**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

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

public class Main {
    static int broken = -1;

    static void solution() {
        int n = in.nextInt();
        Integer[] a = new Integer[n];
        for (int i = 0; i < n; i++) {
            a[i] = in.nextInt();
        }
        ArrayList<Integer> S = new ArrayList<>();
        int maxSoFar = 0;
        for (int i = 0; i < n; i++) {
            if (a[i] >= maxSoFar) {
                S.add(a[i]);
                maxSoFar = a[i];
            }
        }
        int len = S.size();
        int[] dp = new int[len];
        int answer = len;
        for (int i = 0; i < len; i++) {
            dp[i] = 1;
            for (int j = 0; j < i; j++) {
                if (S.get(j) >= S.get(i)) {
                    dp[i] = Math.max(dp[i], dp[j] + 1);
                }
            }
            answer = Math.min(answer, len - dp[i]);
        }
        System.out.println(answer);
    }

    static int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }

    public static void main(String[] args) {
        in = new Scuffer();
        int n = in.nextInt();
        while (n-- > 0) {
            solution();
        }
        out.close();
    }

    public static PrintWriter out;
    public static Scuffer in;

    public static class Scuffer {
        BufferedReader br;
        StringTokenizer st;

        public Scuffer() {
            br = new BufferedReader(new InputStreamReader(System.in));
            out = new PrintWriter(new BufferedOutputStream(System.out));
        }

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

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

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

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

        String nextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }

    }
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases or scenarios that challenge the logic of the algorithm. The key points to consider are:

1. **Non-increasing Subsequences**: The code is supposed to find the minimum number of elements to remove to make the array vulnerable (i.e., non-increasing by repeatedly applying Stalin Sort to subarrays). The code might fail if it doesn't correctly handle cases where the array has multiple non-increasing subsequences.

2. **Edge Cases**: The code should handle arrays with all elements being the same, arrays with strictly increasing elements, and arrays with strictly decreasing elements.

3. **Complex Interactions**: The code should handle arrays where the elements are not in a straightforward increasing or decreasing order, but rather have complex interactions that could lead to incorrect results.

Given these considerations, let's create a small-scale test input that can expose the bug:

```plaintext
4
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
```

### Explanation:
1. **Test Case 1**: `5 4 4 2 2`
   - This array is already non-increasing, so no elements need to be removed. The correct output should be `0`.

2. **Test Case 2**: `2 2 4 4 6 6 10 10`
   - This array is strictly increasing, and the minimum number of elements to remove to make it non-increasing is `6`. The correct output should be `6`.

3. **Test Case 3**: `6 8 9 10 12 9 7 5 4`
   - This array has a complex interaction where the elements are not in a straightforward increasing or decreasing order. The minimum number of elements to remove to make it non-increasing is `4`. The correct output should be `4`.

4. **Test Case 4**: `300000000 600000000 400000000 900000000 200000000 400000000 200000000`
   - This array has large numbers and a complex interaction similar to the previous case. The minimum number of elements to remove to make it non-increasing is `2`. The correct output should be `2`.

This test input covers a variety of scenarios that can help 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: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

Buggy Code:
import java.util.*;
import java.io.*;
public class Main {
    static class Scan {
        private byte[] buf=new byte[1024];
        private int index;
        private InputStream in;
        private int total;
        public Scan()
        {
            in=System.in;
        }
        public int scan()throws IOException
        {
            if(total<0)
            throw new InputMismatchException();
            if(index>=total)
            {
                index=0;
                total=in.read(buf);
                if(total<=0)
                return -1;
            }
            return buf[index++];
        }
        public int scanInt()throws IOException
        {
            int integer=0;
            int n=scan();
            while(isWhiteSpace(n))
            n=scan();
            int neg=1;
            if(n=='-')
            {
                neg=-1;
                n=scan();
            }
            while(!isWhiteSpace(n))
            {
                if(n>='0'&&n<='9')
                {
                    integer*=10;
                    integer+=n-'0';
                    n=scan();
                }
                else throw new InputMismatchException();
            }
            return neg*integer;
        }
        public double scanDouble()throws IOException
        {
            double doub=0;
            int n=scan();
            while(isWhiteSpace(n))
            n=scan();
            int neg=1;
            if(n=='-')
            {
                neg=-1;
                n=scan();
            }
            while(!isWhiteSpace(n)&&n!='.')
            {
                if(n>='0'&&n<='9')
                {
                    doub*=10;
                    doub+=n-'0';
                    n=scan();
                }
                else throw new InputMismatchException();
            }
            if(n=='.')
            {
                n=scan();
                double temp=1;
                while(!isWhiteSpace(n))
                {
                    if(n>='0'&&n<='9')
                    {
                        temp/=10;
                        doub+=(n-'0')*temp;
                        n=scan();
                    }
                    else throw new InputMismatchException();
                }
            }
            return doub*neg;
        }
        public String scanString()throws IOException
        {
            StringBuilder sb=new StringBuilder();
            int n=scan();
            while(isWhiteSpace(n))
            n=scan();
            while(!isWhiteSpace(n))
            {
                sb.append((char)n);
                n=scan();
            }
            return sb.toString();
        }
        private boolean isWhiteSpace(int n)
        {
            if(n==' '||n=='\n'||n=='\r'||n=='\t'||n==-1)
            return true;
            return false;
        }
    }

    public static void sort(int arr[],int l,int r) {    //sort(arr,0,n-1);
        if(l==r) {
            return;
        }
        int mid=(l+r)/2;
        sort(arr,l,mid);
        sort(arr,mid+1,r);
        merge(arr,l,mid,mid+1,r);
    }
    public static void merge(int arr[],int l1,int r1,int l2,int r2) {
        int tmp[]=new int[r2-l1+1];
        int indx1=l1,indx2=l2;
        //sorting the two halves using a tmp array
        for(int i=0;i<tmp.length;i++) {
            if(indx1>r1) {
                tmp[i]=arr[indx2];
                indx2++;
                continue;
            }
            if(indx2>r2) {
                tmp[i]=arr[indx1];
                indx1++;
                continue;
            }
            if(arr[indx1]<arr[indx2]) {
                tmp[i]=arr[indx1];
                indx1++;
                continue;
            }
            tmp[i]=arr[indx2];
            indx2++;
        }
        //Copying the elements of tmp into the main array
        for(int i=0,j=l1;i<tmp.length;i++,j++) {
            arr[j]=tmp[i];
        }
    }

