**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: GCD-sequence
time_limit_ms: 2000
memory_limit_mb: 256
Description: GCD (Greatest Common Divisor) of two integers $x$ and $y$ is the maximum integer $z$ by which both $x$ and $y$ are divisible. For example, $GCD(36, 48) = 12$, $GCD(5, 10) = 5$, and $GCD(7,11) = 1$.

Kristina has an array $a$ consisting of exactly $n$ positive integers. She wants to count the GCD of each neighbouring pair of numbers to get a new array $b$, called GCD-sequence.

So, the elements of the GCD-sequence $b$ will be calculated using the formula $b_i = GCD(a_i, a_{i + 1})$ for $1 \le i \le n - 1$.

Determine whether it is possible to remove exactly one number from the array $a$ so that the GCD sequence $b$ is non-decreasing (i.e., $b_i \le b_{i+1}$ is always true).

For example, let Khristina had an array $a$ = [$20, 6, 12, 3, 48, 36$]. If she removes $a_4 = 3$ from it and counts the GCD-sequence of $b$, she gets:

  * $b_1 = GCD(20, 6) = 2$    * $b_2 = GCD(6, 12) = 6$    * $b_3 = GCD(12, 48) = 12$    * $b_4 = GCD(48, 36) = 12$

The resulting GCD sequence $b$ = [$2,6,12,12$] is non-decreasing because $b_1 \le b_2 \le b_3 \le b_4$.

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

This is followed by the descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($3 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($1 \le a_i \le 10^9$) — the elements of array $a$.

It is guaranteed that the sum of $n$ over all test case does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single line:

  * "YES" if you can remove exactly one number from the array $a$ so that the GCD-sequence of $b$ is non-decreasing;    * "NO" otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will all be recognized as a positive answer).

Examples:
input:
12
6
20 6 12 3 48 36
4
12 6 3 4
3
10 12 3
5
32 16 8 4 2
5
100 50 2 10 20
4
2 4 8 1
10
7 4 6 2 4 5 1 4 2 8
7
5 9 6 8 5 9 2
6
11 14 8 12 9 3
9
5 7 3 10 6 3 12 6 3
3
4 2 4
8
1 6 11 12 6 12 3 6
output:
YES
NO
YES
NO
YES
YES
NO
YES
YES
YES
YES
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
def sorted_check(v):
    return v == sorted(v)

def gcd(a, b):
    while b:
        a, b = b, a % b
    return a

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

    b = [gcd(a[i], a[i + 1]) for i in range(n - 1)]

    if sorted_check(b):
        print("YES")
        return

    found = False
    for i in range(n):
        temp = []
        if i == 0:
            temp = [gcd(a[j], a[j + 1]) for j in range(1, n - 1)]
        elif i == n - 1:
            temp = [gcd(a[j], a[j + 1]) for j in range(n - 2)]
        else:
            for j in range(n - 1):
                if j == i - 1:
                    continue
                if j == i:
                    temp.append(gcd(a[j - 1], a[j + 1]))
                else:
                    temp.append(gcd(a[j], a[j + 1]))

        if sorted_check(temp):
            found = True
            break

    if found:
        print("YES")
    else:
        print("NO")

def main():
    import sys
    input = sys.stdin.read
    data = input().split()
    t = int(data[0])
    index = 1
    for _ in range(t):
        n = int(data[index])
        index += 1
        a = list(map(int, data[index:index + n]))
        index += n
        solve_single_case(a)

def solve_single_case(a):
    n = len(a)
    b = [gcd(a[i], a[i + 1]) for i in range(n - 1)]

    if sorted_check(b):
        print("YES")
        return

    found = False
    for i in range(n):
        temp = []
        if i == 0:
            temp = [gcd(a[j], a[j + 1]) for j in range(1, n - 1)]
        elif i == n - 1:
            temp = [gcd(a[j], a[j + 1]) for j in range(n - 2)]
        else:
            for j in range(n - 1):
                if j == i - 1:
                    continue
                if j == i:
                    temp.append(gcd(a[j - 1], a[j + 1]))
                else:
                    temp.append(gcd(a[j], a[j + 1]))

        if sorted_check(temp):
            found = True
            break

    if found:
        print("YES")
    else:
        print("NO")

if __name__ == "__main__":
    main()

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider the following aspects:

1. **Edge Cases**: These include arrays with the minimum and maximum allowed lengths, arrays with all elements being the same, arrays with elements that are consecutive numbers, and arrays with elements that have specific GCD properties.
2. **Non-decreasing GCD Sequences**: We need to create an array where removing any single element will make the GCD sequence non-decreasing.
3. **Specific Patterns**: Arrays where the GCD sequence has specific patterns that can be disrupted by removing a single element.

