**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
#include<bits/stdc++.h>
#define int long long
using namespace std;
#define pb push_back
#define vc vector
#define pii pair <int, int>

void solve(){
    int a,b,c,d;cin>>a>>b>>c>>d;
    if(min(a,b)>max(c,d))cout<<4<<endl;
    else if(max(a,b)>max(c,d) && min(a,b)>min(c,d))cout<<2<<endl;
    else cout<<0<<endl;
}

signed main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int t;
    cin>>t;
    while(t--){
        solve();
    }
}

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

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

Think step by step.

Response:
To expose the bug in the given code, we need to focus on scenarios where the logic in the `solve` function might fail. The current logic in the `solve` function is:

1. If the minimum of Suneet's cards is greater than the maximum of Slavic's cards, it outputs 4.
2. If the maximum of Suneet's cards is greater than the maximum of Slavic's cards and the minimum of Suneet's cards is greater than the minimum of Slavic's cards, it outputs 2.
3. Otherwise, it outputs 0.

This logic fails to consider all possible outcomes of the game. Specifically, it does not account for cases where Suneet can win in exactly one of the two rounds.

Let's generate a test case that exposes this bug:

### Test Case Analysis
- **Suneet's cards:** 5, 7
- **Slavic's cards:** 4, 6

### Possible Outcomes
1. Suneet flips 5 and Slavic flips 4. Suneet wins the first round.
2. Suneet flips 7 and Slavic flips 6. Suneet wins the second round.

In this case, Suneet wins both rounds, so Suneet wins the game.

3. Suneet flips 5 and Slavic flips 6. Slavic wins the first round.
4. Suneet flips 7 and Slavic flips 4. Suneet wins the second round.

In this case, Suneet and Slavic each win one round, so no one wins the game.

5. Suneet flips 7 and Slavic flips 4. Suneet wins the first round.
6. Suneet flips 5 and Slavic flips 6. Slavic wins the second round.

In this case, Suneet and Slavic each win one round, so no one wins the game.

7. Suneet flips 7 and Slavic flips 6. Suneet wins the first round.
8. Suneet flips 5 and Slavic flips 4. Suneet wins the second round.

In this case, Suneet wins both rounds, so Suneet wins the game.

### Expected Output
Suneet wins the game in 2 out of the 4 possible scenarios. Therefore, the expected output should be 2.

### Test Input
```plaintext
1
5 7 4 6
```

This test case should expose the bug in the given code, as the current logic will incorrectly output 0 instead of 2.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
// Source: https://usaco.guide/general/io

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

int main() {
	int t;cin>>t;int ans[t];
	for(int i=0;i<t;i++)
	{
		int a[4];
		for(int j=0;j<4;j++)	cin>>a[j];
		if(min(a[0],a[1])>max(a[2],a[3]))	ans[i]=4;
		else if(max(a[0],a[1])>max(a[2],a[3]))	ans[i]=2;
		else ans[i]=0;
	}
	for(int i=0;i<t;i++)	cout<<ans[i]<<endl;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, we need to focus on edge cases where the logic in the buggy code might fail. The buggy code has the following issues:

1. It only checks if the minimum of Suneet's cards is greater than the maximum of Slavic's cards or if the maximum of Suneet's cards is greater than the maximum of Slavic's cards.
2. It does not consider all possible combinations of card flips, which is necessary to determine the number of ways Suneet can win.

Let's consider a test case where the cards are such that the minimum and maximum checks in the buggy code do not provide the correct result. Specifically, we need a case where Suneet can win in some scenarios but the buggy code incorrectly determines that he cannot win.

### Test Case Analysis
Consider the following cards:
- Suneet: 5, 7
- Slavic: 4, 6

Let's analyze the possible outcomes:
1. Suneet flips 5, Slavic flips 4. Suneet wins the first round.
2. Suneet flips 7, Slavic flips 6. Suneet wins the second round.
   - Suneet wins the game.

3. Suneet flips 5, Slavic flips 6. Slavic wins the first round.
4. Suneet flips 7, Slavic flips 4. Suneet wins the second round.
   - The game is a tie.

5. Suneet flips 7, Slavic flips 4. Suneet wins the first round.
6. Suneet flips 5, Slavic flips 6. Slavic wins the second round.
   - The game is a tie.

7. Suneet flips 7, Slavic flips 6. Suneet wins the first round.
8. Suneet flips 5, Slavic flips 4. Suneet wins the second round.
   - Suneet wins the game.

From the above analysis, Suneet wins the game in 2 out of 4 possible scenarios.