    public static void sort(long arr[],int l,int r) {    //sort(arr,0,n-1);
        if(l==r) {
            return;
        }
        int mid=(l+r)/2;
        sort(arr,l,mid);
        sort(arr,mid+1,r);
        merge(arr,l,mid,mid+1,r);
    }
    public static void merge(long arr[],int l1,int r1,int l2,int r2) {
        long tmp[]=new long[r2-l1+1];
        int indx1=l1,indx2=l2;
        //sorting the two halves using a tmp array
        for(int i=0;i<tmp.length;i++) {
            if(indx1>r1) {
                tmp[i]=arr[indx2];
                indx2++;
                continue;
            }
            if(indx2>r2) {
                tmp[i]=arr[indx1];
                indx1++;
                continue;
            }
            if(arr[indx1]<arr[indx2]) {
                tmp[i]=arr[indx1];
                indx1++;
                continue;
            }
            tmp[i]=arr[indx2];
            indx2++;
        }
        //Copying the elements of tmp into the main array
        for(int i=0,j=l1;i<tmp.length;i++,j++) {
            arr[j]=tmp[i];
        }
    }

    public static void main(String args[]) throws IOException {
        Scan input=new Scan();
        StringBuilder ans=new StringBuilder("");
        int test=input.scanInt();
        for(int tt=1;tt<=test;tt++) {
            int n=input.scanInt();
            int arr[]=new int[n];
            int max=0;
            for(int i=0;i<n;i++) {
                arr[i]=input.scanInt();
                max=Math.max(max,arr[i]);
            }
            int count=0,prev=-1;
            for(int i=0;i<n;i++) {
                if(max==arr[i]) {
                    break;
                }
                if(prev==-1) {
                    count++;
                    prev=arr[i];
                    continue;
                }
                if(arr[i]<prev) {
                    continue;
                }
                count++;
                prev=arr[i];
            }
            ans.append(count+"\n");
        }
        System.out.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
3
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

Buggy Code:
results = []

for testcase in range(int(input())):

    n = int(input())
    a = []
    for i in input().split():
        a.append(int(i))

    r = 5000
    sa = a
    for i in range(0,len(a)):
        sa.pop(0)
        r = min(r, i+sum(x>sa[0] for x in sa))
    if r != 0:
        r += 1

    results.append(r)

for i in results:
    print(i)

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

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

Think step by step.

Response:
```plaintext
3
5
1 2 3 4 5
4
10 9 8 7
6
1 2 2 1 2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

Buggy Code:
for test in range(int(input())):
    n = int(input())
    arr = list(map(int,input().split()))
    ans,m,p = 0,0,0
    for i in range(n):
        if arr[i]>=m:
            ans+=1
            if arr[i] == m:
                p+=1
            else:
                m = arr[i]
                p = 0
    print(ans-p-1)

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

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

Think step by step.

Response:
```plaintext
3
5
1 2 3 4 5
4
5 4 3 2
6
1 2 3 3 2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

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

public class Ez {

  static IOHandler sc = new IOHandler();
  static PrintWriter out = new PrintWriter(System.out);

  public static void main(String[] args) throws IOException {
    new Thread(null, new Solution(), "solution", 1 << 26).start();
  }

  private static class Solution implements Runnable {

    public void run() {
      int testCases = sc.nextInt();
      //int testCases = 1;
      for (int i = 1; i <= testCases; ++i) {
        solve();
      }
      out.flush();
    }

    private void solve() {
      int n = sc.nextInt();
      int[] a = sc.readArray(n);
      int max = Integer.MIN_VALUE;
      int ans = 0;
      for (int i = 0; i < n; i++) {
        max = Math.max(max, a[i]);
      }
      int current = a[0];
      for (int i = 0; i < n; i++) {
        if (a[i] == max) {
          break;
        }
        if (current <= a[i]) {
          current = a[i];
          ans++;
        }
      }
      out.println(ans);
    }

    int find(int[] a, int n) {
      int cnt = 0;
      for (int i = 1; i < n; i++) {
        if (a[i - 1] == a[i]) {
          cnt++;
        }
      }
      return cnt;
    }

    int minTravel(int r, int c, int[][] a, int n) {
      int min = Integer.MAX_VALUE;
      while (r < n && c < n) {
        min = Math.min(min, a[r][c]);
        r++;
        c++;
      }
      return min;
    }

    boolean ask(String s) {
      out.println("? " + s);
      out.flush();
      return sc.nextInt() == 1;
    }

    private boolean isTrue(long sum, int n, long[] a) {
      double midVal = (double) sum / n / 2;
      int l = -1, r = n, mid;
      while (l + 1 < r) {
        mid = (l + r) / 2;
        if (midVal > a[mid]) {
          l = mid;
        } else {
          r = mid;
        }
      }
      return l >= (n / 2);
    }

    private long getVal(long num) {
      long min = 0, max = Integer.MAX_VALUE, mid;
      while (max - min > 1) {
        mid = (max + min) >> 1;
        if (mid * mid <= num) {
          min = mid;
        } else {
          max = mid;
        }
      }
      return min;
    }
  }

  private static class IOHandler {
    BufferedReader br;
    StringTokenizer st;

    public IOHandler() {
      br = new BufferedReader(new InputStreamReader(System.in));
    }

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

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

    int[] readArray(int n) {
      int[] res = new int[n];

      for (int i = 0; i < n; ++i)
        res[i] = nextInt();

      return res;
    }

    int[] readArray2(int n) {
      int[] res = new int[n + 1];

      for (int i = 1; i <= n; ++i)
        res[i] = nextInt();

      return res;
    }

    int[][] readGraph(int n, int m, int c) {
      int[][] graph = new int[n + 1][];

      int[] count = new int[n + 1];
      int[][] arr = new int[m][1 + c];

      int a, b;

      for (int i = 0; i < m; ++i) {
        a = sc.nextInt();
        b = sc.nextInt();

        arr[i][0] = a;
        arr[i][1] = b;

        ++count[a];
        ++count[b];

        for (int j = 2; j <= c; ++j) {
          arr[i][j] = sc.nextInt();
        }
      }

      for (int i = 0; i <= n; ++i) {
        graph[i] = new int[count[i] * c];
        count[i] = 0;
      }
      int swap;

      for (int[] tArr : arr) {
        for (int j = 1; j < tArr.length; ++j) {
          graph[tArr[0]][count[tArr[0]]++] = tArr[j];
        }

        swap = tArr[0];
        tArr[0] = tArr[1];
        tArr[1] = swap;

        for (int j = 1; j < tArr.length; ++j) {
          graph[tArr[0]][count[tArr[0]]++] = tArr[j];
        }
      }

      return graph;
    }

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

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

    String nextLine() {
      String str = "";
      try {
        str = br.readLine();
      } catch (IOException e) {
        e.printStackTrace();
      }
      return str;
    }

  }

}

class Pair {
  int a;
  int b;

  public Pair(int a, int b) {
    this.a = a;
    this.b = b;
  }

  long sum() {
    return a + b;
  }
}

**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, let's analyze the code and the problem requirements:

### Analysis of the Buggy Code:
1. **Initialization and Input Reading**:
   - The code reads the number of test cases and the arrays.
   - It initializes `max` to `Integer.MIN_VALUE` and `ans` to 0.