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

1. **Array with All Elements the Same**: This should always result in a non-decreasing GCD sequence.
2. **Array with Elements that Have Specific GCD Properties**: For example, an array where the GCD sequence is non-decreasing except for one element.
3. **Array with Consecutive Numbers**: This can help test the GCD calculation and the non-decreasing condition.

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

```plaintext
3
3
10 10 10
5
10 15 20 25 30
4
1 2 3 4
```

Explanation:
1. **First Test Case**: The array [10, 10, 10] will always result in a GCD sequence [10, 10], which is non-decreasing. Removing any element will still result in a non-decreasing GCD sequence.
2. **Second Test Case**: The array [10, 15, 20, 25, 30] has a GCD sequence [5, 5, 5, 5], which is non-decreasing. Removing any element will still result in a non-decreasing GCD sequence.
3. **Third Test Case**: The array [1, 2, 3, 4] has a GCD sequence [1, 1, 1], which is non-decreasing. Removing any element will still result in a non-decreasing GCD sequence.

This test input should help expose any issues in the code, particularly in handling edge cases and specific patterns.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: GCD-sequence
time_limit_ms: 2000
memory_limit_mb: 256
Description: GCD (Greatest Common Divisor) of two integers $x$ and $y$ is the maximum integer $z$ by which both $x$ and $y$ are divisible. For example, $GCD(36, 48) = 12$, $GCD(5, 10) = 5$, and $GCD(7,11) = 1$.

Kristina has an array $a$ consisting of exactly $n$ positive integers. She wants to count the GCD of each neighbouring pair of numbers to get a new array $b$, called GCD-sequence.

So, the elements of the GCD-sequence $b$ will be calculated using the formula $b_i = GCD(a_i, a_{i + 1})$ for $1 \le i \le n - 1$.

Determine whether it is possible to remove exactly one number from the array $a$ so that the GCD sequence $b$ is non-decreasing (i.e., $b_i \le b_{i+1}$ is always true).

For example, let Khristina had an array $a$ = [$20, 6, 12, 3, 48, 36$]. If she removes $a_4 = 3$ from it and counts the GCD-sequence of $b$, she gets:

  * $b_1 = GCD(20, 6) = 2$    * $b_2 = GCD(6, 12) = 6$    * $b_3 = GCD(12, 48) = 12$    * $b_4 = GCD(48, 36) = 12$

The resulting GCD sequence $b$ = [$2,6,12,12$] is non-decreasing because $b_1 \le b_2 \le b_3 \le b_4$.

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

This is followed by the descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($3 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($1 \le a_i \le 10^9$) — the elements of array $a$.

It is guaranteed that the sum of $n$ over all test case does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single line:

  * "YES" if you can remove exactly one number from the array $a$ so that the GCD-sequence of $b$ is non-decreasing;    * "NO" otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will all be recognized as a positive answer).

Examples:
input:
12
6
20 6 12 3 48 36
4
12 6 3 4
3
10 12 3
5
32 16 8 4 2
5
100 50 2 10 20
4
2 4 8 1
10
7 4 6 2 4 5 1 4 2 8
7
5 9 6 8 5 9 2
6
11 14 8 12 9 3
9
5 7 3 10 6 3 12 6 3
3
4 2 4
8
1 6 11 12 6 12 3 6
output:
YES
NO
YES
NO
YES
YES
NO
YES
YES
YES
YES
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
from math import gcd

def is_non_decreasing(arr):
    return all(arr[i] <= arr[i + 1] for i in range(len(arr) - 1))

def can_form_non_decreasing_gcd_sequence(a):
    n = len(a)
    if n <= 2:
        return "YES"

    # Precompute the GCDs for adjacent pairs
    gcd_seq = [gcd(a[i], a[i + 1]) for i in range(n - 1)]

    # Check if the initial GCD sequence is already non-decreasing
    if is_non_decreasing(gcd_seq):
        return "YES"

    # Try removing each element and check the new GCD sequence
    for i in range(n):
        new_gcd_seq = []
        if i > 0:
            new_gcd_seq.extend(gcd_seq[:i-1])
        if i > 0 and i < n - 1:
            new_gcd_seq.append(gcd(a[i-1], a[i+1]))
        if i < n - 2:
            new_gcd_seq.extend(gcd_seq[i+1:])

        if is_non_decreasing(new_gcd_seq):
            return "YES"

    return "NO"

# Read input
for _ in range(int(input().strip())):
    n = int(input().strip())
    a = list(map(int, input().strip().split()))
    print(can_form_non_decreasing_gcd_sequence(a))