### Test Input
```plaintext
1
5 7 4 6
```

This test input should expose the bug in the buggy code because the buggy code will incorrectly determine that Suneet cannot win, while the correct number of ways Suneet can win is 2.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
def main():
    t = int(input())
    for i in range(t):
        answer = 0
        a1, a2, b1, b2 = list(map(int, input().split()))
        if (a1 > b1 and a2 >= b2):
            answer += 1
        if (a1 >= b1 and a2 > b2):
            answer += 1
        if (a1 > b2 and a2 >= b1):
            answer += 1
        if (a1 >= b2 and a2 > b1):
            answer += 1
        print(answer)
main()

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on scenarios where the code might incorrectly count the number of ways Suneet can win the game. The bug in the code lies in the conditions used to determine when Suneet wins a round. Specifically, the conditions are not correctly handling cases where both players have the same card value in one of the rounds.

Let's consider a test case where Suneet and Slavic have cards that can lead to such a situation:

1. **Case with equal cards in one round:**
   - Suneet's cards: 5, 5
   - Slavic's cards: 5, 3

In this case, the possible outcomes are:
- Suneet flips 5 and Slavic flips 5: No one wins this round.
- Suneet flips 5 and Slavic flips 3: Suneet wins this round.
- Suneet flips 5 and Slavic flips 5: No one wins this round.
- Suneet flips 5 and Slavic flips 3: Suneet wins this round.

For Suneet to win the game, he needs to win both rounds. However, the current code will incorrectly count the scenario where both players flip a 5 as a win for Suneet, leading to an incorrect result.

Here is the test input that will expose this bug:

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

This test case will show that the current code incorrectly counts the number of ways Suneet can win the game. The correct output should be 1, but the buggy code will output 2.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
t = int(input())

while t > 0:
    t -= 1

    a, b, c, d = map(int, input().split())

    result = 0

    # suneet - a, b
    # slavic - c, d

    # suneet - a, slavic - c
    if a > c:
        result += 1
    else:
        result -= 1

    # suneet - a, slavic - d
    if a > d:
        result += 1
    else:
        result -= 1

    # suneet - b, slavic - c
    if b > c:
        result += 1
    else:
        result -= 1

    # suneet - b, slavic - d
    if b > d:
        result += 1
    else:
        result -= 1

    if result < 0:
        print(0)
    else:
        print(result)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's analyze the logic of the code and the possible edge cases where it might fail.

### Analysis of the Buggy Code

The buggy code calculates the result by considering all possible combinations of card flips and incrementing or decrementing the result based on which player wins each round. However, the logic is flawed because it does not correctly account for the total number of games Suneet wins. Instead, it simply adds or subtracts 1 for each round, which does not reflect the correct number of games Suneet wins.

### Edge Cases to Consider

1. **Equal Cards**: When both players have the same cards, the result should be 0.
2. **High and Low Cards**: When one player has significantly higher cards and the other has significantly lower cards.
3. **One High and One Low**: When one player has one high card and one low card, and the other player has two cards in between.
4. **Symmetric Cases**: When the cards are symmetric (e.g., Suneet has 1, 2 and Slavic has 2, 1).

### Challenging Test Input

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

```plaintext
5
1 1 1 1
10 10 1 1
1 10 2 9
5 5 5 5
3 8 7 2
```

### Explanation

1. **1 1 1 1**: Both players have the same cards, so no one wins. The result should be 0.
2. **10 10 1 1**: Suneet has significantly higher cards, so he should win all possible games. The result should be 4.
3. **1 10 2 9**: Suneet has one high card and one low card, and Slavic has two cards in between. The result should be 2.
4. **5 5 5 5**: Both players have the same cards, so no one wins. The result should be 0.
5. **3 8 7 2**: This is a case where the cards are not symmetric, and Suneet has a chance to win. The result should be 2.

This test input should effectively expose the bug in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
import java.util.*;
public class card_game {
    public static void main(String[] args){
        Scanner sc=new Scanner(System.in);
        int tests=sc.nextInt();
        while(tests>0){
            int a=sc.nextInt();
            int b=sc.nextInt();
            int c=sc.nextInt();
            int d=sc.nextInt();
            int wins=0;
            if(a>c && b>d || a>c && b==d)wins++;
            if(a>d && b>c || a>d && b==c)wins++;
            if(b>c && a>d || b>c && a==d)wins++;
            if(b>d && a>c || b>d && a==c)wins++;
            System.out.println(wins);
            tests--;
        }
    }
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on the logic used to determine the number of games Suneet wins. The current implementation has a logical flaw in how it counts the wins. Specifically, it seems to be counting the number of valid combinations where Suneet wins both rounds, but it does not correctly account for the scenario where Suneet wins one round and Slavic wins the other, which should not count as a win for Suneet.

Let's consider the following edge cases:

1. **Equal Cards**: When both players have the same cards, no one should win.
2. **One Player Dominates**: When one player's cards are strictly greater than the other's cards in all possible combinations.
3. **Mixed Outcomes**: When the outcomes of the rounds are mixed (one player wins the first round, the other wins the second).

A good test input to expose the bug would be a scenario where the cards are such that Suneet wins one round, and Slavic wins the other, which should not count as a win for Suneet. However, the current code might incorrectly count this as a win.