2. **First Loop**:
   - The first loop iterates through the array to find the maximum value in the array and assigns it to `max`.

3. **Second Loop**:
   - The second loop iterates through the array again.
   - It breaks the loop if it encounters the maximum value.
   - It increments `ans` if the current element is greater than or equal to the previous element and updates `current`.

### Problem with the Buggy Code:
- The code only counts the number of elements that are in non-decreasing order up to the first occurrence of the maximum value.
- It does not consider the entire array and does not correctly handle cases where the array can be made non-increasing by removing elements.

### Challenging Test Case:
To expose the bug, we need an array where the maximum value is not at the end and the array can be made non-increasing by removing a few elements. Here is a small-scale test input that should expose the bug:

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

### Explanation:
1. **Test Case 1**: `[5, 4, 3, 2, 1]`
   - This array is already non-increasing.
   - The expected output is `0`.

2. **Test Case 2**: `[3, 6, 4, 9, 2, 5, 2]`
   - This array can be made non-increasing by removing the elements `3` and `9`.
   - The expected output is `2`.

3. **Test Case 3**: `[1, 2, 3, 4]`
   - This array is in non-decreasing order and cannot be made non-increasing without removing all elements.
   - The expected output is `4`.

### Expected Output:
```plaintext
0
2
4
```

This test input should effectively expose the bug in the provided buggy code by highlighting its failure to correctly handle arrays that can be made non-increasing by removing elements.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

Buggy Code:
import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.math.BigInteger;
import java.util.*;

import static java.lang.Math.*;

public class Main {
    public static long lowerBound(long n, long k) {
        long low = 0, high = n;
        long ans = 0;

        while (low <= high) {
            long mid = (low + high) / 2;
            long val = (long) pow(k,mid);
            if (val <= n) {
                ans = val;
                low = mid + 1;
            } else {
                high = mid - 1;
            }
        }
        return ans;
    }
    public  static void shuffleArray(int[] arr,int n){
        Random rnd = new Random();
        for(int i=0; i<n; ++i){
            int tmp = arr[i];
            int randomPos = i + rnd.nextInt(n-i);
            arr[i] = arr[randomPos];
            arr[randomPos] = tmp;
        }
    }

    private static void local() throws FileNotFoundException {
        FileInputStream fileInputStream = new FileInputStream("./src/input.txt");
        System.setIn(fileInputStream);
    }
    public static void main(String[] args) throws FileNotFoundException {
        Scanner sc = new Scanner(System.in);
        int t = Integer.parseInt(sc.nextLine());
        while(t-- > 0){
            int n = sc.nextInt();
            List<Integer>a = new ArrayList<>();
            for(int i  = 0;i < n;i++){
                a.add(sc.nextInt());
            }
            boolean operation = true;
            while(operation){
                List<Integer>b = new ArrayList<>();
                b.add(a.get(0));
                int idx = 0;
                operation = false;
                for(int i = 1;i < a.size();i++){
                    if(a.get(i) < b.get(idx)){
                        operation = true;
                        continue;
                    }else{
                        b.add(a.get(i));
                        idx++;
                    }
                }
                a = b;
            }
            Map<Integer,Integer> hp = new HashMap<>();
            for(int x : a){
                hp.put(x,hp.getOrDefault(x,0)+1);
            }
            int max_count = 0;
            for(Map.Entry<Integer, Integer> val : hp.entrySet()){
                max_count = Math.max(max_count,val.getValue());
            }
            System.out.println(a.size()-max_count);
        }
    }

}

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

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

Think step by step.

Response:
To expose the bug in the given code, we need to focus on the logic of the Stalin Sort and how the code handles it. The main issue in the code is that it does not correctly handle the scenario where the array can be made vulnerable by removing the minimum number of elements. The code's approach of repeatedly applying Stalin Sort and then counting the maximum frequency of any element in the resulting array is flawed.

Here are the steps to create a challenging test input:

1. **Identify the Flaw**: The code incorrectly assumes that the number of elements to be removed is simply the size of the final array minus the maximum frequency of any element in the final array. This does not always hold true, especially when the array has multiple elements that can be removed to make it vulnerable.

2. **Create a Test Case**:
   - **Small Size**: Ensure the test case is small enough to fit within the constraints and be easily readable.
   - **Vulnerable Array**: The array should be such that it can be made non-increasing by removing the minimum number of elements.
   - **Edge Cases**: Include cases where the array has multiple elements that can be removed in different ways to make it vulnerable.

3. **Example Test Case**:
   - **Test Case 1**: An array where the minimum number of elements to be removed is not simply the size of the final array minus the maximum frequency of any element.
   - **Test Case 2**: An array that is already non-increasing to ensure the code handles this edge case correctly.

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

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

**Explanation**:
- **Test Case 1**: The array `[5, 4, 3, 2, 1]` is already non-increasing, so no elements need to be removed. The expected output is `0`.
- **Test Case 2**: The array `[3, 6, 2, 5, 4, 1]` can be made non-increasing by removing `3` and `6`, resulting in `[2, 5, 4, 1]`. The expected output is `2`.
- **Test Case 3**: The array `[1, 2, 3, 4, 5, 6, 7]` is already non-decreasing, and to make it non-increasing, all elements except the maximum one must be removed. The expected output is `6`.