**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
12 6 3 4 2
```

### Explanation:
This test case is designed to expose the bug in the faulty solution by focusing on a sequence where removing any single element will not result in a non-decreasing GCD sequence.

- The array `a` is `[12, 6, 3, 4, 2]`.
- The initial GCD sequence `b` is `[gcd(12, 6), gcd(6, 3), gcd(3, 4), gcd(4, 2)]`, which is `[6, 3, 1, 2]`.

If we remove any element:
- Removing `12`: GCD sequence becomes `[gcd(6, 3), gcd(3, 4), gcd(4, 2)] = [3, 1, 2]` (not non-decreasing).
- Removing `6`: GCD sequence becomes `[gcd(12, 3), gcd(3, 4), gcd(4, 2)] = [3, 1, 2]` (not non-decreasing).
- Removing `3`: GCD sequence becomes `[gcd(12, 6), gcd(6, 4), gcd(4, 2)] = [6, 2, 2]` (not non-decreasing).
- Removing `4`: GCD sequence becomes `[gcd(12, 6), gcd(6, 3), gcd(3, 2)] = [6, 3, 1]` (not non-decreasing).
- Removing `2`: GCD sequence becomes `[gcd(12, 6), gcd(6, 3), gcd(3, 4)] = [6, 3, 1]` (not non-decreasing).

Since no single removal results in a non-decreasing GCD sequence, the correct output should be "NO". However, if the faulty solution incorrectly handles this case, it might return "YES" and expose the bug.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: GCD-sequence
time_limit_ms: 2000
memory_limit_mb: 256
Description: GCD (Greatest Common Divisor) of two integers $x$ and $y$ is the maximum integer $z$ by which both $x$ and $y$ are divisible. For example, $GCD(36, 48) = 12$, $GCD(5, 10) = 5$, and $GCD(7,11) = 1$.

Kristina has an array $a$ consisting of exactly $n$ positive integers. She wants to count the GCD of each neighbouring pair of numbers to get a new array $b$, called GCD-sequence.

So, the elements of the GCD-sequence $b$ will be calculated using the formula $b_i = GCD(a_i, a_{i + 1})$ for $1 \le i \le n - 1$.

Determine whether it is possible to remove exactly one number from the array $a$ so that the GCD sequence $b$ is non-decreasing (i.e., $b_i \le b_{i+1}$ is always true).

For example, let Khristina had an array $a$ = [$20, 6, 12, 3, 48, 36$]. If she removes $a_4 = 3$ from it and counts the GCD-sequence of $b$, she gets:

  * $b_1 = GCD(20, 6) = 2$    * $b_2 = GCD(6, 12) = 6$    * $b_3 = GCD(12, 48) = 12$    * $b_4 = GCD(48, 36) = 12$

The resulting GCD sequence $b$ = [$2,6,12,12$] is non-decreasing because $b_1 \le b_2 \le b_3 \le b_4$.

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

This is followed by the descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($3 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($1 \le a_i \le 10^9$) — the elements of array $a$.

It is guaranteed that the sum of $n$ over all test case does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single line:

  * "YES" if you can remove exactly one number from the array $a$ so that the GCD-sequence of $b$ is non-decreasing;    * "NO" otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will all be recognized as a positive answer).

Examples:
input:
12
6
20 6 12 3 48 36
4
12 6 3 4
3
10 12 3
5
32 16 8 4 2
5
100 50 2 10 20
4
2 4 8 1
10
7 4 6 2 4 5 1 4 2 8
7
5 9 6 8 5 9 2
6
11 14 8 12 9 3
9
5 7 3 10 6 3 12 6 3
3
4 2 4
8
1 6 11 12 6 12 3 6
output:
YES
NO
YES
NO
YES
YES
NO
YES
YES
YES
YES
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
import java.util.*;

public class Main {

    private static int gcd(int x, int y) {
        while (y != 0) {
            int temp = y;
            y = x % y;
            x = temp;
        }
        return x;
    }

    private static boolean isNonDecreasingGCDSequence(List<Integer> a) {
        List<Integer> b = new ArrayList<>();
        for (int i = 0; i < a.size() - 1; i++) {
            b.add(gcd(a.get(i), a.get(i + 1)));
        }
        for (int i = 1; i < b.size(); i++) {
            if (b.get(i) < b.get(i - 1)) {
                return false;
            }
        }
        return true;
    }

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int t = scanner.nextInt();
        while (t-- > 0) {
            int n = scanner.nextInt();
            List<Integer> a = new ArrayList<>();
            for (int i = 0; i < n; i++) {
                a.add(scanner.nextInt());
            }

            boolean possible = false;
            for (int i = 0; i < n; i++) {
                List<Integer> temp = new ArrayList<>(a);
                temp.remove(i);
                if (isNonDecreasingGCDSequence(temp)) {
                    possible = true;
                    break;
                }
            }

            if (possible) {
                System.out.println("YES");
            } else {
                System.out.println("NO");
            }
        }
        scanner.close();
    }
}

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

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

Think step by step.

Response:
```plaintext
1
6
12 6 12 3 12 6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: GCD-sequence
time_limit_ms: 2000
memory_limit_mb: 256
Description: GCD (Greatest Common Divisor) of two integers $x$ and $y$ is the maximum integer $z$ by which both $x$ and $y$ are divisible. For example, $GCD(36, 48) = 12$, $GCD(5, 10) = 5$, and $GCD(7,11) = 1$.