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

```plaintext
3
5 5 5 5
6 4 5 5
7 3 5 5
```

### Explanation:
1. **5 5 5 5**: Both players have the same cards, so no one should win any games. The output should be `0`.
2. **6 4 5 5**: Suneet has a higher card (6) and a lower card (4). In one scenario, Suneet wins the first round (6 > 5) and Slavic wins the second round (5 > 4). This should not count as a win for Suneet, but the current code might incorrectly count it as a win.
3. **7 3 5 5**: Suneet has a higher card (7) and a lower card (3). In one scenario, Suneet wins the first round (7 > 5) and Slavic wins the second round (5 > 3). This should not count as a win for Suneet, but the current code might incorrectly count it as a win.

This test input should help identify the logical error in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
import java.util.Scanner;

public class secsum
{
    public static void main(String args[])
    {
        Scanner o = new Scanner(System.in);
        int t= o.nextInt();
        for(int i=0;i<t;i++)
        {
          int a1=o.nextInt();
          int a2=o.nextInt();
          int b1=o.nextInt();
          int b2=o.nextInt();

          int c=0;
          if (a1 > b1 && a2 > b2) c++;
          if (a1 > b2 && a2 > b1) c++;
          if (a2 > b1 && a1 > b2) c++;
          if (a2 > b2 && a1 > b1) c++;

            System.out.println(c);

        }
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases where the code's logic fails to correctly count the number of ways Suneet can win. The buggy code checks for specific conditions where Suneet wins both rounds, but it does not account for the fact that the order of the cards can affect the outcome.

Here are some steps to think through the problem:

1. **Identify the Bug**: The code checks for specific conditions where both of Suneet's cards are greater than both of Slavic's cards. It does not consider all possible permutations of card flips.

2. **Create a Test Case**:
   - **Case 1**: Suneet has cards that are both greater than Slavic's cards.
   - **Case 2**: Suneet has one card that is greater than one of Slavic's cards, and another card that is greater than the other card.
   - **Case 3**: Suneet has cards that are equal to Slavic's cards in some permutations but not in others.

3. **Edge Cases**:
   - **Equal Cards**: Test cases where some cards are equal to expose the logic that only checks for strict inequality.
   - **Small and Large Values**: Test cases with small and large values to ensure the logic handles all possible card values.

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

```plaintext
3
5 5 4 4
6 4 5 5
10 1 10 1
```

### Explanation:
1. **Test Case 1: 5 5 4 4**
   - Suneet: 5, 5
   - Slavic: 4, 4
   - Suneet wins both rounds in all permutations: (5, 5) vs (4, 4) and (5, 5) vs (4, 4).
   - Expected output: 4 (since there are 4 permutations where Suneet wins both rounds).

2. **Test Case 2: 6 4 5 5**
   - Suneet: 6, 4
   - Slavic: 5, 5
   - Suneet wins one round and loses one round in two permutations: (6, 4) vs (5, 5) and (4, 6) vs (5, 5).
   - Expected output: 2 (since there are 2 permutations where Suneet wins one round and loses one round).

3. **Test Case 3: 10 1 10 1**
   - Suneet: 10, 1
   - Slavic: 10, 1
   - Suneet wins one round and loses one round in two permutations: (10, 1) vs (10, 1) and (1, 10) vs (10, 1).
   - Expected output: 2 (since there are 2 permutations where Suneet wins one round and loses one round).

This test input should expose the bug in the provided code by testing different scenarios where the order of card flips matters.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int T = 0;
void solve()
{
    int a1, a2, b1, b2;
    cin >> a1 >> a2 >> b1 >> b2;
    int res = 0, a = 0, b = 0, c = 0, d = 0;
    if (a1 - b1 > 0)
        a = 1;
    else if (a1 - b1 < 0)
        a = -1;
    else
        a = 0;

    if (a1 - b2 > 0)
        b = 1;
    else if (a1 - b2 < 0)
        b = -1;
    else
        b = 0;

    if (a2 - b1 > 0)
        c = 1;
    else if (a2 - b1 < 0)
        c = -1;
    else
        c = 0;

    if (a2 - b2 > 0)
        d = 1;
    else if (a2 - b2 < 0)
        d = -1;
    else
        d = 0;

    res = a + b + c + d;
    if (res == 1)
        res *= 2;
    if (res < 0)
        res = 0;
    cout << res << endl;
}

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

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

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

Think step by step.

Response:
To create a test input that exposes the bug in the provided buggy code, we need to focus on scenarios where the logic of the code might fail. Specifically, we need to look at cases where the sum of the results of the comparisons is not correctly interpreted.

The buggy code has a flaw in how it calculates the number of ways Suneet can win the game. It simply sums up the results of the comparisons and then applies some incorrect logic to determine the number of wins. This approach fails to correctly handle the different possible outcomes of the game.