This test input should help expose the bug in the given code by challenging its logic in handling different scenarios.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

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

#define int long long
#define pb push_back
#define ff first
#define ss second
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define endl "\n"
#define yes cout << "YES" << endl
#define no cout << "NO" << endl
#define fastio                                                                 \
  ios_base::sync_with_stdio(0);                                                \
  cin.tie(0);

#define vi vector<int>
#define vc vector<char>
#define vii vector<vector<int>>

#define mii map<int, int>
#define mci map<char, int>
#define mic map<int, char>

#define f(i, a, b) for (int i = (a); i < (b); ++i)
#define rf(i, a, b) for (int i = (a); i >= (b); --i)
#define si set<int>
#define sc set<char>

#ifndef ONLINE_JUDGE
#define debug(x)                                                               \
  cerr << #x << " ";                                                           \
  _print(x);                                                                   \
  cerr << endl;
#else
#define debug(x) ;
#endif

void _print(int t) { cerr << t; }
void _print(string t) { cerr << t; }
void _print(char t) { cerr << t; }
void _print(double t) { cerr << t; }
void _print(uint64_t t) { cerr << t; }
template <class T, class V> void _print(pair<T, V> p);
template <class T> void _print(vector<T> v);
template <class T> void _print(set<T> v);
template <class T, class V> void _print(map<T, V> v);
template <class T> void _print(multiset<T> v);
template <class T, class V> void _print(pair<T, V> p) {
  cerr << "{";
  _print(p.ff);
  cerr << ",";
  _print(p.ss);
  cerr << "}";
}
template <class T> void _print(vector<T> v) {
  cerr << "[ ";
  for (T i : v) {
    _print(i);
    cerr << " ";
  }
  cerr << "]";
}
template <class T> void _print(set<T> v) {
  cerr << "[ ";
  for (T i : v) {
    _print(i);
    cerr << " ";
  }
  cerr << "]";
}
template <class T> void _print(multiset<T> v) {
  cerr << "[ ";
  for (T i : v) {
    _print(i);
    cerr << " ";
  }
  cerr << "]";
}
template <class T, class V> void _print(map<T, V> v) {
  cerr << "[ ";
  for (auto i : v) {
    _print(i);
    cerr << " ";
  }
  cerr << "]";
}

const int MOD = (int)1e9 + 7;
const int MOD2 = 998244353;
const int MAX = 1e5 + 5;
const int INF = 1e9;
const int INFLL = 1e18;
void solve() {
  int n;
  cin >> n;
  vi a(n);
  for (auto &x : a)
    cin >> x;
  int cnt = 0;
  int i = 0;
  for (i; i < n - 1; i++) {
    if (a[i] < a[i + 1])
      cnt++;
    else
      break;
  }
  int ans = a[i];
  i++;
  for (i; i < n; i++) {
    if (a[i] > ans)
      cnt++;
  }
  cout << cnt << endl;
}
signed main() {

#ifndef ONLINE_JUDGE
  freopen("Error.txt", "w", stderr);
#endif
  fastio;

  int t = 1;
  cin >> t;

  while (t--) {
    solve();
  }

  return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided buggy code, we need to craft a test input that highlights the issue in the algorithm. The buggy code incorrectly counts the number of elements that need to be removed to make the array vulnerable. Specifically, it only counts elements that are strictly increasing and then stops, which is not the correct approach for this problem.

### Step-by-Step Analysis:
1. **Understanding the Bug**:
   - The code only checks for strictly increasing sequences and stops when it finds a non-increasing element.
   - It does not handle cases where the array can be made vulnerable by removing elements in a more complex manner.

2. **Edge Cases to Consider**:
   - Arrays with repeated elements.
   - Arrays with a mix of increasing and decreasing sequences.
   - Arrays that are already non-increasing.
   - Arrays that require multiple removals to become vulnerable.

3. **Crafting the Test Input**:
   - We need an array that requires more than one removal to become vulnerable and where the current logic will fail to identify the correct number of removals.

### Test Input:
Let's consider an array where the current logic will fail to correctly count the number of elements to be removed. For example, an array that has a mix of increasing and decreasing sequences and requires multiple removals to become vulnerable.

```plaintext
1
6
1 3 2 4 3 5
```

### Explanation:
- The array `[1, 3, 2, 4, 3, 5]` has a mix of increasing and decreasing sequences.
- To make it vulnerable, we need to remove elements such that the remaining array is non-increasing.
- The correct number of elements to remove is 3, resulting in the array `[1, 2, 3, 3]` which is non-increasing.
- The buggy code will likely stop at the first non-increasing element and fail to correctly count the required removals.

This test input should 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: Stalin Sort
time_limit_ms: 1000
memory_limit_mb: 256
Description: Stalin Sort is a humorous sorting algorithm designed to eliminate elements which are out of place instead of bothering to sort them properly, lending itself to an $\mathcal{O}(n)$ time complexity.

It goes as follows: starting from the second element in the array, if it is strictly smaller than the previous element (ignoring those which have already been deleted), then delete it. Continue iterating through the array until it is sorted in non-decreasing order. For example, the array $[1, 4, 2, 3, 6, 5, 5, 7, 7]$ becomes $[1, 4, 6, 7, 7]$ after a Stalin Sort.

We define an array as vulnerable if you can sort it in non-increasing order by repeatedly applying a Stalin Sort to any of its subarrays$^{\text{∗}}$, as many times as is needed.

Given an array $a$ of $n$ integers, determine the minimum number of integers which must be removed from the array to make it vulnerable.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

Input: Each test consists of several test cases. The first line contains a single integer $t$ ($1 \le t \le 500$) — the number of test cases. This is followed by descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 2000$) — the size of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2000$.

Output: For each test case, output a single integer — the minimum number of integers which must be removed from the array to make it vulnerable.

Examples:
input:
6
7
3 6 4 9 2 5 2
5
5 4 4 2 2
8
2 2 4 4 6 6 10 10
1
1000
9
6 8 9 10 12 9 7 5 4
7
300000000 600000000 400000000 900000000 200000000 400000000 200000000
output:
2
0
6
0
4
2

Note: In the first test case, the optimal answer is to remove the numbers $3$ and $9$. Then we are left with $a = [6, 4, 2, 5, 2]$. To show this array is vulnerable, we can first apply a Stalin Sort on the subarray $[4, 2, 5]$ to get $a = [6, 4, 5, 2]$ and then apply a Stalin Sort on the subarray $[6, 4, 5]$ to get $a = [6, 2]$, which is non-increasing.

In the second test case, the array is already non-increasing, so we don't have to remove any integers.

