small and large values.

This input will test the algorithm's ability to handle a moderate number of car models and a moderate number of cars per customer, ensuring that it correctly calculates the minimum number of customers needed to sell all the cars.

Here is the test input:

```plaintext
1
5 3
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import sys
import heapq

def min_customers(test_cases):
    results = []

    for n, x, a in test_cases:
        # Create a max heap with negative counts
        max_heap = [-count for count in a]
        heapq.heapify(max_heap)

        customers = 0

        while max_heap:
            # We can serve one customer
            customers += 1

            # Temporary list to store the counts of the cars sold
            temp = []

            # Sell up to x cars
            for _ in range(x):
                if max_heap:
                    max_cars = -heapq.heappop(max_heap)  # Get the largest count
                    if max_cars > 1:  # If there's more than 1 car, reduce it and keep it
                        temp.append(max_cars - 1)

            # Push back the remaining cars into the heap
            for remaining in temp:
                heapq.heappush(max_heap, -remaining)

        results.append(customers)

    return results

def main():
    input = sys.stdin.read
    data = input().splitlines()

    t = int(data[0])  # number of test cases
    index = 1
    test_cases = []

    for _ in range(t):
        n, x = map(int, data[index].split())
        a = list(map(int, data[index + 1].split()))
        test_cases.append((n, x, a))
        index += 2

    results = min_customers(test_cases)

    # Output all results at once
    print("\n".join(map(str, results)))

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 generate a challenging test input that exposes the bug in the given buggy code, we need to consider edge cases and scenarios that maximize the failure probability. The bug in the code is that it does not handle the case where the number of cars in the heap is less than `x` correctly. Specifically, it should be able to handle cases where the number of available models is less than `x`.

Here are the steps to generate a challenging test input:

1. **Test Case with Few Models and Small `x`:**
   - This will test the basic functionality and ensure that the code can handle small inputs correctly.

2. **Test Case with Many Models and Small `x`:**
   - This will test the code's ability to handle a large number of models but with a small `x`.

3. **Test Case with Few Models and Large `x`:**
   - This will test the code's ability to handle a small number of models but with a large `x`.

4. **Test Case with Many Models and Large `x`:**
   - This will test the code's ability to handle a large number of models and a large `x`.

5. **Test Case with Uneven Distribution of Cars:**
   - This will test the code's ability to handle cases where the distribution of cars across models is highly uneven.

6. **Test Case with All Models Having the Same Number of Cars:**
   - This will test the code's ability to handle cases where all models have the same number of cars.

Based on these considerations, here is a small-scale test input that covers these scenarios:

```plaintext
5
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
10 10
1 1 1 1 1 1 1 1 1 1
```

This test input includes:
- A basic test case with few models and small `x`.
- A test case with many models and small `x`.
- A test case with few models and large `x`.
- A test case with many models and large `x`.
- A test case with an uneven distribution of cars.
- A test case with all models having the same number of cars.

This should help expose any bugs in the given code, particularly the issue with handling cases where the number of available models is less than `x`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import java.util.*;

public class B{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while (t-- > 0) {
            int n = sc.nextInt();
            int x = sc.nextInt();
            long sum = 0,max = 0;
            long arr[] = new long[n];
            for (int i = 0; i < n; i++){
                long b = sc.nextLong();
                arr[i] = b;
                max = Math.max(max,b);
                sum = sum + b;
            }
            System.out.println((int)Math.max(max,Math.ceil(((double)sum)/x)));
        }
        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:
To create a challenging test input that exposes the bug in the given code, let's consider the following points:

1. **Overflow and Large Values**: The code uses `long` for the array and sum, which is good for handling large values, but the division and ceiling operation should be carefully handled.

2. **Max Value Misinterpretation**: The code incorrectly uses the maximum value of a single model (`max`) in the final output. This can lead to incorrect results if the maximum value is significantly higher than the average.

3. **Division and Rounding**: The code uses `Math.ceil` to round up the division result, which is generally correct, but it should be applied correctly to the sum of all cars divided by the number of models a customer can buy.

4. **Edge Cases**: Consider cases where the number of models is large, the number of cars per model is large, and the maximum number of cars a customer can buy is small.

Given these points, a challenging test input would be:

- A large number of models with a small number of cars per model, but the sum of cars is just slightly above a multiple of the maximum number of cars a customer can buy.
- A large number of models with one model having a significantly higher number of cars than the others.

Here is a small-scale test input that should expose the bug:

```plaintext
2
5 3
1 1 1 1 1
5 3
1 1 1 1 10
```

Explanation:
- In the first test case, there are 5 models with 1 car each, and a customer can buy up to 3 cars. The correct output should be 2 (one customer buys 3 cars, and the other buys 2 cars).
- In the second test case, there are 5 models with 1 car each except for one model with 10 cars, and a customer can buy up to 3 cars. The correct output should be 5 (one customer buys 3 cars, another buys 3 cars, and the last customer buys 4 cars).

This input should help identify the issue with the current code, which incorrectly uses the maximum value of a single model in the final output.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Kar Salesman
time_limit_ms: 1000
memory_limit_mb: 256
Description: Karel is a salesman in a car dealership. The dealership has $n$ different models of cars. There are $a_i$ cars of the $i$-th model. Karel is an excellent salesperson and can convince customers to buy up to $x$ cars (of Karel's choice), as long as the cars are from different models.      Determine the minimum number of customers Karel has to bring in to sell all the cars.

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

The first line of each test case contains two integers $n$ and $x$ ($1 \le n \le 5 \cdot 10^5$; $1 \le x \le 10$) — the number of different models of cars and the maximum number of cars Karel can convince a customer to buy.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the number of cars of each model.

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

Output: For each test case, output the minimum possible number of customers needed to sell all the cars.

Examples:
input:
4
3 2
3 1 2
3 3
2 1 3
5 3
2 2 1 9 2
7 4
2 5 3 3 5 2 5
output:
3
3
9
7

Note: For the first case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $2$ cars with model $1$ and $2$.    * Customer $3$ buys $2$ cars with model $1$ and $3$.

For the second case, Karel only needs to lure in $3$ customers. He will convince the customers to buy the following models of cars:

  * Customer $1$ buys $2$ cars with model $1$ and $3$.    * Customer $2$ buys $3$ cars with model $1$, $2$ and $3$.    * Customer $3$ buys $1$ car with model $3$.

Buggy Code:
import java.util.*;
public class Main {
    static long mod = 1000000007;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int tc = sc.nextInt();
        while(tc-->0) {
            int n = sc.nextInt(),r=sc.nextInt();
            long max=0;
            long sum=0;
            for(int i=0;i<n;i++){
                int val = sc.nextInt();
                max=Math.max(val,max);
                sum+=val;
            }
            System.out.println(Math.max((sum+r-1)/r,max));
        }
    }
}

**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
10 1
1 1 1 1 1 1 1 1 1 1000000000
```

**Explanation:**
- This test case is designed to challenge the buggy code by providing a scenario where the number of models is relatively large and the maximum number of cars Karel can sell per customer is 1.
- The array `a` contains 9 models with 1 car each and 1 model with a very large number of cars (1,000,000,000).
- The buggy code calculates the result as `Math.max((sum + r - 1) / r, max)`. In this case, `sum` will be 1,000,000,009 and `max` will be 1,000,000,000.
- The correct output should be 1,000,000,009 because Karel needs to bring in 1,000,000,009 customers to sell all the cars, as each customer can only buy 1 car.
- However, the buggy code will output 1,000,000,000, which is incorrect. This exposes the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Ultra-Meow
time_limit_ms: 2500
memory_limit_mb: 256
Description: K1o0n gave you an array $a$ of length $n$, consisting of numbers $1, 2, \ldots, n$. Accept it? Of course! But what to do with it? Of course, calculate $\text{MEOW}(a)$.

Let $\text{MEX}(S, k)$ be the $k$-th positive (strictly greater than zero) integer in ascending order that is not present in the set $S$. Denote $\text{MEOW}(a)$ as the sum of $\text{MEX}(b, |b| + 1)$, over all distinct subsets $b$ of the array $a$.

Examples of $\text{MEX}(S, k)$ values for sets:

  * $\text{MEX}(\\{3,2\\}, 1) = 1$, because $1$ is the first positive integer not present in the set;    * $\text{MEX}(\\{4,2,1\\}, 2) = 5$, because the first two positive integers not present in the set are $3$ and $5$;    * $\text{MEX}(\\{\\}, 4) = 4$, because there are no numbers in the empty set, so the first $4$ positive integers not present in it are $1, 2, 3, 4$.

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

In a single line of each test case, an integer $n$ ($1 \le n \le 5000$) is entered, the size of the array of gifted numbers.

It is guaranteed that the sum of $n^2$ over all test cases does not exceed $25 \cdot 10^6$.

Output: For each test case, output a single number — $\text{MEOW}(a)$. Since it may be very large, output it modulo $10^9 + 7$.

Examples:
input:
5
2
3
4999
5
1
output:
12
31
354226409
184
4

Note:

Buggy Code:
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;

template <typename T>
using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;

#define endl "\n"
#define fo(i, n) for (i = 0; i < n; i++)
#define Fo(i, k, n) for (i = k; k < n; k++)
#define pb push_back
#define ll long long
#define mp make_pair
#define ff first
#define ss second
#define all(x) x.begin(), x.end()
#define clr(x) memset(x, 0, sizeof(x))
#define sortall(x) sort(all(x))
#define tr(it, arr) for (auto it = arr.begin(); it != arr.end(); it++)
#define PI 3.1415926535897932384626
#define suprit                    \
    ios_base::sync_with_stdio(0); \
    cout.tie(0);                  \
    cin.tie(0);
#define line cout << endl;