Kristina has an array $a$ consisting of exactly $n$ positive integers. She wants to count the GCD of each neighbouring pair of numbers to get a new array $b$, called GCD-sequence.

So, the elements of the GCD-sequence $b$ will be calculated using the formula $b_i = GCD(a_i, a_{i + 1})$ for $1 \le i \le n - 1$.

Determine whether it is possible to remove exactly one number from the array $a$ so that the GCD sequence $b$ is non-decreasing (i.e., $b_i \le b_{i+1}$ is always true).

For example, let Khristina had an array $a$ = [$20, 6, 12, 3, 48, 36$]. If she removes $a_4 = 3$ from it and counts the GCD-sequence of $b$, she gets:

  * $b_1 = GCD(20, 6) = 2$    * $b_2 = GCD(6, 12) = 6$    * $b_3 = GCD(12, 48) = 12$    * $b_4 = GCD(48, 36) = 12$

The resulting GCD sequence $b$ = [$2,6,12,12$] is non-decreasing because $b_1 \le b_2 \le b_3 \le b_4$.

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

This is followed by the descriptions of the test cases.

The first line of each test case contains a single integer $n$ ($3 \le n \le 2 \cdot 10^5$) — the number of elements in the array $a$.

The second line of each test case contains exactly $n$ integers $a_i$ ($1 \le a_i \le 10^9$) — the elements of array $a$.

It is guaranteed that the sum of $n$ over all test case does not exceed $2 \cdot 10^5$.

Output: For each test case, output a single line:

  * "YES" if you can remove exactly one number from the array $a$ so that the GCD-sequence of $b$ is non-decreasing;    * "NO" otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes", and "YES" will all be recognized as a positive answer).

Examples:
input:
12
6
20 6 12 3 48 36
4
12 6 3 4
3
10 12 3
5
32 16 8 4 2
5
100 50 2 10 20
4
2 4 8 1
10
7 4 6 2 4 5 1 4 2 8
7
5 9 6 8 5 9 2
6
11 14 8 12 9 3
9
5 7 3 10 6 3 12 6 3
3
4 2 4
8
1 6 11 12 6 12 3 6
output:
YES
NO
YES
NO
YES
YES
NO
YES
YES
YES
YES
YES

Note: The first test case is explained in the problem statement.

Buggy Code:
import java.util.*;
import java.io.*;
public class GCDSequence {
    public static int gcd(int a, int b) {
        if (b == 0) return a;
        return gcd(b, a % b);
    }
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while (t-- > 0) {
            int n = sc.nextInt();
            int[] a = new int[n];
            for (int i = 0; i < n; i++) {
                a[i] = sc.nextInt();
            }
            boolean alreadyNon = true;
            int[] b = new int[n-1];
            for (int i = 0; i < n - 1; i++) {
                b[i] = gcd(a[i], a[i+1]);
                if (i > 0 && b[i] < b[i-1]) {
                    alreadyNon = false;
                }
            }
            if (alreadyNon) {
                System.out.println("YES");
                continue;
            }
            boolean possible = false;
            for (int i = 0; i < n; i++){
                List<Integer> newArray = new ArrayList<>();
                for (int j = 0; j < n; j++) {
                    if (j != i) newArray.add(a[j]);
                }
                boolean isNonDecreasing = true;
                for (int j = 0; j < newArray.size() - 1; j++) {
                    int cGCD = gcd(newArray.get(j), newArray.get(j + 1));
                    if (j > 0 && cGCD < gcd(newArray.get(j - 1), newArray.get(j))) {
                        isNonDecreasing = false;
                        break;
                    }
                }
                if (isNonDecreasing) {
                    possible = true;
                    break;
                }
            }
            if (possible) {
                System.out.println("YES");
            } else {
                System.out.println("NO");
            }
        }

        sc.close();
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that are likely to cause the algorithm to fail. The buggy code has a few potential issues:

1. **Inefficient Check for Non-Decreasing Sequence**: The code checks if the sequence is already non-decreasing and then tries removing each element to see if the sequence becomes non-decreasing. This approach can be inefficient and may miss some edge cases.
2. **Handling of Special Cases**: The code might not handle special cases, such as arrays with repeated elements or arrays where the GCD sequence has a specific pattern that makes it difficult to determine the non-decreasing property.