Buggy Code:
// Problem: B. Stalin Sort
// Contest: Codeforces - Codeforces Round 982 (Div. 2)
// URL: https://codeforces.com/contest/2027/problem/B
// Memory Limit: 256 MB
// Time Limit: 1000 ms
//
// Powered by CP Editor (https://cpeditor.org)

#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;

#define int long long

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

class fenwick_tree{
	int n;
	vector<int> bit;
	public:
	fenwick_tree(int _n){
		n = _n;
		bit.resize(n + 1);
	}

	void update(int ind){
		for(int i = ind; i <= n; i += i&-i)bit[i]++;
	}
	int query(int l, int r){
	 	int result = 0;
	 	for(int i = r; i > 0; i -= i&-i)result += bit[i];
	 	for(int i = l - 1; i > 0; i -= i&-i)result -= bit[i];
	 	return result;
	 }
};
void solve(){
	int n; cin >> n;
	int a[n], indices[n];
	for(int i = 0; i < n; i++)cin >> a[i];
	iota(indices, indices + n, 0);

	sort(indices, indices + n, [&](int x, int y){
		return a[x] > a[y];
	});

	fenwick_tree ft(n);
	int result = n;
	for(int &ind: indices){
		int index = ind + 1;
		int query = ft.query(index, n);
		ft.update(index);
		result = min(result, query + index - 1);
	}
	cout << result << endl;
}

signed main(){
	int t = 1; cin >> t;
	while(t--)solve();
	return 0;
}

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the provided code, we need to consider edge cases and scenarios where the code's logic might fail. The provided code uses a Fenwick Tree (or Binary Indexed Tree) to keep track of the indices and their positions, and it calculates the minimum number of elements to remove to make the array vulnerable. However, the code might fail in scenarios where the array has multiple elements with the same value or where the array is already in a specific order that the code does not handle correctly.

Let's consider the following scenarios:

1. **Array with Multiple Identical Elements**: The code might not handle identical elements correctly, especially when they are in a non-decreasing order.
2. **Array Already in Non-Increasing Order**: The code might not recognize that no elements need to be removed.
3. **Array with a Single Element**: The code should handle this case trivially.
4. **Array with Elements in Reverse Order**: The code should correctly identify that no elements need to be removed.
5. **Array with Elements in a Pattern that Requires Specific Removals**: The code should correctly identify the minimum number of elements to remove.

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

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

### Explanation:
1. **Array with Multiple Identical Elements**: `[3, 3, 3, 3]` - The array is already in non-decreasing order, and no elements need to be removed.
2. **Array Already in Non-Increasing Order**: `[3, 2, 1]` - The array is already in non-increasing order, and no elements need to be removed.
3. **Array with a Single Element**: `[5]` - The array is trivially vulnerable, and no elements need to be removed.
4. **Array with Elements in Reverse Order**: `[5, 4, 3, 2, 1]` - The array is already in non-increasing order, and no elements need to be removed.
5. **Array with Elements in a Pattern that Requires Specific Removals**: `[3, 6, 4, 9, 2, 5, 2]` - This array requires removing 2 elements to make it vulnerable.

This test input should help expose any issues in the code's logic, especially in handling identical elements and already sorted arrays.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include<iostream>
using namespace std;
#define ll long long
const int mod=1e9+7;

ll xpow(ll a,ll b){
    if(b==1)return a%mod;

    ll x=xpow(a,b/2)%mod;

    ll v1=(x*x)%mod;
    ll v2=(v1*(a%mod))%mod;

    if(b%2==0)return v1;
    else return v2;
}

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

    pair<ll,ll> a[n];
    for(int i=0;i<n;++i) cin>>a[i].first;
    for(int i=0;i<n;++i) cin>>a[i].second;
    ll ans=0;
    ll root[n];
    ll s1=0;
    ll s2=s1;
    ll total=0;
    for(int i=0;i<n;++i){
        root[i]=a[0].first*(a[i].first+a[i].second) + a[0].second*a[i].first;

        total+=root[i];
        total%=mod;

        s1+=a[i].first; s2+=a[i].second;

    }

    //cout<<total<<'\n';

    for(int i=0;i<n;++i){

        ans+= xpow(  ((root[i])*((a[i].first*(s1+s2)) + a[i].second*s1)) ,m-1)%mod;
        ans%=mod;
    }
    ans%=mod;
    cout<<ans;
}

int main(){
    ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
    //freopen("task.inp","r",stdin);freopen("task.out","w",stdout);

    int t=1;
    while(t--)solve();
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases or scenarios that maximize the failure probability. Here are the steps to create such a test input:

1. **Understand the Bug**: The buggy code has issues with the power calculation and the way it handles the combinations of trails. Specifically, the power function `xpow` and the way the combinations are calculated might not handle large values correctly.

2. **Edge Cases**:
   - **Small Number of Days (n)**: Ensure that the code works correctly for small values of `n`.
   - **Large Number of Days (n)**: Test the code with a large value of `n` to see if the power function and modulo operations handle it correctly.
   - **Zero Short or Long Trails**: Test cases where some cabins have zero short or long trails.
   - **Equal Short and Long Trails**: Test cases where the number of short and long trails are equal for some cabins.

3. **Small-Scale Test Input**:
   - We will choose a small number of cabins (`m`) and a moderate number of days (`n`) to keep the input size manageable.
   - We will include a mix of cabins with different numbers of short and long trails to test various combinations.

Given these considerations, here is a small-scale test input that should expose the bug:

```plaintext
3 3
1 0 2
0 1 1
```

Explanation:
- **m = 3**: There are 3 cabins.
- **n = 3**: Harry walks for 3 days.
- **s = [1, 0, 2]**: The number of short trails for each cabin.
- **l = [0, 1, 1]**: The number of long trails for each cabin.

This test input includes:
- A cabin with only short trails.
- A cabin with only long trails.
- A cabin with both short and long trails.
- A moderate number of days to test the power function and combination calculations.

This should help in exposing any issues with the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include<iostream>
using namespace std;
#define ll long long
const ll mod=1e9+7;

ll xpow(ll a,ll b){
    if(b==1)return a%mod;

    ll x=xpow(a,b/2)%mod;

    ll v1=(x*x)%mod;
    ll v2=(v1*a)%mod;

    if(b%2==0)return v1;
    else return v2;
}

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

    pair<ll,ll> a[n];
    for(int i=0;i<n;++i) cin>>a[i].first;
    for(int i=0;i<n;++i) cin>>a[i].second;
    ll ans=0;
    ll root[n];
    ll s1=0;
    ll s2=s1;
    ll total=0;
    for(int i=0;i<n;++i){
        root[i]=a[0].first*(a[i].first+a[i].second) + a[0].second*a[i].first;

        total+=root[i];
        total%=mod;

        s1+=a[i].first; s2+=a[i].second;

    }

    //cout<<total<<'\n';

    for(int i=0;i<n;++i){

        ans+= xpow(  ((root[i]%mod)*(a[i].first*(s1+s2) + a[i].second*s1)%mod)%mod ,m-1)%mod;
        ans%=mod;
    }

    //ans*=total;
    ans%=mod;
    cout<<ans;
}

int main(){
    ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
    //freopen("task.inp","r",stdin);freopen("task.out","w",stdout);

    int t=1;
    while(t--)solve();
}

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that can maximize the failure probability. Here are the steps to think about:

1. **Edge Cases for Input Values:**
   - Use the smallest and largest possible values for `m` and `n` within the constraints.
   - Use the smallest and largest possible values for `s_i` and `l_i` within the constraints.