### Key Observations:
1. **Equality Cases**: The code does not correctly handle cases where the cards are equal.
2. **Multiple Wins**: The code does not correctly count the number of ways Suneet can win if he wins both rounds.
3. **Single Win**: The code incorrectly multiplies the result by 2 if Suneet wins one round, which is not always correct.

### Test Input:
To expose the bug, we need a test case where the sums of the comparisons do not correctly reflect the number of ways Suneet can win. A good test case is one where Suneet and Slavic have cards that can result in both wins, a single win, and no wins.

```plaintext
3
5 5 4 4
1 10 2 9
3 8 7 2
```

### Explanation:
1. **Test Case 1: 5 5 4 4**
   - Possible outcomes:
     - Suneet flips 5 and Slavic flips 4: Suneet wins (1 point)
     - Suneet flips 5 and Slavic flips 4: Suneet wins (1 point)
   - Suneet wins both rounds, so the result should be 1 (since there is only one way Suneet can win both rounds).

2. **Test Case 2: 1 10 2 9**
   - Possible outcomes:
     - Suneet flips 1 and Slavic flips 2: Slavic wins (0 points)
     - Suneet flips 10 and Slavic flips 9: Suneet wins (1 point)
     - Suneet flips 10 and Slavic flips 2: Suneet wins (1 point)
     - Suneet flips 1 and Slavic flips 9: Slavic wins (0 points)
   - Suneet wins one round in two of the four possible outcomes, so the result should be 2.

3. **Test Case 3: 3 8 7 2**
   - Possible outcomes:
     - Suneet flips 3 and Slavic flips 7: Slavic wins (0 points)
     - Suneet flips 8 and Slavic flips 2: Suneet wins (1 point)
     - Suneet flips 8 and Slavic flips 7: Suneet wins (1 point)
     - Suneet flips 3 and Slavic flips 2: Suneet wins (1 point)
   - Suneet wins two rounds in one of the four possible outcomes, so the result should be 1.

This test input should effectively expose the bug in the buggy code by highlighting the incorrect handling of the comparison results and the logic for determining the number of ways Suneet can win.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
#include <bits/stdc++.h>
#define int long long
using namespace std;
const int N=1e5+5;
const int mod=998244353;
const int INF=0x3f3f3f3f;
int arr[N];
signed main()
{
    ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
    int t;
    cin>>t;
    while(t--)
    {
        int a1,a2,b1,b2,ans=0;
        cin>>a1>>a2>>b1>>b2;
        if(a1>b1&&a1>b2&&a2>b2&&a2>b1)
            cout<<4<<endl;
        else if(a1>b1&&a1<b2&&a2>b1&&a2>b2)
        {
            cout<<2<<endl;
        }
        else if(a1>b2&&a1<b1&&a2>b1&&a2>b2)
        {
            cout<<2<<endl;
        }
        else if(a1>b1&&a1>b2&&a2>b1&&a2<b2)
        {
            cout<<2<<endl;
        }
        else if(a1>b1&&a1>b2&&a2>b2&&a2<b1)
        {
            cout<<2<<endl;
        }
        else
            cout<<0<<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
3
5 5 5 5
6 6 5 5
9 1 10 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
for i in range(int(input())):
    a1,a2,b1,b2 =map(int,input().split())
    c=0
    if a1>b1 and a2>b2:
        c+=2
    if a2>b1 and a1>b2:
        c+=2
    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:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider the following:

1. **Edge Cases**: Scenarios where the cards have the same values or are very close in value.
2. **Symmetric Cases**: Cases where the cards are symmetrically distributed to test the code's ability to handle equal values.
3. **Asymmetric Cases**: Cases where the cards are not symmetrically distributed to test the code's ability to handle different values.

The buggy code only increments the count `c` by 2 if both conditions are met for each pair of cards. This means it fails to handle scenarios where only one round is won by Suneet. We need to create a test case where Suneet wins exactly one round in both possible scenarios.

Let's consider the following test input:

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

### Explanation:

1. **Test Case 1: 5 3 4 2**
   - Possible outcomes:
     - Suneet flips 5, Slavic flips 4: Suneet wins.
     - Suneet flips 3, Slavic flips 2: Suneet wins.
   - Suneet wins both rounds, so the expected output is 2.

2. **Test Case 2: 4 2 3 1**
   - Possible outcomes:
     - Suneet flips 4, Slavic flips 3: Suneet wins.
     - Suneet flips 2, Slavic flips 1: Suneet wins.
   - Suneet wins both rounds, so the expected output is 2.

3. **Test Case 3: 6 4 5 3**
   - Possible outcomes:
     - Suneet flips 6, Slavic flips 5: Suneet wins.
     - Suneet flips 4, Slavic flips 3: Suneet wins.
   - Suneet wins both rounds, so the expected output is 2.

However, the buggy code will only output 2 for each of these cases, which is correct in these specific instances. To expose the bug, we need a case where Suneet wins exactly one round.

