**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
import java.io.*;
import java.lang.*;
import java.math.BigInteger;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.math.MathContext;
import java.util.*;

public class D_Counting_Pairs
{
    static class Pair {
        char c;
        int fre;
        Pair(char c, int fre){
            this.c = c;
            this.fre = fre;
        }
    }
    static class FastReader
    {
        BufferedReader br;
        StringTokenizer st;

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

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

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

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

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

        String nextLine()
        {
            String str = "";
            try
            {
                str = br.readLine();
            }
            catch (IOException e)
            {
                e.printStackTrace();
            }
            return str;
        }
    }
    static void sort(long[] arr) {
        ArrayList<Long> a = new ArrayList<>();
        for(long i: arr) {
            a.add(i);
        }
        Collections.sort(a);
        for(int i = 0; i < a.size(); i++) {
            arr[i] = a.get(i);
        }
    }
    static int LongestIncreasingSubsequenceLength(long v[])
    {
        if (v.length == 0)
            return 0;

        long[] tail = new long[v.length];
        int length = 1;
        tail[0] = v[0];

        for (int i = 1; i < v.length; i++) {

            if (v[i] > tail[length - 1]) {
                // v[i] extends the largest subsequence
                tail[length++] = v[i];
            }
            else {

                int idx = Arrays.binarySearch(
                        tail, 0, length - 1, v[i]);

                if (idx < 0)
                    idx = -1 * idx - 1;

                tail[idx] = v[i];
            }
        }
        return length;
    }
    static ArrayList<Long> factors(long x) {
        ArrayList<Long> list = new ArrayList<>();
        HashSet<Long> set = new HashSet<>();
        long i = 1;
        long sum = 0;
        while (i * i <= x) {

            if (x % i == 0) {

                set.add(i);
                sum+=i;
                if (x / i != i) {

                    set.add(x/i);
                    sum += (x/i);
                }
            }
            i++;

        }
        for(long ii: set){
            list.add(ii);
        }
        Collections.sort(list);

        return list;
    }
    public static String collapseZeros(String binaryString) {
        StringBuilder collapsedBuilder = new StringBuilder();

        for (int i = 0; i < binaryString.length(); i++) {
            char currentChar = binaryString.charAt(i);

            if (currentChar == '1') {
                collapsedBuilder.append(currentChar);
            } else {

                while (i < binaryString.length() && binaryString.charAt(i) == '0') {
                    i++;
                }

                collapsedBuilder.append('0');

                i--;
            }
        }

        return collapsedBuilder.toString();
    }
    public static int findIndex(long[] arr, long val) {
        int left = 0, right = arr.length - 1;
        int result = -1;

        while (left <= right) {
            int mid = left + (right - left) / 2;
            if (arr[mid] >= val) {
                result = mid;
                right = mid - 1;
            } else {
                left = mid + 1;
            }
        }
        return result;
    }
    public static int findIndexLast(long[] arr, long val) {
        int left = 0, right = arr.length - 1;
        int result = -1;

        while (left <= right) {
            int mid = left + (right - left) / 2;
            if (arr[mid] <= val) {
                result = mid;
                left = mid + 1;
            } else {
                right = mid - 1;
            }
        }
        return result;
    }
    public static int findMin(long[] arr, long val) {
        int left = 0, right = arr.length - 1;
        int result = -1;

        while (left <= right) {
            int mid = left + (right - left) / 2;
            if (arr[mid] <= val) {
                result = mid;
                left = mid + 1;
            } else {
                right = mid - 1;
            }
        }
        return result;
    }

    public static void main(String[] args) throws Exception {
        FastReader sc = new FastReader();
        PrintWriter out = new PrintWriter(System.out);
        int t =sc.nextInt();
        while(t-- > 0){
            int n = sc.nextInt();
            long x = sc.nextLong();
            long y = sc.nextLong();
            long[] arr = new long[n];
            long sum = 0;
            HashMap<Long,Long> map = new HashMap<>();
            for(int i = 0; i < n; i++){
                arr[i] = sc.nextLong();
                sum += arr[i];
                map.put(arr[i],map.getOrDefault(arr[i], 0l) +1);
            }

            sort(arr);
            long ans = 0;
            for(int i = 0; i < n; i++){
                long val = arr[i];
                long dummy = sum - val;
                if(dummy < x){
                    continue;
                }
                if(dummy >= y){
                    long a = dummy - y;
                    long b = dummy - x;
                    long min = Math.min(a,b);
                    long max = Math.max(a,b);
                    int minIndex = findIndex(arr, min);
                    int maxIndex = findIndexLast(arr, max);
                    if(minIndex == -1 || maxIndex == -1){
                        continue;
                    }
                    ans += (maxIndex - minIndex +1);
                    if(arr[minIndex] <= val && arr[maxIndex] >= val){
                        ans -= (map.get(val) == 1 ? 1 : map.get(val)-1);
                    }
                   // out.println(minIndex +" "+maxIndex+ " "+ ans+ " "+min);

                }else{
                    long min = dummy - x;
                    int minIndex = findMin(arr, min);
                    if(minIndex == -1){
                        continue;
                    }

                    ans += minIndex+1;
                    if(val <= min){
                        ans -= (map.get(val) == 1 ? 1 : map.get(val)-1);

                    }
                 //   out.println(minIndex + " "+ ans+ " "+min);

                }

            }
            out.println(ans/2);

        }
        out.flush();
    }

}

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

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

Think step by step.

Response:
```plaintext
1
3 6 10
1 2 3
```

### Explanation:
This test case is designed to expose a potential bug in the logic for handling the minimum and maximum bounds for the sums of the remaining elements after removing a pair.

- **Input Details:**
  - `n = 3`: The sequence has 3 elements.
  - `x = 6`: The minimum sum of the remaining elements after removing a pair should be at least 6.
  - `y = 10`: The maximum sum of the remaining elements after removing a pair should be at most 10.
  - `a = [1, 2, 3]`: The sequence of elements.

- **Expected Output:**
  - The only possible pair is (1, 3), because:
    - Removing elements at positions 1 and 3 leaves the sequence [2], and the sum is 2, which is not in the range [6, 10].
    - Removing elements at positions 1 and 2 leaves the sequence [3], and the sum is 3, which is not in the range [6, 10].
    - Removing elements at positions 2 and 3 leaves the sequence [1], and the sum is 1, which is not in the range [6, 10].

Since no pairs satisfy the conditions, the expected output is `0`.

- **Potential Bug Exposure:**
  - The code should correctly handle the scenario where no pairs satisfy the conditions. If the code incorrectly counts any pairs, it will fail this test case.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

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

public class Template {
    public static void main(String[] args) throws FileNotFoundException {
        Locale.setDefault(Locale.ROOT);
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
//        PrintWriter out = new PrintWriter(new FileOutputStream("lca.out"));
//        InputReader in = new InputReader(new FileInputStream("lca.in"));

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

    public static void solve(InputReader in, PrintWriter out) {
        int t = in.nextInt();

        for (int i = 0; i < t; i++) {
            int n = in.nextInt();

            long x = in.nextLong();
            long y = in.nextLong();

            long sum = 0;

            int[] arr = new int[n];

            for (int j = 0; j < n; j++) {
                int num = in.nextInt();
                arr[j] = num;
                sum += num;
            }

            Arrays.sort(arr);

            int l = 0;
            int r = n-1;
            long ans = 0;

            for (int j = 0; j < n; j++) {
                long b = sum - arr[j];
                if (l == j) {
                    l++;
                }
                if (l > r) {
                    break;
                }
                while (y < b - arr[l]) {
                    l++;
                    if (l > r) {
                        break;
                    }
                }
                while (x > b - arr[r]) {
                    r--;
                    if (r < l) {
                        break;
                    }
                }
                if (l > r) {
                    break;
                }
                ans += r - l + 1;
            }

            out.println(ans);

        }

    }

    static class InputReader {
        public BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32078);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

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

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

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

        String nextLine() {
            try {
                return reader.readLine();
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }
    }
}

**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 craft a test case that highlights the issues in the logic. The main logic in the code involves calculating the sum of the array and then using two pointers to find pairs that meet the conditions. Here are the steps to think about the test case:

1. **Edge Cases**:
   - When the array is very small (e.g., 3 elements).
   - When the array has many duplicate values.
   - When the values of `x` and `y` are very close to each other.
   - When the values of `x` and `y` are very large compared to the sum of the array.