2. **Special Cases:**
   - All cabins have only short trails.
   - All cabins have only long trails.
   - Mixed cases where some cabins have both short and long trails.

3. **Large Number of Days:**
   - Use a large value for `n` to test the power function and modular arithmetic.

4. **Specific Combinations:**
   - Test with a combination that might cause overflow or incorrect modular arithmetic.

Given these considerations, let's create a test input:

```plaintext
3 1000000000
1000 0 1000
0 1000 1000
```

Explanation:
- `m = 3`: Three cabins.
- `n = 1000000000`: A very large number of days.
- `s_1 = 1000`, `s_2 = 0`, `s_3 = 1000`: Cabin 1 and Cabin 3 have 1000 short trails, Cabin 2 has no short trails.
- `l_1 = 0`, `l_2 = 1000`, `l_3 = 1000`: Cabin 1 has no long trails, Cabin 2 and Cabin 3 have 1000 long trails.

This input will test the code's ability to handle large numbers and mixed trail types, potentially exposing any issues with modular arithmetic and power calculations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
// LUOGU_RID: 171526305
#include<bits/stdc++.h>

using namespace std;

const int mod = 1e9 + 7;
int n, m;
int s[105], l[105], sum[105], dp[1005][105], ans;

signed main()
{
	cin >> m >> n;
	for (int i = 1; i <= m; i++) cin >> s[i];
	for (int i = 1; i <= m; i++)
	{
		cin >> l[i];
		sum[i] = l[i] + s[i];
	}
	dp[0][1] = 1;
	for (int i = 1; i <= n; i++)
		for (int j = 1; j <= m; j++)
			for (int k = 1; k <= m; k++)
				dp[i][j] = (dp[i][j] + ((sum[k] * sum[j] - l[k] * l[j]) % mod) * dp[i - 1][k]) % mod;
	for (int i = 1; i <= m; i++) ans = (ans + dp[n][i]) % mod;
	cout << ans % mod << 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 create a challenging test input that exposes the bug in the provided code, we need to consider edge cases that might cause issues. The code appears to have a logical error in how it calculates the number of valid trail combinations, particularly in the way it handles the modulo operation and the transition between states.

### Step-by-Step Analysis:

1. **Edge Cases:**
   - **Small \( m \) and \( n \):** This helps in verifying the basic logic.
   - **Large \( n \):** This tests the efficiency and correctness of the modulo operation.
   - **Zero Trails:** This tests the handling of cabins with no trails.
   - **Single Trail:** This tests the handling of cabins with only short or only long trails.
   - **All Short Trails:** This tests the handling of cabins where all trails are short.
   - **All Long Trails:** This tests the handling of cabins where all trails are long.

2. **Challenging Scenario:**
   - **Combination of Short and Long Trails:** This tests the logic for ensuring at least one trail is short.

### Chosen Test Input:
Let's create a small-scale test input that includes a mix of short and long trails, and a large \( n \) to test the modulo operation.

```plaintext
3 1000
1 0 1
0 1 1
```

### Explanation:
- **\( m = 3 \):** We have 3 cabins.
- **\( n = 1000 \):** Harry walks for 1000 days. This is a large number to test the efficiency and correctness of the modulo operation.
- **Short Trails:**
  - Cabin 1: 1 short trail
  - Cabin 2: 0 short trails
  - Cabin 3: 1 short trail
- **Long Trails:**
  - Cabin 1: 0 long trails
  - Cabin 2: 1 long trail
  - Cabin 3: 1 long trail

This input ensures that:
- Cabin 1 has only one short trail.
- Cabin 2 has only one long trail.
- Cabin 3 has one short and one long trail.

The combination of these trails will challenge the logic in the code, especially the part where it ensures at least one trail is short each day. The large \( n \) will also test the efficiency and correctness of the modulo operation.

This test input should help expose any issues 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: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
MOD = 10**9 + 7

def matrix_mult(A, B, m):
    # Multiplies two matrices A and B of size m x m
    C = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            C[i][j] = sum(A[i][k] * B[k][j] for k in range(m)) % MOD
    return C

def matrix_exponentiation(T, n, m):
    # Exponentiates matrix T to the power of n
    result = [[1 if i == j else 0 for j in range(m)] for i in range(m)]
    base = T
    while n > 0:
        if n % 2 == 1:
            result = matrix_mult(result, base, m)
        base = matrix_mult(base, base, m)
        n //= 2
    return result

def count_trail_combinations(m, n, s, l):
    # Construct the transition matrix T
    T = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            T[i][j] = (s[i] * s[j] + s[i] * l[j] + l[i] * s[j]) % MOD

    # Exponentiate the matrix T to the power of n
    T_n = matrix_exponentiation(T, n, m)

    # The result is the sum of the first row of T^n
    result = sum(T_n[0][j] for j in range(m)) % MOD
    return result

# Read input
m, n = map(int, input().split())
s = list(map(int, input().split()))
l = list(map(int, input().split()))

# Calculate the number of possible combinations
result = count_trail_combinations(m, n, s, l)

# Output the result
print(result)

**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 highlight potential issues in the matrix construction and exponentiation steps. Here are some key points to consider:

1. **Small but Non-trivial Number of Cabins (m):**
   - A small number of cabins (e.g., 2 or 3) can help us quickly verify the correctness of the matrix operations without overwhelming complexity.

2. **Large Number of Days (n):**
   - Using a large number of days (e.g., \(10^9\)) can test the efficiency and correctness of the matrix exponentiation.

3. **Edge Cases for Trails:**
   - Cabins with only short trails.
   - Cabins with only long trails.
   - Cabins with both short and long trails.
   - Cabins with zero trails (though this might be less relevant given the constraints).

Given these considerations, let's construct a test input that includes a mix of these scenarios:

```plaintext
3 1000000000
1 0 1
0 1 1
```

### Explanation:
- **m = 3**: We have 3 cabins.
- **n = 1000000000**: A very large number of days to test the matrix exponentiation.
- **s = [1, 0, 1]**: Cabin 1 and Cabin 3 have 1 short trail each, and Cabin 2 has no short trails.
- **l = [0, 1, 1]**: Cabin 1 has no long trails, Cabin 2 has 1 long trail, and Cabin 3 has 1 long trail.

This input ensures that:
- There are cabins with only short trails.
- There are cabins with only long trails.
- There are cabins with both short and long trails.
- The large number of days will test the efficiency and correctness of the matrix exponentiation.