### Challenging Test Case:

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

### Explanation:

- **Test Case: 6 3 5 4**
  - Possible outcomes:
    - Suneet flips 6, Slavic flips 5: Suneet wins.
    - Suneet flips 3, Slavic flips 4: Slavic wins.
  - Suneet wins one round, so the expected output is 0 (since he needs to win both rounds to win the game).

This test case will expose the bug because the buggy code will incorrectly output 2 instead of 0.

### Final Output:

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
for _ in range(int(input())):
    a1, a2, b1, b2=map(int, input().split())
    # if a1==b1 or a1==b2:
    	# a1-=0.5
    # if a2==b1 or a2==b2:
    	# a2-=0.5
    a1, a2=min(a1, a2), max(a1, a2)
    b1, b2 = min(b1, b2), max(b1, b2)

    if min(a1, a2) > max(b1, b2):
        print(4)
    elif a1>b1 and a1<b2 and a2>=b2:
    	print(2)
    elif a1==b1 and a2>b2:
    	print(2)
    else:
    	print(0)

    # elif min(a1, a2)>min(b1, b2) and min(a1, a2)<max(b1, b2) and max(a1, a2)>max(b1, b2):
        # print(2)
    # elif min(a1, a2)>min(b1, b2) and max(a1, a2)<max(b1, b2):
        # print(0)
    # elif min(a1, a2)<min(b1, b2) and max(a1, a2)>max(b1, b2):
    	# print(0)
    # elif min(a1, a2)<min(b1, b2) and max(a1, a2)>min(b1, b2) and max(a1, a2)<max(b1, b2):
    	# print(0)
    # elif min(a1, a2)<min(b1, b2) and max(a1, a2)<min(b1, b2):
    	# print(0)
    # else:
    	# print(0)

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

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

Think step by step.

Response:
```plaintext
3
5 5 5 4
5 5 4 5
5 4 5 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
import java.util.*;
public class CardGame_CFdiv4 {
    public static void main(String[] args) {
        Scanner sc=new Scanner(System.in);
        int t=sc.nextInt();
        while(t-->0){
            int arr[]=new int[4];
            for(int i=0;i<4;i++){
                arr[i]=sc.nextInt();
            }
            int sum=0;
            if(arr[0]>arr[2] && arr[1]>arr[3]){sum+=1;}
            if(arr[0]>arr[3] && arr[1]>arr[2]){sum+=1;}
            if(arr[1]>arr[3] && arr[0]>arr[2]){sum+=1;}
            if(arr[1]>arr[2] && arr[0]>arr[3]){sum+=1;}
            System.out.println(sum);

        }
        sc.close();
    }
}

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

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

Think step by step.

Response:
```plaintext
1
5 5 5 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Card Game
time_limit_ms: 2000
memory_limit_mb: 256
Description: Suneet and Slavic play a card game. The rules of the game are as follows:

  * Each card has an integer value between $1$ and $10$.    * Each player receives $2$ cards which are face-down (so a player doesn't know their cards).    * The game is turn-based and consists exactly of two turns. In a round, both players pick a random unflipped card and flip it. The player who flipped a card with a strictly greater number wins the round. In case of equality, no one wins the round.    * A player wins a game if he wins the most number of rounds (i.e. strictly greater than the other player). In case of equality, no one wins the game.

Since Suneet and Slavic aren't best friends, you need to calculate the number of ways the game could happen that Suneet would end up as the winner.

For a better understanding, please check the notes section.

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

The first and only line of each test case contains $4$ integers $a_1$, $a_2$, $b_1$, $b_2$ ($1 \leq a_1, a_2, b_1, b_2 \leq 10$) where $a_1$ and $a_2$ represent the cards Suneet has, and $b_1$ and $b_2$ represent the cards Slavic has, respectively.

Output: For each test case, output a single integer — the number of games Suneet would win considering all possible games.

Examples:
input:
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
3 8 7 2
output:
2
0
4
0
2

Note: Consider the first test case when Slavic starts with the cards that have the values $2$ and $6$, and Suneet starts with cards that have the values $3$ and $8$. The game could happen in $4$ different ways:

  * Suneet flips $3$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $8$ and Slavic flips $6$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $3$ and Slavic flips $6$. Slavic wins the first round. Then, Suneet flips $8$ and Slavic flips $2$. Suneet wins the second round. Nobody wins since both players won an equal amount of rounds.

  * Suneet flips $8$ and Slavic flips $6$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $2$. Suneet wins the second round as well. Since Suneet won $2$ rounds, he wins the game.

  * Suneet flips $8$ and Slavic flips $2$. Suneet wins the first round. Then, Suneet flips $3$ and Slavic flips $6$. Slavic wins the round. Nobody wins since both players won an equal amount of rounds.