2. **Behavior of the Code**:
   - The code sorts the array and uses two pointers to find valid pairs.
   - The code might have issues when the sum of the array minus the current element is very close to `x` or `y`.

3. **Test Case**:
   - We need a test case where the sum of the array minus a few elements is very close to `x` or `y`, and the code might incorrectly count or miss some pairs.

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

```plaintext
1
5 10 15
5 5 5 5 5
```

**Explanation**:
- The array `[5, 5, 5, 5, 5]` has a total sum of 25.
- Removing any two elements will result in a sum of 15, which is exactly within the range `[10, 15]`.
- There are 10 possible pairs: (1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5).

This test case should expose any issues in the logic where the code might incorrectly count or miss pairs.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

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

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

    for (int i = 0; i < n; i++)
    {
        long long n,x,y;
        cin >> n >> x >> y;

        int s = 0;

        long long arr[n];

        for (long long i = 0; i < n; i++)
        {
            cin >> arr[i];
            s+= arr[i];
        }

        sort(arr,arr+n);
        int ans = 0;

        for (long long i = 0; i < n-1; i++)
        {
            if (arr[i] > s-x)
            {
                break;
            }
            if (arr[i] + arr[n-1] < s-y)
            {
                continue;
            }

            for (long long j= i+1; j < n; j++)
            {
                if (arr[i] + arr[j] > s - x)
                {
                    break;
                }
                if (arr[i] + arr[j] < s - y)
                {
                    continue;
                }

                ans++;

            }

        }
        cout << ans<< 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:
```plaintext
1
5 10 15
1 2 3 4 5
```

**Explanation:**
- The test case is designed to check the edge condition where the sum of the remaining elements is exactly at the boundary values of `x` and `y`.
- The sequence is `[1, 2, 3, 4, 5]` and the sum of all elements is `15`.
- If we remove the elements at positions `(1, 2)`, the remaining elements are `[3, 4, 5]` with a sum of `12`, which is within the range `[10, 15]`.
- Similarly, removing elements at positions `(1, 3)`, `(1, 4)`, `(2, 3)`, `(2, 4)`, and `(3, 4)` also results in sums within the range `[10, 15]`.
- The expected output is `6`, but the buggy code might miss some of these pairs due to incorrect logic in the inner loops.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

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

using namespace std;

void solve();

int main(){
    ios_base::sync_with_stdio(false), cin.tie(NULL);
    stringstream outputBuffer;
    streambuf* originalCoutBuffer=cout.rdbuf();
    cout.rdbuf(outputBuffer.rdbuf());
    cout << fixed << setprecision(0);
    int t; cin >> t; t--; while (t--) solve();
    solve();
    cout.rdbuf(originalCoutBuffer);
    cout<<outputBuffer.str();
    system("pause > nul");
}

#define endl '\n'
#define all(x) x.begin(), x.end()
#define rall(x) x.rbegin(), x.rend()
#define ll long long
#define mod 1000000007
#define npos string::npos
#define nfor(a, b) for (int i = a; i < b; i++)
#define rfor(b, a) for (int i = b - 1; i >= a; i--)
#define lower_bound_index(v, x) lower_bound(all(v), x) - v.begin()
#define upper_bound_index(v, x) upper_bound(all(v), x) - v.begin()
#define bitcount(x) __builtin_popcountll(x)
#define sz(x) ((int)(x).size())
#define pb push_back

int sum(int x) {
    return x * (x + 1) / 2;
}

int sum(int x, int y) {
    if (x <= 0) return sum(y);
    else return sum(y) - sum(x - 1);
}

bool isPrime(int n) {
    if (n <= 1) return false;
    if (n <= 3) return true;
    if (n % 2 == 0 || n % 3 == 0) return false;
    for (int i = 5; i * i <= n; i += 6) if (n % i == 0 || n % (i + 2) == 0) return false;
    return true;
}

void solve() {
    int n, x, y; cin >> n >> x >> y;
    vector<int> a(n);
    for (int i = 0; i < n; i++) cin >> a[i];
    int tot = a.back();
    int cnt = 0;
    vector<map<int, int>> sto(n);
    for (int i = n - 2; i >= 0; i--) {
        tot += a[i];
        sto[i] = sto[i + 1];
        sto[i][a[i + 1]]++;
    }
    for (int i = 0; i < n - 1; i++) {
        for (auto &j : sto[i]) {
            int curr = tot - a[i] - j.first;
            if (x <= curr && curr <= y) {
                cnt += j.second;
            }
        }
    }
    cout << cnt << endl;
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases where the code might fail. The code seems to handle the general logic of counting pairs, but there are a few potential issues:

1. **Edge Case with Very Small or Large Values**: The code might not handle very small or very large values of `x` and `y` correctly.
2. **Edge Case with All Elements Being the Same**: This can cause issues in the way the `sto` map is populated and used.
3. **Edge Case with a Single Element in the Sequence**: The code might not handle this correctly.
4. **Edge Case with a Large Number of Elements**: The code might have performance issues with a large number of elements.

Given the constraints and the need for a small-scale test input, let's focus on the edge case with all elements being the same, as this can simplify the problem and potentially expose the bug.

### Test Input
```plaintext
1
3 1 1
1 1 1
```

### Explanation
- **Number of Test Cases**: 1
- **Sequence Length**: 3
- **Range for Sum**: 1 to 1
- **Sequence**: [1, 1, 1]

### Expected Output
```plaintext
0
```

### Reasoning
- The sequence consists of three 1s.
- The sum of the remaining elements after removing any two elements is always 1.
- Since the range for the sum is also 1 to 1, the sum of the remaining elements is always within the range.
- However, the code might not correctly handle the case where all elements are the same, leading to an incorrect count of pairs.

This test input is small and complete, and it should 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: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

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

public class Q2 {
    // For fast input output
    static class FastReader {
        BufferedReader br;
        StringTokenizer st;

        public FastReader() {
            try {
                br = new BufferedReader(new FileReader("input.txt"));
                PrintStream out = new PrintStream(new FileOutputStream("output.txt"));
                System.setOut(out);
            } catch (Exception e) {
                br = new BufferedReader(new InputStreamReader(System.in));
            }
        }

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

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

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

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

        String nextLine() {
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }
    }
    // end of fast i/o code

    public static void main(String[] args) {
        FastReader reader = new FastReader();
        int t = reader.nextInt();
        int n;

        while (t > 0) {
            n = reader.nextInt();
            int x = reader.nextInt();
            int y = reader.nextInt();
            long[] a = new long[n];
            for (int i = 0; i < n; i++) {
                a[i] = reader.nextInt();
            }

            t--;
            solve(n, x, y, a);
        }
    }

    static void solve(int n, int x, int y, long[] a) {
        long sum = 0;

        for (long k : a) {
            sum += k;
        }

        long ans = 0;
        HashMap<Long, Long> map = new HashMap<>();

        for (int i = 0; i < n; i++) {
            long dif = sum - a[i];
            if((dif-x) >= (dif-y)) {
                for (long j = (dif- y); j <= (dif - x); j++) {
                    ans += map.getOrDefault(j, 0L);
                }
            }
            map.put(a[i], map.getOrDefault(a[i], 0L) + 1);
        }
        System.out.println(ans);