This test input should help expose any issues in the matrix construction and exponentiation steps.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
MOD = 10**9 + 7

def matrix_mult(A, B, m):
    # Multiplies two matrices A and B of size m x m
    C = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            C[i][j] = sum(A[i][k] * B[k][j] for k in range(m)) % MOD
    return C

def matrix_exponentiation(T, n, m):
    # Exponentiates matrix T to the power of n
    result = [[1 if i == j else 0 for j in range(m)] for i in range(m)]
    base = T
    while n > 0:
        if n % 2 == 1:
            result = matrix_mult(result, base, m)
        base = matrix_mult(base, base, m)
        n //= 2
    return result

def solve(m, n, s, l):
    # Construct the transition matrix T
    T = [[0] * m for _ in range(m)]
    for i in range(m):
        for j in range(m):
            T[i][j] = (s[i] * s[j] + s[i] * l[j] + l[i] * s[j]) % MOD

    # Exponentiate the matrix T to the power of n
    T_n = matrix_exponentiation(T, n, m)

    # The result is the sum of the first row of T^n
    result = sum(T_n[0][j] for j in range(m)) % MOD
    return result

# Read input
import sys
input = sys.stdin.read
data = input().split()

m = int(data[0])
n = int(data[1])
s = list(map(int, data[2:m+2]))
l = list(map(int, data[m+2:2*m+2]))

# Solve the problem
print(solve(m, n, s, l))

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to consider edge cases and scenarios that can highlight potential issues. Here are some key points to consider:

1. **Zero Trails**: Cases where cabins have zero short or long trails.
2. **Large Number of Days**: Cases where `n` is very large, which can cause issues in matrix exponentiation.
3. **Single Cabin**: Cases where there is only one cabin.
4. **Symmetry in Trails**: Cases where the number of short and long trails is the same for all cabins.

Given the constraints, a small-scale test input that can expose potential issues is:

```plaintext
2 1000000000
1 0
0 1
```

### Explanation:

- **m = 2**: Two cabins.
- **n = 1000000000**: A very large number of days, which can stress the matrix exponentiation.
- **s = [1, 0]**: Cabin 1 has 1 short trail, Cabin 2 has 0 short trails.
- **l = [0, 1]**: Cabin 1 has 0 long trails, Cabin 2 has 1 long trail.

This input ensures that:
- There is a clear asymmetry in the trails.
- The large number of days can potentially expose issues in the matrix exponentiation logic.
- The combination of short and long trails is such that the transition matrix should be carefully constructed to ensure the correct number of valid paths.

This test input should help identify any bugs in the matrix construction and exponentiation logic.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
t=[int(n) for n in input().split()]
s=[int(n) for n in input().split()]
l=[int(n) for n in input().split()]
def trails(n,s,l,count):
    total=0
    if count==t[1]:
        return total
    count+=1
    for i in range(s[n]):
        for x in range(t[0]):
            if s[x]+l[x]>0:
                total+=s[x]+l[x]+trails(x,s,l,count)
    for i in range(l[n]):
        for x in range(t[0]):
            if s[x]>0:
                total+=s[x]+trails(x,s,l,count)
    return total

print(trails(0,s,l,0)+2%(10**9+7))

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

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

Think step by step.

Response:
```plaintext
2 3
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: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

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

public class Main {
    private static final int MOD = 1000000007;
    //private static final int MOD = 998244353;
    private static int add(long a, long b) { return (int) (((a+MOD)%MOD+(b+MOD)%MOD)%MOD); }
    private static int multiply(long a, long b) { return (int) (((a%MOD)*(b%MOD))%MOD); }
    public static int[][] multiplyMat(int[][] a, int[][] b) {
        int[][] c = new int[a.length][b[0].length];
        for(int i = 0; i < a.length; i++) {
            for(int j = 0; j < b[0].length; j++) {
                for(int k = 0; k < a[0].length; k++) {
                    c[i][j] = add(c[i][j], multiply(a[i][k], b[k][j]));
                }
            }
        }
        return c;
    }
    public static int[][] powerMat(int[][] a, long b) {
        int[][] c = new int[a.length][a.length];
        for(int i = 0; i < a.length; i++) {
            c[i][i] = 1;
        }
        while(b > 0) {
            if((b&1) > 0) {
                c = multiplyMat(c, a);
            }
            a = multiplyMat(a, a);
            b >>= 1;
        }
        return c;
    }
    public static void main(String[] args) throws IOException {
        //BufferedReader f = new BufferedReader(new FileReader("uva.in"));
        BufferedReader f = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter out = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
        StringTokenizer st = new StringTokenizer(f.readLine());
        int m = Integer.parseInt(st.nextToken());
        int n = Integer.parseInt(st.nextToken());
        st = new StringTokenizer(f.readLine());
        int[] s = new int[m];
        for(int i = 0; i < m; i++) {
            s[i] = Integer.parseInt(st.nextToken());
        }
        st = new StringTokenizer(f.readLine());
        int[] l = new int[m];
        for(int i = 0; i < m; i++) {
            l[i] = Integer.parseInt(st.nextToken());
        }
        int[][] matrix = new int[m][m];
        for(int i = 0; i < m; i++) {
            for(int j = 0; j < m; j++) {
                matrix[i][j] = s[i]*l[j]+s[j]*l[i]+s[i]*s[j];
            }
        }
        matrix = powerMat(matrix, n);
        int ans = 0;
        for(int i = 0; i < m; i++) {
            ans = add(ans, matrix[0][i]);
        }
        out.println(ans);
        f.close();
        out.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 generate a challenging test input that exposes the bug in the given buggy code, let's consider the following aspects:

1. **Edge Cases**:
   - **Single Cabin**: What if there is only one cabin? The algorithm should handle this case correctly.
   - **No Short Trails**: What if there are cabins with no short trails? This could lead to issues in the matrix construction.
   - **No Long Trails**: Similarly, what if there are cabins with no long trails?
   - **All Short Trails**: What if all cabins have only short trails? This could simplify the problem but still needs to be handled correctly.
   - **All Long Trails**: What if all cabins have only long trails? This should be impossible due to the constraint that at least one trail must be short.

2. **Challenging Combinations**:
   - **Mixed Trails**: What if some cabins have a mix of short and long trails, and others have only one type of trail? This can test the matrix multiplication and power functions.

3. **Large Input**:
   - **Large Number of Days**: Use a large value for `n` to test the efficiency and correctness of the matrix exponentiation.

Given these considerations, let's create a test input that includes a mix of cabins with different trail configurations and a large number of days:

```plaintext
5 1000000000
1 0 1 0 1
0 1 0 1 0
```

Explanation:
- **5 Cabins**: We have 5 cabins to ensure the matrix is of a manageable size but still non-trivial.
- **Day Count**: `n = 1000000000` to test the efficiency and correctness of the matrix exponentiation.
- **Trail Configuration**:
  - Cabin 1: 1 short, 0 long
  - Cabin 2: 0 short, 1 long
  - Cabin 3: 1 short, 0 long
  - Cabin 4: 0 short, 1 long
  - Cabin 5: 1 short, 0 long

This configuration ensures that there are cabins with only short trails, only long trails, and a mix of both. It should expose any issues in the matrix construction and the matrix exponentiation process.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
/**
 * E_2_Trails_Medium
 */
import java.util.*;
public class E_2_Trails_Medium {

    static int mod = 1000000009;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int m = sc.nextInt();
        int n = sc.nextInt();
        long s[] = new long[m];
        long l[] = new long[m];
        for(int i=0; i<m; i++){
            s[i] = sc.nextLong();
        }
        for(int i=0; i<m; i++){
            l[i] = sc.nextLong();
        }
        long arr[][] = new long[m][m];
        for(int i=0; i<m; i++){
            for(int j=0; j<m; j++){
                arr[i][j] = (l[i]+s[i])*(l[j]+s[j]) - l[j]*l[i];
                arr[i][j] %= mod;
            }
        }
        long dp[] = new long[m];
        for(int i=0; i<m; i++){
            dp[i] =0;
        }
        dp[0] = 1;
        for(int i=1;i<=n; i++)
        {
            int dp1[] = new int[m];
            for(int j=0; j<m; j++){
                dp1[j]=0;
            }
            for(int j=0; j<m; j++){
                for(int k=0; k<m; k++){
                    long val = dp[k];
                    val *= arr[k][j];
                    val %= mod;
                    dp1[j] += val;
                    dp1[j] %= mod;
                }
            }
            for(int j=0; j<m;j++){
                dp[j] = dp1[j];
            }
        }
        long ans =0;
        for(int i=0; i<m; i++){
            ans += dp[i];
            ans %= mod;
        }
        System.out.println(ans);
        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 generate a challenging test input that exposes the bug in the given code, we need to focus on edge cases or scenarios where the code might fail. Here are the steps to think through this:

1. **Edge Cases for Input Constraints**:
   - Minimum and maximum values for `m` and `n`.
   - Minimum and maximum values for `s_i` and `l_i`.

2. **Behavior of the Code**:
   - The code constructs an adjacency matrix `arr` where each element `arr[i][j]` represents the number of valid trail combinations between cabin `i` and cabin `j`.
   - The code then uses dynamic programming to calculate the number of valid trail combinations over `n` days.

3. **Potential Bugs**:
   - The modulo operation might be applied incorrectly, leading to overflow issues.
   - The initialization of the DP array might be incorrect.
   - The transition between days in the DP array might be incorrect.

4. **Challenging Input**:
   - Use a small `m` to keep the input small but complex enough to expose the bug.
   - Use a large `n` to test the long-term behavior of the DP array.
   - Use a mix of `s_i` and `l_i` values to test the trail combination logic.

Given these considerations, a challenging test input could be:

```plaintext
3 1000
1 0 1
0 1 1
```

This input:
- Has a small `m` (3) to keep the input manageable.
- Uses a large `n` (1000) to test the long-term behavior.
- Uses a mix of `s_i` and `l_i` values to test the trail combination logic.

This input should help expose any issues in the modulo operations, initialization, or transition logic in the DP array.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Trails (Medium)
time_limit_ms: 1000
memory_limit_mb: 256
Description: Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.

Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.

How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?

Give the answer modulo $$$10^9 + 7$$$.

Input: The first line contains the integers $$$m$$$ and $$$n$$$.

The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.

The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.

We have the following constraints:

$$$0 \le s_i, l_i \le 10^3$$$.

$$$1 \le m \le 10^2$$$.

$$$1 \le n \le 10^9$$$.

Output: The number of possible combinations of trails, modulo $$$10^9 + 7$$$.

Examples:
input:
3 2
1 0 1
0 1 1
output:
18

Note:

Buggy Code:
#include <bits/stdc++.h>
#include <cassert>

typedef unsigned short u16;
typedef short i16;
typedef unsigned int u32;
typedef int i32;
typedef unsigned long long u64;
typedef long long i64;

const u32 MOD = 1e9 + 7;

u32 iadd(u32 a, u32 b) { return (a + b) % MOD; }
u32 imul(u32 a, u32 b) { return (a * b) % MOD; }

class Matrix {
protected:
  u32 row, col;
  std::vector<std::vector<u32>> mtx;

public:
  Matrix(u32 _n, u32 _m) : row(_n), col(_m), mtx(row, std::vector<u32>(col)) {}
  const u32 &at(u32 x, u32 y) const { return mtx[x][y]; }
  u32 &at(u32 x, u32 y) { return mtx[x][y]; }
  Matrix mul(const Matrix &other) {
    assert(col == other.row);
    Matrix res(row, other.col);
    for (u32 i = 0; i < row; ++i) {
      for (u32 k = 0; k < col; ++k) {
        for (u32 j = 0; j < other.col; ++j) {
          res.at(i, j) = iadd(res.at(i, j), imul(at(i, k), other.at(k, j)));
        }
      }
    }
    return res;
  }
  static Matrix identity(u32 n) {
    Matrix res(n, n);
    while (n--) {
      res.at(n, n) = 1;
    }
    return res;
  }
  Matrix pow(u32 k) {
    assert(row == col);
    Matrix res = Matrix::identity(row);
    for (auto exp = *this; k; k >>= 1, exp = exp.mul(exp)) {
      if (k & 1) {
        res = res.mul(exp);
      }
    }
    return res;
  }
};

int main(void) {
  u32 m, n;
  std::cin >> m >> n;
  Matrix mtx(m, m);
  std::vector<u32> s(m), l(m);
  for (auto &i : s) {
    std::cin >> i;
  }
  for (auto &i : l) {
    std::cin >> i;
  }
  for (u32 i = 0; i < m; ++i) {
    for (u32 j = 0; j < m; ++j) {
      mtx.at(i, j) =
          imul(iadd(s[i], l[i]), iadd(s[j], l[j])) - imul(l[i], l[j]);
    }
  }
  Matrix res = mtx.pow(n);
  u32 ans = 0;
  for (u32 i = 0; i < m; ++i) {
    ans += res.at(0, i);
  }
  std::cout << ans << '\n';
}

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

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

Think step by step.

Response:
```plaintext
2 1
1 0
0 1
```