Let's consider the following scenarios:

1. **Array with Repeated Elements**: An array where all elements are the same or have a common divisor.
2. **Array with a Single Decreasing Step**: An array where the GCD sequence has only one decreasing step.
3. **Array with Multiple Decreasing Steps**: An array where the GCD sequence has multiple decreasing steps.

### Test Input

```plaintext
5
3
10 10 10
4
10 5 15 30
5
20 10 5 15 30
6
4 8 12 16 20 24
7
100 50 25 50 100 50 25
```

### Explanation

1. **Test Case 1:**
   - Array: [10, 10, 10]
   - GCD Sequence: [10, 10]
   - This is already non-decreasing.
   - Expected Output: "YES"

2. **Test Case 2:**
   - Array: [10, 5, 15, 30]
   - GCD Sequence: [5, 5, 15]
   - This is already non-decreasing.
   - Expected Output: "YES"

3. **Test Case 3:**
   - Array: [20, 10, 5, 15, 30]
   - GCD Sequence: [10, 5, 5, 15]
   - Removing the second element (10) results in the array [20, 5, 15, 30] with GCD Sequence [5, 5, 15].
   - Expected Output: "YES"

4. **Test Case 4:**
   - Array: [4, 8, 12, 16, 20, 24]
   - GCD Sequence: [4, 4, 4, 4, 4]
   - This is already non-decreasing.
   - Expected Output: "YES"

5. **Test Case 5:**
   - Array: [100, 50, 25, 50, 100, 50, 25]
   - GCD Sequence: [50, 25, 25, 50, 50, 25]
   - Removing the fourth element (50) results in the array [100, 50, 25, 100, 50, 25] with GCD Sequence [50, 25, 25, 50, 25].
   - Expected Output: "NO" (because no single removal can make the sequence non-decreasing)

This test input includes a variety of scenarios to test the robustness of the algorithm and should help expose any bugs in the implementation.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
#include<iostream>
#include<algorithm>
#include<cstring>
#include<cmath>
#include<queue>

#define int long long

using namespace std;

const int N=1000010;
int arr[N];

void solve()
{
	memset(arr,0,sizeof(arr));
	int n;
	cin>>n;
	int x;
	for(int i=1;i<=n;i++)
	{
		cin>>x;
		arr[x]++;
	}
	int cnt=0;
	for(int i=1;i<=n;i++)
	{
		if(arr[i]==2)
		{
			cnt++;
		}
	}
	cout<<cnt<<'\n';
}