    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
import java.util.Scanner;
public class Solution{
	static long countPairs(int[] nums, long n, long x, long y){
		long sum = 0;
		long count = 0;
		for(int num : nums){
			sum += num;
		}

		for(int i = 0; i < n; i++){
			for(int j = i+1; j < n; j++){
				long hold = sum - (nums[i]+nums[j]);
				if(hold >= x && hold <= y){
					count++;
				}
			}
		}
		return count;
	}
	public static void main(String[] args){
		Scanner scn = new Scanner(System.in);
		long inputs = scn.nextInt();

		for(int k = 0; k < inputs; k++){
			int n = scn.nextInt();
			long x = scn.nextInt();
			long y = scn.nextInt();

			int[] nums = new int[n];
			for(int i = 0; i < n; i++){
				nums[i] = scn.nextInt();
			}
			System.out.println(countPairs(nums, n, x, y));
		}

	}
}

**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 code, we need to consider edge cases and scenarios that might cause the code to fail. Here are the steps to think through this:

1. **Edge Cases for Input Constraints**:
   - Test with the minimum and maximum values for `n`, `x`, and `y`.
   - Test with sequences where all elements are the same.
   - Test with sequences where elements are at the boundary values (1 and 10^9).

2. **Special Cases for Pairs**:
   - Test with sequences that have no interesting pairs.
   - Test with sequences that have all possible pairs as interesting pairs.
   - Test with sequences that have a mix of interesting and non-interesting pairs.

3. **Performance Considerations**:
   - Test with large `n` to ensure the code handles large inputs efficiently.

Given these considerations, a small-scale test input that can expose potential bugs is:

```plaintext
3
3 10 15
1 2 3
4 1 10
1 2 3 4
6 1 10
1 1 2 2 2 3
```

### Explanation:
1. **Test Case 1**:
   - `n = 3`, `x = 10`, `y = 15`
   - Sequence: `1, 2, 3`
   - Sum of the sequence: `1 + 2 + 3 = 6`
   - No pairs can be removed to make the remaining sum between 10 and 15.
   - Expected output: `0`

2. **Test Case 2**:
   - `n = 4`, `x = 1`, `y = 10`
   - Sequence: `1, 2, 3, 4`
   - Sum of the sequence: `1 + 2 + 3 + 4 = 10`
   - Possible pairs: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)
   - All pairs are interesting because the remaining sum will be between 1 and 10.
   - Expected output: `6`

3. **Test Case 3**:
   - `n = 6`, `x = 1`, `y = 10`
   - Sequence: `1, 1, 2, 2, 2, 3`
   - Sum of the sequence: `1 + 1 + 2 + 2 + 2 + 3 = 11`
   - Possible pairs: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)
   - Most pairs are interesting because the remaining sum will be between 1 and 10.
   - Expected output: `15`

This test input covers a range of scenarios, including sequences with no interesting pairs, sequences with all possible pairs as interesting pairs, and a sequence with a mix of interesting and non-interesting pairs. This should help in exposing any bugs 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: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
def s1(w,x,y,z):
  if y-x==1:
    if w[x]>=z:
      return x
    else:
      return y
  else:
    mid = (x+y)//2
    if w[mid]>=z:
      return s1(w,x,mid,z)
    else:
      return s1(w,mid,y,z)
def s2(w,x,y,z):
  if y-x==1:
    if w[y]<=z:
      return y
    else:
      return x
  else:
    mid = (x+y)//2
    if w[mid]<=z:
      return s2(w,mid,y,z)
    else:
      return s2(w,x,mid,z)
for t in range(int(input())):
  n,x,y = map(int,input().split())
  a = list(map(int,input().split()))
  a.sort()
  s = 0
  su = sum(a)
  for i in range(n):
    if a[0]<=(su-a[i]-x) and a[-1]>=(su-a[i]-y):
      l = s1(a,0,n-1,su-a[i]-y)
      r = s2(a,0,n-1,su-a[i]-x)
      s += max(0,r-l+1)

      if x<=(su-a[i]-a[i]) and y>=(su-a[i]-a[i]):
        s -= 1

  print(s//2)

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

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

Think step by step.

Response:
```plaintext
3
4 9 11
1 3 5 7
3 1 1
1 1 1
6 12 15
1 2 3 4 5 6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
for _ in range(int(input())):
    n,x,y = list(map(lambda x: int(x), input().split(" ")))
    ar = list(map(lambda x: int(x), input().split(" ")))
    s = sum(ar)
    count = 0
    for i in range(n):
        for j in range(i+1, n):
            if x <= s-(ar[i]+ar[j]) <= y:
                count += 1
    print(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 generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that can cause the code to fail. Here are the steps to consider:

1. **Edge Case with Small Values**: Test with small values of `n`, `x`, and `y` to ensure the code handles small inputs correctly.
2. **Edge Case with Large Values**: Test with large values of `n`, `x`, and `y` to ensure the code handles large inputs efficiently.
3. **Boundary Values**: Test with boundary values for `x` and `y` to ensure the code handles these correctly.
4. **Special Cases**: Test with special cases like all elements being the same, or elements being very close to the boundary values of `x` and `y`.

Given the constraints, we will create a small-scale test input that is complete and valid. Here is a challenging test input:

```plaintext
3
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
```

This test input includes:
1. A small case with `n = 3` and `x = 3`, `y = 6` where the elements are 3, 2, 1.
2. A case with `n = 4` and `x = 4`, `y = 12` where the elements are 3, 3, 2, 1.
3. A case with `n = 6` and `x = 8`, `y = 8` where the elements are 1, 1, 2, 2, 2, 3.

These cases are designed to test the code's ability to handle small and large values, boundary conditions, and special cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
#include <bits/stdc++.h>
using i64 = long long;
void solve() {
	i64 n, x, y;
	std::cin >> n >> x >> y;
	std::vector<int> a(n);
	i64 sum = 0;
	for (int i = 0; i < n; i++) {
		std::cin >> a[i];
		sum += a[i];
	}
	int ans = 0;
	std::ranges::sort(a);
	for (int i = 0, j = n - 1; i < n && i < j; i++) {
		i64 now = sum - a[i];
		if (now <= x) {
			continue;
		}
		while (i < j && now - a[j] < x) {
			j--;
		};
		int l = i + 1, r = j, res = j + 1;
		while (l <= r) {
			int mid = l + r >> 1;
			if (now - a[mid] <= y) {
				r = (res = mid) - 1;
			} else {
				l = mid + 1;
			}
		}
		ans += j - res + 1;
		// std::cout << "res " << res << '\n';
		// std::cout << i << ' ' << ans << '\n';
	}
	std::cout << ans << '\n';
}
int main() {
	std::ios::sync_with_stdio(false);
	std::cin.tie(0);
	int t;
	std::cin >> t;
	while (t--) {
		solve();
	}
	return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given code, we need to consider edge cases that might cause the algorithm to fail. The code provided has a logic to find pairs of indices such that the sum of the remaining elements falls within a specified range. Here are some potential edge cases to consider:

1. **All elements are the same**: This can lead to incorrect binary search results.
2. **Elements are very close to the boundary values**: This can cause the sum to be just outside the range.
3. **Very large or very small elements**: This can cause overflow or underflow issues.
4. **Elements that are just slightly above or below the range limits**: This can test the precision of the comparison logic.

Let's construct a small-scale test case that includes these elements:

```plaintext
4
5 10 15
1 2 3 4 5
5 10 15
5 5 5 5 5
5 10 15
1 10 10 10 10
5 10 15
1 100 1000 10000 100000
```

### Explanation:
1. **Test Case 1**: All elements are distinct and within a small range.
   - Sequence: `1 2 3 4 5`
   - Range: `10 15`
   - Expected Output: `4` (Pairs: (1, 4), (1, 5), (2, 5), (3, 5))

2. **Test Case 2**: All elements are the same.
   - Sequence: `5 5 5 5 5`
   - Range: `10 15`
   - Expected Output: `0` (No pairs meet the criteria)

3. **Test Case 3**: One element is very small, and the rest are large.
   - Sequence: `1 10 10 10 10`
   - Range: `10 15`
   - Expected Output: `4` (Pairs: (1, 2), (1, 3), (1, 4), (1, 5))

4. **Test Case 4**: Elements are very large.
   - Sequence: `1 100 1000 10000 100000`
   - Range: `10 15`
   - Expected Output: `0` (No pairs meet the criteria)

This test input should help expose any issues in the given code, particularly in handling edge cases and precision in comparisons.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define loop(n,i) for(decltype(n) i=0;i<n;i++)
#define print output cout<<output<<endl;
#define set(a,n,k) for(decltype(n) lambda=0;lambda<n;lambda++){a[lambda]=k;}
#define out(a,n,k) for(decltype(n) lambda=0;lambda<n;lambda++){cout<<a[lambda]<<k;}
#define in(a,n) for(decltype(n) lambda=0;lambda<n;lambda++){cin>>a[lambda];}
#define f first
#define s second
#define pb push_back
#define mp make_pair
#define pii pair<int, int>
#define pll pair<long long, long long>
#define vi vector<int>
#define vll vector<long long>
#define mii map<int, int>
#define mll map<long long>
#define si set<int>
#define sc set<char>
const ll mod = 1e9+7;
//gcd
int gcd(int a, int b) {
if (b == 0) {
return a;
}
return gcd(b, a % b);
}
//lcm
int lcm(int a, int b) {
return a * b / gcd(a, b);
}
//comp
bool comp(pair<int, int>& a, pair<int, int>& b){
return a.second < b.second;
}
bool comp2(pair<int, int>& a, pair<int, int>& b){
if(a.second==b.second)
return a.first<b.first;
return a.second < b.second;
}
//generating all subsequences using power set
//loop(1<<n, num){
//sub=;
//loop(n,i){
//if(num & (1<<i)) sub+=s[i];
//}
//}
//nCr
ll combi(ll n, ll r){
ll ans=1;
for(ll i=0; i<r; i++){
ans*=(n-i);
ans/=(i+1);
}
return ans;
}
//modular exponentiation
ll pov(ll base, ll exp, ll mod){
ll result = 1;
base = base % mod;
while (exp > 0){
if (exp % 2 == 1){
result = (result * base) % mod;
}
exp = exp >> 1;
base = (base * base) % mod;
}
return result;
}
void solve(){
    ll n,x,y;
    cin>>n>>x>>y;
    ll arr[n];
    ll Tsum=0;
    loop(n,i){
        cin>>arr[i];
        Tsum+=arr[i];
    }
    ll cnt=0;
    for(ll i=0; i<n; i++){
        for(ll j=i+1; j<n; j++){
            ll Psum = arr[i]+arr[j];
            if(Psum>= Tsum-y && Psum<=Tsum-x) cnt++;
        }
    }
    cout<<cnt<<endl;
}
int main()
{
ios::sync_with_stdio(0), cin.tie(0), cout.tie(0);
int t;
cin >> t;
while (t--)
{
solve();
}
 return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that are likely to cause issues. Here are the key points to consider:

1. **Edge Cases for `x` and `y`**: Test cases where `x` and `y` are very close or where `x` is equal to `y`.
2. **Edge Cases for `n`**: Test cases where `n` is very small (e.g., 3) or very large (e.g., 200,000).
3. **Edge Cases for `a_i`**: Test cases where all elements in the array are the same, or where the array has a mix of very large and very small values.
4. **Edge Cases for Sum Constraints**: Test cases where the sum of the array is just slightly greater than `y` or just slightly less than `x`.

Let's create a test input that includes these edge cases:

```plaintext
3
4 10 10
5 5 5 5
3 8 8
1 2 7
200000 200000000000000 200000000000000
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
t=int(input())
for i in range(t):
    l=list(map(int,input().split()))
    n=l[0]
    x=l[1]
    y=l[2]
    a=list(map(int,input().split()))
    speed=sorted(a)
    s=sum(a)
    if s-speed[-1]-speed[-2]>y:
        print(0)
    elif s-speed[0]-speed[1]<x:
        print(0)
    else:
        arr=speed
        count=0
        n = len(arr)
        total_count = 0

        for i in range(n - 1):
            target = s - arr[i]
            start, end = -1, -1

            left, right = i + 1, n - 1
            while left <= right:
                mid = left + (right - left) // 2
                if x <= target - arr[mid] <= y:
                    start = mid
                    right = mid - 1
                elif target - arr[mid] < x:
                    right = mid - 1
                else:
                    left = mid + 1

            left, right = i + 1, n - 1
            while left <= right:
                mid = left + (right - left) // 2
                if x <= target - arr[mid] <= y:
                    end = mid
                    left = mid + 1
                elif target - arr[mid] < x:
                    right = mid - 1
                else:
                    left = mid + 1

            if start != -1 and end != -1 and start <= end:
                total_count += end - start + 1

        print(total_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 generate a challenging test input that exposes the bug in the buggy code, we need to focus on edge cases where the code's logic might fail. The code has issues with handling the correct range of sums and the correct identification of valid pairs. Here are some considerations:

1. **Edge Case with Small `n`**: Test with the minimum possible value of `n` (3) to see if the code handles small sequences correctly.
2. **Edge Case with Large `n`**: Test with a sequence that is close to the maximum limit of `n` (200,000) to check performance and correctness.
3. **Edge Case with `x` and `y` Close to Each Other**: Test with `x` and `y` values that are very close to each other to see if the code can handle tight constraints.
4. **Edge Case with All Elements Being the Same**: Test with a sequence where all elements are the same to see if the code can handle uniform sequences.
5. **Edge Case with a Sequence Where No Pairs Are Interesting**: Test with a sequence where no pairs meet the criteria to see if the code correctly identifies such cases.
6. **Edge Case with a Sequence Where All Pairs Are Interesting**: Test with a sequence where all possible pairs meet the criteria to see if the code can handle such cases.

Given the constraints and the need for a small-scale test input, let's focus on a few of these edge cases:

### Test Input
```plaintext
4
3 10 10
5 5 5
4 10 10
1 2 3 4
4 1 100
1 2 3 4
6 10 10
1 2 3 4 5 6
```

### Explanation
1. **Test Case 1**: `3 10 10 5 5 5`
   - All elements are the same, and the only possible pair is (1, 2), (1, 3), and (2, 3). The sum of the remaining element (5) is 5, which is not within the range [10, 10]. Expected output: 0.
2. **Test Case 2**: `4 10 10 1 2 3 4`
   - Possible pairs: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4).
   - Only (1, 4) and (2, 3) meet the criteria. Expected output: 2.
3. **Test Case 3**: `4 1 100 1 2 3 4`
   - All pairs should be valid because the sum of the remaining elements will always be within the wide range [1, 100]. Expected output: 6.
4. **Test Case 4**: `6 10 10 1 2 3 4 5 6`
   - Possible pairs: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6).
   - Only (1, 6) and (2, 5) meet the criteria. Expected output: 2.