Buggy Code:
import java.util.*;

public class div2 {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while (t-- > 0) {
            int a1=sc.nextInt();
            int a2=sc.nextInt();
            int b1=sc.nextInt();
            int b2=sc.nextInt();
            int count=0,count1=0;
            if(a1>b1)
            {
                if(a2>=b2)
                count+=2;
            }
            if(a1>b2)
            {
                if(a2>=b1)
                count+=2;
            }
            System.out.println(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:
```plaintext
5
3 8 2 6
1 1 1 1
10 10 2 2
1 1 10 10
5 5 5 5
```

### Explanation:
1. **Test Case 1: 3 8 2 6**
   - This is a straightforward case where Suneet can win both rounds in two out of four possible scenarios. The expected output is 2.

2. **Test Case 2: 1 1 1 1**
   - Both players have the same cards, so no one wins any round. The expected output is 0.

3. **Test Case 3: 10 10 2 2**
   - Suneet always wins both rounds because his cards are always greater. The expected output is 4.

4. **Test Case 4: 1 1 10 10**
   - Slavic always wins both rounds because his cards are always greater. The expected output is 0.

5. **Test Case 5: 5 5 5 5**
   - Both players have the same cards, so no one wins any round. The expected output is 0.

This test input includes a variety of scenarios to ensure the code handles different edge cases and common scenarios correctly. The buggy code might fail in cases where the logic for counting the number of wins is incorrect, especially when the cards have equal values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
#include<iostream>
using namespace std;
int main(){
	int n;
	scanf("%d",&n);
	for (int i = 0; i < n; i++) {
		int a, b;scanf("%d%d",&a,&b);
		if (a < b)swap(a, b);

		for (int j = b+1;; j++) {
			if (j%a == j % b) { printf("%d\n",j); break; }
		}
	}
	return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
1 1
2 2
1000 1000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
#include <iostream>
using namespace std;
int main() {
  int t;
  cin >> t;
  while(t--){
     long a,b;
      cin >> a>>b;
      for(int m=0; m<a*b;m++){
          if((m>=a||m>=b)&&(m%a==m%b)){
              cout << m <<endl;
              break;
          }

      }

 } return 0;}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code might fail. The buggy code has a loop that starts from `m = 0` and checks up to `m < a * b`. This approach can fail in several scenarios, especially when `a` and `b` are relatively prime or when `a` and `b` are very close to each other.

Let's consider the following scenarios:
1. **When `a` and `b` are equal**: The smallest `m` that satisfies the conditions should be `a` (or `b`).
2. **When `a` and `b` are relatively prime**: The smallest `m` that satisfies the conditions will be `a * b`.
3. **When `a` and `b` are close to each other**: The smallest `m` might be a value between `a` and `b`.

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

```plaintext
3
4 4
3 5
10 11
```

### Explanation:
1. **Test Case 1 (4, 4)**:
   - The smallest `m` that satisfies the conditions is `4`.
   - The buggy code will correctly output `4` in this case, but it's a good starting point.

2. **Test Case 2 (3, 5)**:
   - The smallest `m` that satisfies the conditions is `15` (since `3 * 5 = 15` and `15 % 3 = 0` and `15 % 5 = 0`).
   - The buggy code might fail to find `15` within the loop because it only checks up to `m < 15`.

3. **Test Case 3 (10, 11)**:
   - The smallest `m` that satisfies the conditions is `110` (since `10 * 11 = 110` and `110 % 10 = 0` and `110 % 11 = 0`).
   - The buggy code might fail to find `110` within the loop because it only checks up to `m < 110`.

This test input should help expose the bug in the code by covering a range of scenarios where the loop might not find the correct `m` due to its limited range.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
#include <iostream>
using namespace std;
int main()
{
    int t;
    cin>>t;
    for(int i=0;i<t;i++){
    int a,b,max,min;
    cin>>a>>b;
    if(a>b)
    max=a,min=b;
    else
    max=b,min=a;
    for(int j=max;;j+=min){
    if(j%a==j%b){
    for(int k=j;;k--){
    if((k-1)%a!=(k-1)%b){
    cout<<k<<endl;
    break;
    }
    }
    break;
    }
    }
    }
    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
1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
def YES(a, b):
    m = a * b
    max_ab = max(a, b)
    valid_numbers = []

    while m >= max_ab:
        if m % a == m % b:
            valid_numbers.append(m)
        m //= 2

    if valid_numbers:
        print(min(valid_numbers))
    else:
        print(-1)

t = int(input())
for i in range(t):
    a, b = map(int, input().split())
    YES(a, b)

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

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

Think step by step.

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

1. **Edge Cases**: Consider the smallest and largest possible values for `a` and `b`.
2. **Special Conditions**: Test cases where `a` and `b` are equal, or one is a multiple of the other.
3. **General Cases**: Test cases where `a` and `b` have different remainders for a range of values.

The buggy code has a few issues:
- It starts with `m = a * b`, which might be a very large number and not necessarily the smallest valid `m`.
- It uses `m //= 2` to decrement `m`, which might skip valid values and lead to incorrect results.
- It only prints the minimum of the valid numbers found, but it might not find any valid numbers due to the large initial value and the decrement method.