signed 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:
```plaintext
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
#include <iostream>
#include <vector>
#include <string>

#include <numeric>
#include <functional>
#include <cmath>
#include <set>

using namespace std;

int main()
{

	int t; cin >> t;

	while(t--){

		int n;  cin >> n;

		vector<int> a(n, 0);
		int w;
		int k = 0;

		for (int i = 0; i < n; i++) {
			cin >> w;
			if (a[w] == 1) k++;
			else (a[w] = 1);
		}

		cout << k << 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
4
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
#include<iostream>
#include<cmath>
#include<algorithm>
#include<iomanip>
#include<vector>
#include<stack>
#include<map>
#include<string>
#include<deque>
#include<set>
#define ll long long
using namespace std;
int main() {
	ios_base::sync_with_stdio(0);

	cin.tie(0);

	cout.tie(0);
ll t;
cin>>t;
while(t>0)
{
ll n;
cin>>n;
ll v[1000000]{},c=0;
for(ll i=0;i<n;i++)
{
	ll z;
	cin>>z;
	v[z]++;
	if(v[z]==2)
	c++;

}
cout<<c<<endl;
	t--;
}
	return 0;
}

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

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

Think step by step.

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

This test input is designed to cover a variety of scenarios that can expose the bug in the provided buggy code. Here's the reasoning behind each test case:

1. **Test Case 1:**
   - **n = 4**
   - **Cards: 1 1 2 3**
   - **Explanation:** This case tests the basic scenario where the player has a pair of one card (1) and two unique cards (2 and 3). The expected output is 1 because the player can score 1 point by playing the second 1 after Nene has played the first 1.

2. **Test Case 2:**
   - **n = 8**
   - **Cards: 7 4 1 2 8 8 5 5**
   - **Explanation:** This case includes multiple pairs (8, 8 and 5, 5) and some unique cards. The expected output is 2 because the player can score 2 points by playing the second 8 and the second 5.

3. **Test Case 3:**
   - **n = 8**
   - **Cards: 7 1 4 5 3 4 2 6**
   - **Explanation:** This case has a single pair (4, 4) and several unique cards. The expected output is 1 because the player can score 1 point by playing the second 4.

4. **Test Case 4:**
   - **n = 3**
   - **Cards: 1 2 3**
   - **Explanation:** This case tests the scenario where the player has no pairs. The expected output is 0 because the player cannot score any points.

5. **Test Case 5:**
   - **n = 1**
   - **Cards: 1**
   - **Explanation:** This is the smallest possible case where the player has only one card. The expected output is 0 because the player cannot score any points.

These test cases are designed to cover a range of scenarios, including different numbers of pairs and unique cards, to ensure that the buggy code is thoroughly tested.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
def f(n, a):
    count = 0
    for i in a:
        if a.count(i) == 2:
            count += 1
            a.pop(a.index(i))
            a.pop(a.index(i))
    return count

t = int(input())
ans = []
for i in range(0, t):
    n = int(input())
    if n == 1:
        ans.append(0)
    else:
        a = list(map(int, input().split()))
        ans.append(f(n, a))

[print(i) for i in ans]

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
from collections import Counter

t = int(input())
for _ in range(t):
    n = int(input())
    a = (int(i) for i in input().split())
    cnt = Counter(a)
    bth = sum(v == 2 for v in cnt.values())
    one = len(cnt) - bth
    res = n if bth > one else bth
    print(res)

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

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

Think step by step.

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

This test input includes a variety of cases to cover different scenarios:
- A case where you have a pair and some singles (4 cards).
- A case with multiple pairs and singles (8 cards).
- A case with a mix of pairs and singles, but more singles (8 cards).
- A case with all singles (3 cards).
- A case with a single card (1 card).

These cases are designed to challenge the logic of the buggy code, especially in how it handles the distribution of pairs and singles, and the optimal play strategy for both players.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
t = int(input())
for tt in range(t):
    m = [0]*(2*10**5+1)
    n = int(input())
    a = list(map(int, input().split()))
    ans = 0
    for x in a:
        m[x] += 1
        if m[x] == 2:
            ans += 1
    print(ans)

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the provided code, let's analyze the code and the problem:

### Code Analysis
The provided code has a flaw in how it calculates the points. Specifically, it increments the points whenever a card appears twice in the player's hand, without considering the optimal play strategy of Nene. This means the code fails to account for the fact that Nene will play optimally to maximize her score and minimize the player's score.

### Problem Understanding
- The player and Nene each have `n` cards.
- The player's cards are given in the input.
- Nene will play optimally to maximize her score and minimize the player's score.
- The player needs to determine the maximum number of points they can get.

### Test Input
To expose the bug, we need a test case where the player has cards that can potentially score points, but Nene's optimal play will prevent the player from scoring as many points as the code incorrectly calculates.

### Challenging Test Case
Consider a scenario where the player has a set of cards that include pairs, and Nene has the other pairs. Nene will play optimally to prevent the player from scoring points.

#### Example:
- `n = 4`
- Player's cards: `1 1 2 2`
- Nene's cards: `3 3 4 4`

In this case, the player can only score points if Nene does not play optimally. However, since Nene plays optimally, the player will not score any points.