This test input should help expose any bugs in the code by covering a variety of edge cases.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
def solve(a, x, y):
    n = len(a)
    summ = [0]
    s = 0
    for nm in a:
        s += nm
        summ.append(s)
    ans = 0
    for i in range(n):
        for j in range(i + 1, n):

            b = summ[n] - a[i] - a[j]
            if y >= b >= x:
                ans += 1
    return ans
def slove():
    t = int(input())
    for _ in range(t):
        n, x, y = map(int, input().split())
        a = list(map(int, input().split()))
        print(solve(a, x, y))
if __name__ == "__main__":
    slove()

**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
3 5 10
1 2 3
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
import java.util.*;
public class Main
{
    public static void main(String args[])
    {
        Scanner sc=new Scanner(System.in);
        int t=sc.nextInt();
        while(t-->0)
        {
            int n=sc.nextInt();
            long x=sc.nextLong(),y=sc.nextLong();
            long a[]=new long[n];
            long s=0;
            for(int i=0;i<n;i++)
            {
                a[i]=sc.nextInt();
                s+=a[i];
            }
            long c=0;
            int i=0,j=i+1;
            for(i=0;i<n-1;i++)
            {
                j=i+1;
                while(j<n)
                {
                    long r = s - a[i] - a[j];
                    if (r >= x && r <= y)
                        c++;
                    j++;
                }
            }
            System.out.println(c);
        }
    }
}

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

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

Think step by step.

Response:
```plaintext
1
5 10 10
1 1 1 1 6
```

### Explanation:
- **Test Case:** A single test case with `n = 5`, `x = 10`, and `y = 10`.
- **Sequence:** The sequence is `[1, 1, 1, 1, 6]`.
- **Sum of the sequence:** The total sum `s` is `10`.

### Reasoning:
- **Interesting Pairs:**
  - If we remove the pair `(1, 2)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(1, 3)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(1, 4)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(1, 5)`, the remaining elements are `[1, 1, 1]` with a sum of `3` (not in the range `[10, 10]`).
  - If we remove the pair `(2, 3)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(2, 4)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(2, 5)`, the remaining elements are `[1, 1, 1]` with a sum of `3` (not in the range `[10, 10]`).
  - If we remove the pair `(3, 4)`, the remaining elements are `[1, 1, 6]` with a sum of `8` (not in the range `[10, 10]`).
  - If we remove the pair `(3, 5)`, the remaining elements are `[1, 1, 1]` with a sum of `3` (not in the range `[10, 10]`).
  - If we remove the pair `(4, 5)`, the remaining elements are `[1, 1, 1]` with a sum of `3` (not in the range `[10, 10]`).

- **Expected Output:** The expected number of interesting pairs is `0`.

This test case is designed to expose a potential issue in the buggy code where it might incorrectly count pairs that do not meet the criteria. The sum of the sequence is exactly `10`, but removing any two elements will result in a sum that is less than `10`, thus no pairs should be counted.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Counting Pairs
time_limit_ms: 2000
memory_limit_mb: 256
Description: You are given a sequence $$$a$$$, consisting of $$$n$$$ integers, where the $$$i$$$-th element of the sequence is equal to $$$a_i$$$. You are also given two integers $$$x$$$ and $$$y$$$ ($$$x \le y$$$).

A pair of integers $$$(i, j)$$$ is considered interesting if the following conditions are met:

- $$$1 \le i < j \le n$$$;
- if you simultaneously remove the elements at positions $$$i$$$ and $$$j$$$ from the sequence $$$a$$$, the sum of the remaining elements is at least $$$x$$$ and at most $$$y$$$.

Your task is to determine the number of interesting pairs of integers for the given sequence $$$a$$$.

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

Each test case consists of two lines:

- The first line contains three integers $$$n, x, y$$$ ($$$3 \le n \le 2 \cdot 10^5$$$, $$$1 \le x \le y \le 2 \cdot 10^{14}$$$);
- The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^{9}$$$).

Additional constraint on the input: the sum of $$$n$$$ across all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer — the number of interesting pairs of integers for the given sequence $$$a$$$.

Examples:
input:
7
4 8 10
4 6 3 6
6 22 27
4 9 6 3 4 5
3 8 10
3 2 1
3 1 1
2 3 4
3 3 6
3 2 1
4 4 12
3 3 2 1
6 8 8
1 1 2 2 2 3
output:
4
7
0
0
1
5
6

Note: In the first example, there are $$$4$$$ interesting pairs of integers:

1. $$$(1, 2)$$$;
2. $$$(1, 4)$$$;
3. $$$(2, 3)$$$;
4. $$$(3, 4)$$$.

Buggy Code:
import java.util.*;
public class Main {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-- > 0){
        int n = sc.nextInt();
        long a = sc.nextLong();
        long b = sc.nextLong();
        long []arr = new long[n];
        long sum = 0;
        for(int i = 0; i < n; i++){
            long v = sc.nextLong();
            sum += v;
            arr[i] = v;
        }
        Arrays.sort(arr);
        long c = 0;
        long maxi = sum - a;
        long mini = sum - b;
        for(int i = 0; i < n; i++){
            if(arr[i] > maxi)break;
            else {
                for(int j = i+1; j < n; j++){
                    if(arr[j] > maxi)break;
                    if(arr[i] + arr[j] >= mini && maxi >= arr[i]+arr[j])c++;
                }
            }
        }
        System.out.println(c);

    }
}

    }

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

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

Think step by step.

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

1. **Understand the Bug**: The bug in the code seems to be in the logic for determining the interesting pairs. Specifically, the code calculates the sum of the array and then iterates through the array to find pairs that, when removed, leave a sum within the given range. However, the logic for checking the conditions is flawed.

2. **Edge Cases to Consider**:
   - **Small Array**: Test with a small array to ensure the logic is correct.
   - **Large Array**: Test with a large array to ensure the algorithm handles large inputs efficiently.
   - **Edge Values**: Test with values of `x` and `y` that are very close to each other or very far apart.
   - **Uniform Values**: Test with an array where all elements are the same.
   - **Mixed Values**: Test with an array that has a mix of large and small values.

3. **Generate Test Input**:
   - We will start with a small-scale test input that is complete and valid.
   - We will use a test case where the values of `x` and `y` are close to each other and the array elements are such that the sum of the remaining elements is just within the range.

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

```plaintext
1
5 10 15
1 2 3 4 5
```

Explanation:
- **Test Case 1**:
  - `n = 5`, `x = 10`, `y = 15`
  - Array: `1, 2, 3, 4, 5`
  - The sum of the array is `15`.
  - The pairs to check are:
    - Removing (1, 2) leaves `3, 4, 5` with sum `12` (valid).
    - Removing (1, 3) leaves `2, 4, 5` with sum `11` (valid).
    - Removing (1, 4) leaves `2, 3, 5` with sum `10` (valid).
    - Removing (1, 5) leaves `2, 3, 4` with sum `9` (invalid).
    - Removing (2, 3) leaves `1, 4, 5` with sum `10` (valid).
    - Removing (2, 4) leaves `1, 3, 5` with sum `9` (invalid).
    - Removing (2, 5) leaves `1, 3, 4` with sum `8` (invalid).
    - Removing (3, 4) leaves `1, 2, 5` with sum `8` (invalid).
    - Removing (3, 5) leaves `1, 2, 4` with sum `7` (invalid).
    - Removing (4, 5) leaves `1, 2, 3` with sum `6` (invalid).

The correct output for this test case should be `4` (the valid pairs are (1, 2), (1, 3), (1, 4), (2, 3)). This test case should help expose the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Matrix Transformation
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given two matrices $$$A$$$ and $$$B$$$ of size $$$n \times m$$$, filled with integers between $$$0$$$ and $$$10^9$$$. You can perform the following operations on matrix $$$A$$$ in any order and any number of times:

- &=: choose two integers $$$i$$$ and $$$x$$$ ($$$1 \le i \le n$$$, $$$x \ge 0$$$) and replace each element in row $$$i$$$ with the result of the bitwise AND operation between $$$x$$$ and that element. Formally, for every $$$j \in [1, m]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ & } x$$$;
- |=: choose two integers $$$j$$$ and $$$x$$$ ($$$1 \le j \le m$$$, $$$x \ge 0$$$) and replace each element in column $$$j$$$ with the result of the bitwise OR operation between $$$x$$$ and that element. Formally, for every $$$i \in [1, n]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ | } x$$$.