typedef pair<int, int> pi;
typedef pair<ll, ll> pl;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<vi> vvi;
typedef vector<vl> vvl;
typedef map<ll, ll> ml;
typedef map<string, ll> msl;
typedef map<ll, string> mls;
typedef unordered_map<ll, ll> uml;
typedef unordered_map<string, ll> umsl;
typedef unordered_map<ll, string> umls;
typedef set<ll> sl;
typedef set<pair<ll, ll>> spl;
typedef ordered_set<ll> osl;
typedef ordered_set<pair<ll, ll>> ospl;

const ll mod = 1e9 + 7;

bool comp2(pair<ll, ll> &arr, pair<ll, ll> &b)
{
    if (arr.first == b.first)
        return arr.second < b.second;
    return arr.first < b.first;
};

template <typename T>
void read(T i, T n, vector<T> &arr)
{
    for (T j = i; j < n; j++)
        cin >> arr[j];
}
template <typename T>
void read(vector<T> &arr)
{
    for (auto &j : arr)
        cin >> j;
}

#ifndef ONLINE_JUDGE

template <typename T, typename V>
void print(set<pair<T, V>> &arr)
{
    for (auto &it : arr)
        cout << it.first << " " << it.second << endl;
    line
}
template <typename T, typename V>
void print(pair<T, V> &pa) { cout << pa.first << " " << pa.second << endl; }
template <typename T>
void print(T i, T last, vector<T> &arr)
{
    for (T j = i; j < last; j++)
        cout << arr[j] << " ";
    line
}
template <typename T>
void print(T i, vector<T> &arr)
{
    for (T j = i; j < arr.size(); j++)
        cout << arr[j] << " ";
    line
}
template <typename T>
void print(vector<T> &arr)
{
    for (auto &i : arr)
        cout << i << " ";
    line
}
template <typename T, typename V>
void print(unordered_map<T, V> &arr)
{
    for (auto &it : arr)
        cout << it.first << " " << it.second << endl;
    line
}
template <typename T, typename V>
void print(map<T, V> &arr)
{
    for (auto &it : arr)
        cout << it.first << " " << it.second << endl;
}
template <typename T>
void print(unordered_set<T> &arr)
{
    for (auto &it : arr)
        cout << it << " ";
    line
}
template <typename T>
void print(ordered_set<T> &arr)
{
    for (auto &it : arr)
        cout << it << " ";
    line
}
template <typename T>
void print(set<T> &arr)
{
    for (auto &it : arr)
        cout << it << " ";
    line
}
template <typename T, typename... Args>
void print(T t, Args... args)
{
    cout << t << " ";
    print(args...);
}
template <typename T>
void print(T t) { cout << t << "\n"; }
#define debug(x) cout << #x << " " << x << endl;
#define debug2(x, y) cout << #x << " " << x << " " << #y << " " << y << endl;
#define debug3(x, y, z) cout << #x << " " << x << " " << #y << " " << y << " " << #z << " " << z << endl;
#define debug4(x, y, z, a) cout << #x << " " << x << " " << #y << " " << y << " " << #z << " " << z << " " << #a << " " << a << endl;

#else

#define print(x)
#define debug(x)
#define debug2(x, y)
#define debug3(x, y, z)
#define debug4(x, y, z, a)

#endif

// 5 5 4 3 2 1
// (n-5) c 5
// 4 c 1
const int nmax = 10002;
vl factorial(nmax), inv_fact(nmax);

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

void precalc()
{
    factorial[0] = 1;
    for (ll i = 1; i <= nmax; i++)
        factorial[i] = factorial[i - 1] * i % mod;
    inv_fact[nmax - 1] = binpow(factorial[nmax - 1], mod - 2, mod);
    for (ll i = nmax - 2; i >= 0; i--)
        inv_fact[i] = inv_fact[i + 1] * (i + 1) % mod;
}

ll nCr(ll n, ll r)
{
    if (r == 0 || r == n)
        return 1;
    if (r > n || r < 0)
        return 0;
    ll ans = factorial[n] * inv_fact[r] % mod;
    ans = ans * inv_fact[n - r] % mod;
    return ans;
}

void solve()
{
    ll n;
    cin >> n;
    ll ans = 1;
    for (ll i = 2; i <= n; i++)
    {
        for (ll k = 1; k <= i; k++)
        {
            if(i - k > k - 1) continue;
            ll temp = nCr(i - 1, i - k) * nCr(n - i, 2 * k - i - 1) % mod;
            temp = temp * i % mod;
            // debug3(i, k, temp);
            ans = (ans + temp) % mod;
        }
    }
    for(ll i = n+1; i <= 2*n+1; i++) {
        if(i % 2 == 0) continue;
        ll temp = i * nCr(n, i/2) % mod;
        ans = (ans + temp) % mod;
    }
    // already_calc[n] = ans;
    cout << ans << endl;
}

int main()
{
    suprit;
    clock_t start = clock();

    int t = 1;
    precalc();
    cin >> t;
    while (t--)
    {
        solve();
    }
    clock_t end = clock();

#ifndef ONLINE_JUDGE
    double elapsed = double(end - start) / CLOCKS_PER_SEC;
    cout << setprecision(10) << elapsed << endl;
#endif
    return 0;
}

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

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

Think step by step.

Response:
```plaintext
5
1
2
3
4
5000
```