### Test Input
```plaintext
1
4
1 1 2 2
```

### Explanation
- The player has cards `1, 1, 2, 2`.
- Nene has cards `3, 3, 4, 4`.
- The player cannot score any points because Nene will play her cards optimally, ensuring that the player does not get any points.

This test case should expose the bug in the provided code, which incorrectly calculates the points as 2 (since it counts the pairs in the player's hand without considering Nene's optimal play). The correct answer is 0.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
import java.util.*;
public class one {
	public static void main(String[] args) {
		Scanner sc=new Scanner(System.in);
		int t=sc.nextInt();
		while(t-->0) {
			int n=sc.nextInt();
			int a[]=new int[n];
			int f[]=new int[400005];
			int ans=0;
			for(int i=0;i<n;i++) {
				a[i]=sc.nextInt();
				if(f[a[i]]!=0)	ans++;
				f[a[i]]++;
			}
			System.out.println(ans);
		}
		sc.close();
	}

}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
import java.util.*;

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

        for (int testCase = 0; testCase < t; testCase++) {
            int n = sc.nextInt();
            sc.nextLine();

            String inputString = sc.nextLine();
            Set<Integer> uniqueNumbers = new HashSet<>();
            int duplicateCount = 0;

            for (int j = 0; j < inputString.length(); j += 2) {
                int number = Character.getNumericValue(inputString.charAt(j));
                if (!uniqueNumbers.add(number)) {
                    duplicateCount++;
                }
            }
            System.out.println(duplicateCount);
        }
        sc.close();
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define int long long
#define INF 0x3f3f3f3f3f3f3f3f
#define pii pair<int,int>
constexpr int mod = 1e9 + 7;
constexpr int N = 1e2 + 9;

int a[N], f[N];
int n;
void solve() {
	cin >> n;
	for (int i = 1; i <= n; i++) {
		cin >> a[i];
	}
	sort(a + 1, a + n + 1);
	int ans = 0;
	for (int i = 1; i <= n; i++) {
		if (a[i - 1] == a[i]) {
			ans += 1;
		}
	}
	cout << ans << "\n";
}

constexpr bool test = 1;
signed main() {
	cin.tie(0)->ios::sync_with_stdio(false);
	cout << fixed << setprecision(20);
	int T = 1;
	if (test) cin >> T;
	for (int i = 1; i <= T; i++) solve();
	return 0;
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int main()
{
    int t;
    cin >> t;
    while (t--)
    {
        int arr[100000];
        int sum = 0;
        long long n;
        cin >> n;
        arr[n + 3];
        int x;
        for (int i = 0; i < 1000; i++)
        {
            arr[i] = -1;
        }
        for (int i = 0; i < n; i++)
        {
            cin >> x;
            arr[x]++;
           // cout<<arr[x]<<" ";
            if (arr[x] >= 1)
            {
                sum = sum + arr[x];
            }
        }
        cout << sum << endl;
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
for _ in range(int(input())):
    n = int(input())
    a = list(map(int, input().split()))
    count = 0

    for x in range(n+1):
        if a.count(x) == 2:
            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 where the code's logic might fail. The buggy code counts the number of integers that appear exactly twice in the list and outputs that count as the result. However, this logic is incorrect because it does not consider the optimal strategy for both players, especially Nene, who plays optimally to maximize her score and minimize yours.

### Step-by-Step Analysis:
1. **Understanding the Bug**:
   - The code simply counts the number of integers that appear exactly twice in your hand and outputs that count.
   - This approach ignores the fact that Nene plays optimally, which can significantly affect the final score.

2. **Edge Cases to Consider**:
   - **Case 1**: All cards in your hand appear exactly once.
   - **Case 2**: All cards in your hand appear exactly twice.
   - **Case 3**: Mixed distribution of cards (some appear once, some appear twice).
   - **Case 4**: Small \( n \) values (e.g., \( n = 1 \) or \( n = 2 \)).
   - **Case 5**: Large \( n \) values to test performance.

3. **Challenging Input**:
   - We need an input where the code's output is clearly wrong due to its incorrect logic.
   - A good candidate is a case where you have a mix of cards that appear once and twice, and Nene can strategically play to minimize your score.

### Chosen Test Input:
Let's create a test case where \( n = 5 \) and the cards in your hand are a mix of cards that appear once and twice. This will challenge the code's logic.

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

### Explanation:
- **n = 5**: You and Nene each have 5 cards.
- **Your hand**: [1, 1, 2, 3, 4]
- **Nene's hand**: [2, 3, 4, 5, 5] (since each integer from 1 to 5 appears exactly twice in the deck)

### Optimal Play:
- **Your Turn 1**: Play 1 (no points, table: [1])
- **Nene's Turn 1**: Play 5 (no points, table: [1, 5])
- **Your Turn 2**: Play 1 (1 point, table: [1, 1, 5])
- **Nene's Turn 2**: Play 5 (1 point, table: [1, 1, 5, 5])
- **Your Turn 3**: Play 2 (no points, table: [1, 1, 5, 5, 2])
- **Nene's Turn 3**: Play 2 (1 point, table: [1, 1, 5, 5, 2, 2])
- **Your Turn 4**: Play 3 (no points, table: [1, 1, 5, 5, 2, 2, 3])
- **Nene's Turn 4**: Play 3 (1 point, table: [1, 1, 5, 5, 2, 2, 3, 3])
- **Your Turn 5**: Play 4 (no points, table: [1, 1, 5, 5, 2, 2, 3, 3, 4])
- **Nene's Turn 5**: Play 4 (1 point, table: [1, 1, 5, 5, 2, 2, 3, 3, 4, 4])

### Final Score:
- **You**: 1 point
- **Nene**: 4 points

The correct output should be 1, but the buggy code will output 2 because it counts the number of integers that appear exactly twice in your hand.

### Output:
```plaintext
1
5
1 1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
t=int(input())
for i in range(t):
    c=0
    n=int(input())
    l=list(map(int,input().split()))
    for j in set(l):
        if l.count(j)==2:
            c+=1
    print(c)