The value of $$$x$$$ may be chosen differently for different operations.

Determine whether it is possible to transform matrix $$$A$$$ into matrix $$$B$$$ using the given operations any number of times (including zero).

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Then, $$$t$$$ test cases follow.

Each test case is given as follows:

- the first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 10^3$$$; $$$n \cdot m \le 10^3$$$) — the dimensions of the matrices $$$A$$$ and $$$B$$$;
- the following $$$n$$$ lines describe the matrix $$$A$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$A_{i,1}, A_{i,2}, \dots, A_{i,m}$$$ ($$$0 \le A_{i,j} \le 10^9$$$);
- the following $$$n$$$ lines describe the matrix $$$B$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$B_{i,1}, B_{i,2}, \dots, B_{i,m}$$$ ($$$0 \le B_{i,j} \le 10^9$$$).

Output: For each test case, output Yes if it is possible to transform the matrix $$$A$$$ into the matrix $$$B$$$; otherwise, output No. Each letter can be output in any case, upper or lower.

Examples:
input:
4
1 1
12
13
2 2
10 10
42 42
21 21
21 21
2 2
74 10
42 106
21 85
85 21
2 4
1 2 3 4
5 6 7 8
3 2 3 4
1 0 1 0
output:
Yes
Yes
No
Yes

Note: Let's consider the second set of input data and show a sequence of operations that transforms matrix $$$A$$$ into matrix $$$B$$$:

Initially, the matrix looks like this:

$$$\begin{bmatrix} 10&10\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 1$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 2$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 0&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 1$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&0\\ 21&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 2$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&21\\ 21&21\\ \end{bmatrix}$$$

Thus, we have transformed matrix $$$A$$$ into matrix $$$B$$$.

Buggy Code:
#include <stdio.h>
#include <vector>
using namespace std;

int n,m,a[1010][1010],b[1010][1010],c[1010];
vector <int> edge[2020];
bool v[2020];

bool dfs(int x){
	if(v[x] == 1)
		return 0;
	v[x] = 1;
	for(auto y : edge[x])
		if(!dfs(y)){
			v[x] = 0;
			return 0;
		}
	v[x] = 0;
	return 1;
}

bool check(int x){
	for(int i = 1;i <= n;i++)
		for(int j = 1;j <= m;j++){
			if(b[i][j] & x)
				edge[i].push_back(n + j);
			else
				edge[n + j].push_back(i);
		}
	for(int i = 1;i <= n;i++)
		for(int j = 1;j <= m;j++){
			if((a[i][j] & x) && !(b[i][j] & x) && !dfs(n + j))
				return 0;
			if(!(a[i][j] & x) && (b[i][j] & x) && !dfs(i))
				return 0;
		}
	return 1;
}

bool solve(bool r){
	for(int i = 1;i <= n;i++)
		for(int j = 1;j <= m;j++)
			scanf("%d",&a[i][j]);
	for(int i = 1;i <= n;i++)
		for(int j = 1;j <= m;j++)
			scanf("%d",&b[i][j]);
	for(int x = 0;x <= 30;x++){
		for(int i = 1;i <= n + m;i++)
			edge[i].clear(), v[i] = 0;
		if(!check(1 << x))
			return 0;
	}
	return r;
}

int main(){
	int _; scanf("%d",&_);
	while(_--){
		scanf("%d%d",&n,&m);
		puts(solve(1) ? "Yes" : "No");
	}
}

**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 the logic that checks the feasibility of transforming matrix \(A\) into matrix \(B\). The buggy code uses a depth-first search (DFS) to check the feasibility, but it might have issues with handling certain edge cases or specific bit patterns.

### Key Points to Consider:
1. **Bitwise Operations**: The operations involve bitwise AND and OR, which can lead to complex interactions between the elements of the matrices.
2. **Edge Cases**: Small matrices with specific values that can cause the DFS to fail.
3. **Bit Patterns**: Values that have specific bit patterns that might not be correctly handled by the DFS.

### Test Input:
Let's create a test case that involves a small matrix with values that can expose the bug. Specifically, we will use values that have specific bit patterns that might cause the DFS to fail.

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

### Explanation:
- **Matrix A**:
  \[
  \begin{bmatrix}
  1 & 2 \\
  3 & 4 \\
  \end{bmatrix}
  \]
- **Matrix B**:
  \[
  \begin{bmatrix}
  2 & 3 \\
  1 & 4 \\
  \end{bmatrix}
  \]

### Why This Test Case?
- **Bit Patterns**: The values in the matrices have specific bit patterns. For example, `1` (001), `2` (010), `3` (011), and `4` (100). These bit patterns can cause the DFS to fail if the code does not correctly handle the bitwise operations.
- **Transformation Feasibility**: It is not possible to transform matrix \(A\) into matrix \(B\) using the given operations. Specifically, the transformation from `1` to `2` and from `3` to `1` in the same row or column is not feasible with the given operations.

### Expected Output:
The expected output for this test case should be:
```
No
```

This test case should help expose the bug in the provided code by forcing the DFS to fail in handling the specific bit patterns and transformations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Matrix Transformation
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given two matrices $$$A$$$ and $$$B$$$ of size $$$n \times m$$$, filled with integers between $$$0$$$ and $$$10^9$$$. You can perform the following operations on matrix $$$A$$$ in any order and any number of times:

- &=: choose two integers $$$i$$$ and $$$x$$$ ($$$1 \le i \le n$$$, $$$x \ge 0$$$) and replace each element in row $$$i$$$ with the result of the bitwise AND operation between $$$x$$$ and that element. Formally, for every $$$j \in [1, m]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ & } x$$$;
- |=: choose two integers $$$j$$$ and $$$x$$$ ($$$1 \le j \le m$$$, $$$x \ge 0$$$) and replace each element in column $$$j$$$ with the result of the bitwise OR operation between $$$x$$$ and that element. Formally, for every $$$i \in [1, n]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ | } x$$$.

The value of $$$x$$$ may be chosen differently for different operations.

Determine whether it is possible to transform matrix $$$A$$$ into matrix $$$B$$$ using the given operations any number of times (including zero).

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Then, $$$t$$$ test cases follow.

Each test case is given as follows:

- the first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 10^3$$$; $$$n \cdot m \le 10^3$$$) — the dimensions of the matrices $$$A$$$ and $$$B$$$;
- the following $$$n$$$ lines describe the matrix $$$A$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$A_{i,1}, A_{i,2}, \dots, A_{i,m}$$$ ($$$0 \le A_{i,j} \le 10^9$$$);
- the following $$$n$$$ lines describe the matrix $$$B$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$B_{i,1}, B_{i,2}, \dots, B_{i,m}$$$ ($$$0 \le B_{i,j} \le 10^9$$$).

Output: For each test case, output Yes if it is possible to transform the matrix $$$A$$$ into the matrix $$$B$$$; otherwise, output No. Each letter can be output in any case, upper or lower.

Examples:
input:
4
1 1
12
13
2 2
10 10
42 42
21 21
21 21
2 2
74 10
42 106
21 85
85 21
2 4
1 2 3 4
5 6 7 8
3 2 3 4
1 0 1 0
output:
Yes
Yes
No
Yes

Note: Let's consider the second set of input data and show a sequence of operations that transforms matrix $$$A$$$ into matrix $$$B$$$:

Initially, the matrix looks like this:

$$$\begin{bmatrix} 10&10\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 1$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 2$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 0&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 1$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&0\\ 21&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 2$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&21\\ 21&21\\ \end{bmatrix}$$$

Thus, we have transformed matrix $$$A$$$ into matrix $$$B$$$.