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Nene and the Card Game
time_limit_ms: 1000
memory_limit_mb: 256
Description: You and Nene are playing a card game. The deck with $$$2n$$$ cards is used to play this game. Each card has an integer from $$$1$$$ to $$$n$$$ on it, and each of integers $$$1$$$ through $$$n$$$ appears exactly on $$$2$$$ cards. Additionally, there is a table where cards are placed during the game (initially, the table is empty).

In the beginning of the game, these $$$2n$$$ cards are distributed between you and Nene so that each player receives $$$n$$$ cards.

After it, you and Nene alternatively take $$$2n$$$ turns, i.e. each person takes $$$n$$$ turns, starting with you. On each turn:

- The player whose turn is it selects one of the cards in his hand. Let $$$x$$$ be the number on it.
- The player whose turn is it receives $$$1$$$ point if there is already a card with the integer $$$x$$$ on the table (otherwise, he receives no points). After it, he places the selected card with the integer $$$x$$$ on the table.

Note that turns are made publicly: each player can see all the cards on the table at each moment.

Nene is very smart so she always selects cards optimally in order to maximize her score in the end of the game (after $$$2n$$$ rounds). If she has several optimal moves, she selects the move that minimizes your score in the end of the game.

More formally, Nene always takes turns optimally in order to maximize her score in the end of the game in the first place and to minimize your score in the end of the game in the second place.

Assuming that the cards are already distributed and cards in your hand have integers $$$a_1, a_2, \ldots, a_n$$$ written on them, what is the maximum number of points you can get by taking your turns optimally?

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

The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of cards you and Nene receive in the beginning of the game.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the integers on the cards in your hand. It is guaranteed that each integer from $$$1$$$ through $$$n$$$ appears in the sequence $$$a_1, a_2, \ldots, a_n$$$ at most $$$2$$$ times.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output: For each test case, output one integer: the maximum number of points you can get.

Examples:
input:
5
4
1 1 2 3
8
7 4 1 2 8 8 5 5
8
7 1 4 5 3 4 2 6
3
1 2 3
1
1
output:
1
2
1
0
0

Note: In the first test case, the integers written on your cards are $$$1$$$, $$$1$$$, $$$2$$$ and $$$3$$$. The integers written on Nene's cards are $$$2$$$, $$$3$$$, $$$4$$$ and $$$4$$$. The game may proceed as follows:

1. You select one of the cards with an integer $$$1$$$ written on it and place it on the table.
2. Nene selects one of the cards with an integer $$$4$$$ written on it and places it on the table.
3. You select the card with an integer $$$1$$$ written on it, receive $$$1$$$ point, and place the selected card on the table.
4. Nene selects the card with an integer $$$4$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
5. You select the card with an integer $$$2$$$ written on it and place it on the table.
6. Nene selects the card with an integer $$$2$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.
7. You select the card with an integer $$$3$$$ written on it and place it on the table.
8. Nene selects the card with an integer $$$3$$$ written on it, receive $$$1$$$ point, and places the selected card on the table.

At the end of the game, you scored $$$1$$$ point, and Nene scored $$$3$$$. It can be shown that you cannot score more than $$$1$$$ point if Nene plays optimally, so the answer is $$$1$$$.

In the second test case, if both players play optimally, you score $$$2$$$ points and Nene scores $$$6$$$ points.

Buggy Code:
import java.util.*;
public class Solution {
    public static int countRepeatedElements(int[] array) {

        Map<Integer, Integer> elementCount = new HashMap<>();

        for (int num : array) {
            if (elementCount.containsKey(num)) {
                elementCount.put(num, elementCount.get(num) + 1);
            } else {
                elementCount.put(num, 1);
            }
        }
        int repeatedCount = 0;
        for (int count : elementCount.values()) {
            if (count > 1) {
                repeatedCount++;
            }
        }

        return repeatedCount;
}
public static void main(String[] args) {
    Scanner sc= new Scanner(System.in);
    int t=sc.nextInt();
    int n;
    while(t-- >0){
       n=sc.nextInt();
       int [] array=new int[100];

       for (int i = 0; i < n; i++) {
            array[i] = sc.nextInt();
       }

   int repeatedCount = countRepeatedElements(array);
   System.out.println( repeatedCount -1);
    }
   sc.close();
}
}

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

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

Think step by step.

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