Buggy Code:
#include <iostream>

using namespace std;

const int N = 1e3 + 5, NM = 1e6 + 5;

int n, m, a[N][N], b[N][N];
int h[N << 1], nxt[NM], to[NM], tot;
bool vis[N << 1], in[N << 1], need[N << 1];

void add(int u, int v) {
    nxt[++tot] = h[u], to[h[u] = tot] = v;
}

bool dfs(int u) {
    if(in[u]) {
        return 0;
    }

    in[u] = 1;
    for(int i = h[u]; i; i = nxt[i]) {
        int v = to[i];

        bool flag = dfs(v);

        if(!flag) {
            return 0;
        }
    }
    in[u] = 0;
    vis[u] = 1;

    return 1;
}

bool check() {
    for(int p = 0; p < 30; p++) {
        tot = 0;
        for(int i = 1; i <= n + m; i++) {
            h[i] = 0;
            need[i] = 0;
        }

        for(int i = 1; i <= n; i++) {
            for(int j = 1; j <= m; j++) {
                int aa = ((a[i][j] >> p) & 1), bb = ((b[i][j] >> p) & 1);

                if(bb == 0) {
                    add(j + n, i);
                } else {
                    add(i, j + n);
                }

                if(aa == 0 && bb == 1) {
                    need[j + n] = 1;
                }
                if(aa == 1 && bb == 0) {
                    need[i] = 1;
                }
            }
        }

        for(int i = 1; i <= n + m; i++) {
            if(need[i]) {
                for(int j = 1; j <= n + m; j++) {
                    vis[j] = 0;
                    in[i] = 0;
                }
                if(!dfs(i)) {
                    return 0;
                }
            }
        }
    }

    return 1;
}

void solve() {
    cin >> n >> m;
    for(int i = 1; i <= n; i++) {
        for(int j = 1; j <= m; j++) {
            cin >> a[i][j];
        }
    }
    for(int i = 1; i <= n; i++) {
        for(int j = 1; j <= m; j++) {
            cin >> b[i][j];
        }
    }

    if(check()) {
        cout << "Yes\n";
    } else {
        cout << "No\n";
    }
}

int main() {
    ios::sync_with_stdio(0);
    cin.tie(0), cout.tie(0);

    int _T;

    cin >> _T;

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

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Matrix Transformation
time_limit_ms: 1000
memory_limit_mb: 256
Description: You are given two matrices $$$A$$$ and $$$B$$$ of size $$$n \times m$$$, filled with integers between $$$0$$$ and $$$10^9$$$. You can perform the following operations on matrix $$$A$$$ in any order and any number of times:

- &=: choose two integers $$$i$$$ and $$$x$$$ ($$$1 \le i \le n$$$, $$$x \ge 0$$$) and replace each element in row $$$i$$$ with the result of the bitwise AND operation between $$$x$$$ and that element. Formally, for every $$$j \in [1, m]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ & } x$$$;
- |=: choose two integers $$$j$$$ and $$$x$$$ ($$$1 \le j \le m$$$, $$$x \ge 0$$$) and replace each element in column $$$j$$$ with the result of the bitwise OR operation between $$$x$$$ and that element. Formally, for every $$$i \in [1, n]$$$, the element $$$A_{i,j}$$$ is replaced with $$$A_{i,j} \text{ | } x$$$.

The value of $$$x$$$ may be chosen differently for different operations.

Determine whether it is possible to transform matrix $$$A$$$ into matrix $$$B$$$ using the given operations any number of times (including zero).

Input: The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. Then, $$$t$$$ test cases follow.

Each test case is given as follows:

- the first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 10^3$$$; $$$n \cdot m \le 10^3$$$) — the dimensions of the matrices $$$A$$$ and $$$B$$$;
- the following $$$n$$$ lines describe the matrix $$$A$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$A_{i,1}, A_{i,2}, \dots, A_{i,m}$$$ ($$$0 \le A_{i,j} \le 10^9$$$);
- the following $$$n$$$ lines describe the matrix $$$B$$$, where the $$$i$$$-th line contains $$$m$$$ integers $$$B_{i,1}, B_{i,2}, \dots, B_{i,m}$$$ ($$$0 \le B_{i,j} \le 10^9$$$).

Output: For each test case, output Yes if it is possible to transform the matrix $$$A$$$ into the matrix $$$B$$$; otherwise, output No. Each letter can be output in any case, upper or lower.

Examples:
input:
4
1 1
12
13
2 2
10 10
42 42
21 21
21 21
2 2
74 10
42 106
21 85
85 21
2 4
1 2 3 4
5 6 7 8
3 2 3 4
1 0 1 0
output:
Yes
Yes
No
Yes

Note: Let's consider the second set of input data and show a sequence of operations that transforms matrix $$$A$$$ into matrix $$$B$$$:

Initially, the matrix looks like this:

$$$\begin{bmatrix} 10&10\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 1$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 42&42\\ \end{bmatrix}$$$

Apply an operation of the first type with parameters $$$i = 2$$$ and $$$x = 0$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 0&0\\ 0&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 1$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&0\\ 21&0\\ \end{bmatrix}$$$

Apply an operation of the second type with parameters $$$j = 2$$$ and $$$x = 21$$$. As a result, we get the matrix:

$$$\begin{bmatrix} 21&21\\ 21&21\\ \end{bmatrix}$$$

Thus, we have transformed matrix $$$A$$$ into matrix $$$B$$$.

Buggy Code:
#include<cstdio>
#include<cmath>
#include<map>
#include<vector>
#include<queue>
#include<stack>
#include<algorithm>
#include<string>
#include<iostream>
#include<cstring>
using namespace std;
const int Maxn = 2e5 + 5,Maxm = 1e6 + 5;
#define ll long long
#define ull unsigned long long
const ll INF= (ll)1e17 + 5LL;
const ll Mod = 1e9 + 7;
// ll Mod;
int n,m,h,r,Cnt,q,Cnt0,Cnt1,Flag;
int x,y;
int vis[Maxn];
int head[Maxn],indeg[Maxn],outdeg[Maxn];
// int minx[1005][10][10];
int num[Maxn],num0[Maxn],ans1[Maxn],ans2[Maxn],num1[Maxn];
// int dp[15][15*15],num0[Maxn],head1[Maxn],vis[Maxn],anc[Maxn][25],sum[Maxn],pre[Maxn],pre1[Maxn],suf[Maxn],P[Maxn],indeg1[Maxn];
ll tot;
char str[Maxn],str1[Maxn],result[Maxn];
// int vis[Maxm][Maxm],mark[Maxm][Maxm];
// map<int,int> mp;
int dirx[] = {1,-1,0,0},diry[] = {0,0,1,-1};
// vector<ll> num1(20,0);
// vector<ll> num2(20,0);
// vector<pair<int,int>> res(Maxn);

struct Side{
    int s;
    int e;
    int w;
    int next;
}side[Maxm];

// ll fact(ll x){
//     if(x <= 1)return 1;
//     return x * fact(x-1);
// }

// ll C(ll m,ll n){
//     if(m > n)return 0LL;
//     return fact(n)/(fact(m))/fact(n-m);
// }

void add(int s,int e){
    ++Cnt;
    ++indeg[e];
    side[Cnt].s = s;
    side[Cnt].e = e;
    // side[Cnt].w = w;
    side[Cnt].next = head[s];
    head[s] = Cnt;
}

ll M(ll x){
    return ((x % Mod) + Mod) % Mod;
}

ll max(ll x,ll y){
    return x > y ? x : y;
}

ll sgn(ll x){
    if(x >= 0)return 1;
    else return -1;
}

ll min(ll x,ll y){
    return x < y ? x : y;
}

ll gcd(ll x,ll y){
    return y ? gcd(y,x%y) : x;
}

ll ex_gcd(ll a,ll b,ll &x,ll &y){
    if(b == 0){
        x = 1;
        y = 0;
        return a;
    }
    ll d;
    d = ex_gcd(b,a % b,y,x);
    y = M(y - a / b * x);
    return d;
}

ll Inverse(ll p,ll q){
    ll x,y;
    ex_gcd(q,Mod,x,y);
    return M(M(x) * M(p));
}

ll fact(ll x){
    if(x <= 1)return 1;
    return x * fact(x-1) % Mod;
}

// ll C(ll m,ll n){
//     if(m > n)return 0LL;
//     return Inverse(fact(n),fact(m)*fact(n-m) % Mod);
// }

ll lcm(ll x,ll y){
    return x * y / gcd(x,y);
}

int count_bit(int x){
    int res = 0;
    while(x){
        ++res;
        x >>= 1;
    }
    return res;
}

ll quick_pow(ll x,ll y){
    ll res = 1,unit = x;
    while(y){
        if(y & 1LL)res = res * unit % Mod;
        unit = unit * unit % Mod;
        y >>= 1LL;
    }
    return res;
}

// int find(ll x){
//     // if(x <= 0)return 0;
//     int l = 1,r = n,mid;
//     while(l <= r){
//         mid = (l + r) >> 1;
//         if(num[mid] > x){
//             r = mid-1;
//         }else{
//             l = mid+1;
//         }
//     }
//     return r;
// }

ll highbit(ll x){
    if(x == 0)return 0;
    ll res = 1;
    while(x > 1){
        x >>= 1;
        res <<= 1;
    }
    return res;
}

void swap(int &x,int &y){
    int tmp = x;
    x = y;
    y = tmp;
}

int count1(int x){
    int cnt = 0;
    while(x){
        cnt += (x & 1);
        x >>= 1;
    }
    return cnt;
}

struct Node{
    int x;
    int y;
    int operator < (const Node &w) const &{
        return x > w.x;
    }
}cur,N[Maxn];

// struct Node1{
//     int x;
//     int y;
//     int id;
//     int operator < (const Node1 &w) const &{
//         return x < w.x;
//     }
// }cur1;

// ll getnum(){
//     ll res = 0,f = 1;
//     char ch;
//     while(1){
//         ch = getchar();
//         if(ch == '-')f *= -1;
//         if(!(ch >= '0' && ch <= '9'))continue;
//         else break;
//     }
//     res = res * 10 + ch - '0';
//     while(1){
//         ch = getchar();
//         if(!(ch >= '0' && ch <= '9'))break;
//         res = res * 10 + ch - '0';
//     }
//     return res * f;
// }
int flag;
void dfs(int x,int s){
    vis[x] = s;
    int i;
    for(i = head[x];i;i = side[i].next){
        if(vis[side[i].e] == s){
            flag = 0;
            return;
        }
        if(vis[side[i].e])continue;
        dfs(side[i].e,s);
    }
}

int len;
int ans[Maxn];
ll pre[Maxn];
int main(){
    int i,j,k,v,w;
    // ll x0,y0,k;
    char ch;
    int tmp,tot,unit;
    int tmp1,tmp2,tmp3,tmp4;
    int Cnt1,Cnt2;
    int _;
    scanf("%d",&_);
    // _ = 1;
    int idx = 0;
    while(_--){
        scanf("%d %d",&n,&m);
        vector<vector<ll>> num1(n+5,vector<ll>(m+5,0)),num2(n+5,vector<ll>(m+5,0));

        for(i = 1;i <= n;++i){
            for(j = 1;j <= m;++j){
                scanf("%lld",&num1[i][j]);
            }
        }
        for(i = 1;i <= n;++i){
            for(j = 1;j <= m;++j){
                scanf("%lld",&num2[i][j]);
            }
        }
        flag = 1;
        for(i = 0;i <= 30;++i){
            if(!flag)break;
            ll unit = (1LL << i);
            vector<vector<ll>> mark(n+5,vector<ll>(m+5,0));
            vector<int> mark1(n+5,0),mark2(m+5,0);
            for(j = 1;j <= n + m;++j)vis[j] = 0;
            Cnt = 0;
            for(j = 1;j <= n + m;++j){
                head[j] = 0;
            }
            for(j = 1;j <= n;++j){
                if(!flag)break;
                for(k = 1;k <= m;++k){
                    ll unit1,unit2;
                    unit1 = (num1[j][k] & unit);
                    unit2 = (num2[j][k] & unit);
                    if(unit1 != 0 && unit2 != 0){
                        add(j,k+n);
                    }
                    if(unit1 == 0 && unit2 != 0){
                        add(j,k+n);
                    }
                }
            }
            for(j = 1;j <= n;++j){
                if(!flag)break;
                for(k = 1;k <= m;++k){
                    ll unit1,unit2;
                    unit1 = (num1[j][k] & unit);
                    unit2 = (num2[j][k] & unit);
                    if(unit1 == 0 && unit2 == 0){
                        add(k+n,j);
                    }
                    if(unit1 != 0 && unit2 == 0){
                        add(k+n,j);
                    }
                }
            }
            for(j = 1;j <= n;++j){
                if(!flag)break;
                for(k = 1;k <= m;++k){
                    ll unit1,unit2;
                    unit1 = (num1[j][k] & unit);
                    unit2 = (num2[j][k] & unit);
                    if(unit1 == 0 && unit2 != 0){
                        dfs(k+n,k+n);
                    }
                    if(unit1 != 0 && unit2 == 0){
                        dfs(j,j);
                    }
                }
            }
        }
        if(flag)printf("YES\n");
        else printf("NO\n");
    }
    scanf("%d %d",&n,&k);
    return 0;
}

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

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

Think step by step.

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

1. **Matrix Dimensions**: Use small matrices to ensure the input is manageable.
2. **Bitwise Operations**: Create matrices where the bitwise AND and OR operations can lead to complex transformations.
3. **Edge Cases**: Consider cases where the matrices have specific patterns or values that could cause the algorithm to fail.

Let's think through a few scenarios:

### Scenario 1: Single Element Matrices
- **Input**:
  ```
  1
  1 1
  12
  13
  ```
- **Explanation**:
  - Here, the matrices are of size 1x1.
  - The transformation is not possible because 12 & x cannot become 13 for any x.

### Scenario 2: Row and Column Operations
- **Input**:
  ```
  1
  2 2
  10 10
  42 42
  21 21
  21 21
  ```
- **Explanation**:
  - The initial matrix A is:
    ```
    10 10
    42 42
    ```
  - The target matrix B is:
    ```
    21 21
    21 21
    ```
  - This is a valid transformation as shown in the example.

### Scenario 3: Impossible Transformation
- **Input**:
  ```
  1
  2 2
  74 10
  42 106
  21 85
  85 21
  ```
- **Explanation**:
  - The initial matrix A is:
    ```
    74 10
    42 106
    ```
  - The target matrix B is:
    ```
    21 85
    85 21
    ```
  - This transformation is not possible because the bitwise operations cannot transform the elements as required.

### Scenario 4: Mixed Values
- **Input**:
  ```
  1
  2 4
  1 2 3 4
  5 6 7 8
  3 2 3 4
  1 0 1 0
  ```
- **Explanation**:
  - The initial matrix A is:
    ```
    1 2 3 4
    5 6 7 8
    ```
  - The target matrix B is:
    ```
    3 2 3 4
    1 0 1 0
    ```
  - This transformation is possible by applying the appropriate bitwise operations.

### Chosen Test Input
To ensure the test input is challenging and exposes potential bugs, let's use a combination of the scenarios above, focusing on a case where the transformation is not possible:

```plaintext
1
2 2
74 10
42 106
21 85
85 21
```

This input should be challenging for the buggy code, as it requires the algorithm to correctly determine that the transformation is not possible.