Let's create a test input that exposes these issues:

```plaintext
4
1 1
2 4
1000 1000
1 1000
```

### Explanation:
1. **Test Case 1: `1 1`**
   - Here, both `a` and `b` are 1. The smallest `m` that satisfies the conditions is 1.
   - This tests the edge case where both numbers are the smallest possible.

2. **Test Case 2: `2 4`**
   - Here, `b` is a multiple of `a`. The smallest `m` that satisfies the conditions is 4.
   - This tests the case where one number is a multiple of the other.

3. **Test Case 3: `1000 1000`**
   - Here, both `a` and `b` are 1000. The smallest `m` that satisfies the conditions is 1000.
   - This tests the edge case where both numbers are the largest possible.

4. **Test Case 4: `1 1000`**
   - Here, `a` is 1 and `b` is 1000. The smallest `m` that satisfies the conditions is 1000.
   - This tests the case where one number is the smallest and the other is the largest.

These test cases should help expose the bugs in the given code, particularly the issues with the initial value of `m` and the decrement method.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
def solve_vault_riddle(a, b):
    if b > a and b % a == 0:
        return a
    if a > b and a % b == 0:
        return b

    def gcd(x, y):
        while y:
            x, y = y, x % y
        return x

    def lcm(x, y):
        return x * y // gcd(x, y)

    lcm_value = lcm(a, b)
    m = lcm_value

    return m

def main():
    t = int(input())
    for _ in range(t):
        a, b = map(int, input().split())
        result = solve_vault_riddle(a, b)
        print(result)

if __name__ == "__main__":
    main()

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

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

Think step by step.

Response:
```plaintext
2
4 8
8 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
for _ in range(int(input())):
    a,b = map(int,input().split())
    c = min(a,b)
    for i in range(c, a*b):
        if i%a == i%b:
            print(i)
            break

**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
1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
import java.util.Scanner;

public class Seating_in_a_bus {
    public static void main(String[] args) {
        Scanner sc=new Scanner(System.in);
        int t=sc.nextInt();
        for(int i=0;i<t;i++){
            int a=sc.nextInt();
            int b=sc.nextInt();
            System.out.println(call(a,b));
        }
    }
    public static int call(int a,int b) {
        if(a==b){
            return a+1;
        }
        int m=Math.max(a,b);
        while(true){
            if(m%a==0 && m%b==0){
                return m;
            }
            m++;
        }

    }
}

**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
1000 1000
999 1000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
import java.io.*;

public class A_King_Keykhosrow_s_Mystery {
    public static void main(String[] args) throws Exception {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        int t = Integer.parseInt(br.readLine());
        while(t-- > 0){
            String nm[] = br.readLine().split(" ");
            int n = Integer.parseInt(nm[0]);
            int m = Integer.parseInt(nm[1]);

            for(int i = Math.min(n, m) + 1; i<=n*m; i++){
                if((i % n) == (i % m )){
                    System.out.println(i);
                    break;
                }
            }
        }
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios where the code might fail. The key issues in the buggy code are:

1. The loop starts from `Math.min(n, m) + 1`, which might skip valid solutions.
2. The loop condition `i <= n * m` might be too large, leading to inefficiency and potential timeouts.

To expose these issues, we can use the following test cases:

1. **Equal Values**: When `a` and `b` are equal, the smallest `m` should be `a` or `b` itself.
2. **One Value is a Multiple of the Other**: When one number is a multiple of the other, the smallest `m` should be the larger number.
3. **Relatively Prime Numbers**: When `a` and `b` are relatively prime (i.e., their greatest common divisor is 1), the smallest `m` should be `a * b - a - b + 1`.
4. **Large Values**: To test the efficiency and ensure the loop condition is not too large.

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

```plaintext
4
5 5
3 6
7 9
1000 1000
```

Explanation:
1. **5 5**: Both values are equal, so the smallest `m` should be 5.
2. **3 6**: One value is a multiple of the other, so the smallest `m` should be 6.
3. **7 9**: Relatively prime numbers, so the smallest `m` should be 53 (calculated as `7 * 9 - 7 - 9 + 1`).
4. **1000 1000**: Both values are equal and large, so the smallest `m` should be 1000.

This test input should help expose the bug in the provided buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

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

#define int long long
#define pii pair<int,int>
#define pci pair<char,int>
#define vi vector<int>
#define mii map<int,int>
#define vpii vector<pii>
#define loop(i,n) for(int i=0;i<n;i++)
#define revloop(i,n) for(int i=n-1;i>=0;i--)

using namespace std;

//void debug(auto arr){
//	for(auto i:arr){
//		cout<<i<<" ";
//	}
//	cout<<endl;
//}

signed main(){
ios_base::sync_with_stdio(false);
cin.tie(NULL);
    int t=1,n,ans;
    cin>>t;
    while(t--){
        ans=0;
        int a,b;
        cin>>a>>b;
        for(int i=min(a,b);i<=1e8 + 10;i++){
            if(i%a == i%b){
                ans=i;
                break;
            }
        }
        cout<<ans<<endl;
    }
return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, let's consider the following points:

1. **Edge Cases**:
   - When `a` and `b` are equal.
   - When `a` or `b` is 1.
   - When `a` and `b` are large and have a large least common multiple (LCM).

2. **Scenarios**:
   - When the smallest `m` is very large, which could cause the loop to run for a long time.
   - When the loop might not terminate within a reasonable time due to the large upper limit (`1e8 + 10`).

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

1. **Equal Values**: `a = b`
2. **One of the values is 1**: `a = 1`, `b = 1000`
3. **Large Values with Large LCM**: `a = 997`, `b = 999`

Here is the test input:

```plaintext
3
4 4
1 1000
997 999
```

Explanation:
- The first test case (`4 4`) checks the scenario where `a` and `b` are equal. The smallest `m` should be 4.
- The second test case (`1 1000`) checks the scenario where one of the values is 1. The smallest `m` should be 1.
- The third test case (`997 999`) checks the scenario where `a` and `b` are large and have a large LCM. The smallest `m` should be 997 * 999.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

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

#define IO                        \
    ios_base::sync_with_stdio(0); \
    cin.tie(0);                   \
    cout.tie(0);
#define ll long long
#define ld long double
using namespace std;
const int N = 1e5 + 5, MOD = 1e9 + 7;

void solve()
{
    ll a, b;
    cin >> a >> b;
    for (ll i = min(a, b); i < 1e6; i++)
    {
        if ((i % a) == (i % b))
        {
            cout << i << "\n";
            break;
        }
    }
}

int main()
{
    IO
        ll t;
    cin >> t;
    while (t--)
        solve();
}

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

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

Think step by step.

Response:
To expose the bug in the given buggy code, we need to focus on edge cases or scenarios that can cause the code to fail. The key issues to consider are:

1. **Cases where `a` and `b` are equal**: If `a` and `b` are the same, the smallest `m` that satisfies the conditions is `a` (or `b`).
2. **Cases where `a` and `b` are co-prime**: If `a` and `b` are co-prime (i.e., their greatest common divisor is 1), the smallest `m` that satisfies the conditions is `a * b`.
3. **Cases where `a` and `b` have a common divisor**: If `a` and `b` share a common divisor, the smallest `m` that satisfies the conditions is the least common multiple (LCM) of `a` and `b`.

Let's generate a small-scale test input that includes these edge cases:

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

Explanation:
1. **Test Case 1: `1 1`**
   - Here, `a` and `b` are equal. The smallest `m` that satisfies the conditions is `1`.
2. **Test Case 2: `2 3`**
   - Here, `a` and `b` are co-prime. The smallest `m` that satisfies the conditions is `6`.
3. **Test Case 3: `4 6`**
   - Here, `a` and `b` have a common divisor (2). The smallest `m` that satisfies the conditions is `12`.

This test input will help expose any issues in the buggy code, especially in handling the edge cases mentioned.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: King Keykhosrow's Mystery
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is a tale about the wise King Keykhosrow who owned a grand treasury filled with treasures from across the Persian Empire. However, to prevent theft and ensure the safety of his wealth, King Keykhosrow's vault was sealed with a magical lock that could only be opened by solving a riddle.

The riddle involves two sacred numbers $$$a$$$ and $$$b$$$. To unlock the vault, the challenger must determine the smallest key number $$$m$$$ that satisfies two conditions:

- $$$m$$$ must be greater than or equal to at least one of $$$a$$$ and $$$b$$$.
- The remainder when $$$m$$$ is divided by $$$a$$$ must be equal to the remainder when $$$m$$$ is divided by $$$b$$$.

Only by finding the smallest correct value of $$$m$$$ can one unlock the vault and access the legendary treasures!

Input: The first line of the input contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$), the number of test cases.

Each test case consists of a single line containing two integers $$$a$$$ and $$$b$$$ ($$$1 \leq a, b \leq 1000$$$).

Output: For each test case, print the smallest integer $$$m$$$ that satisfies the conditions above.

Examples:
input:
2
4 6
472 896
output:
12
52864

Note: In the first test case, you can see that:

- $$$4 \bmod 4 = 0$$$ but $$$4 \bmod 6 = 4$$$
- $$$5 \bmod 4 = 1$$$ but $$$5 \bmod 6 = 5$$$
- $$$6 \bmod 4 = 2$$$ but $$$6 \bmod 6 = 0$$$
- $$$7 \bmod 4 = 3$$$ but $$$7 \bmod 6 = 1$$$
- $$$8 \bmod 4 = 0$$$ but $$$8 \bmod 6 = 2$$$
- $$$9 \bmod 4 = 1$$$ but $$$9 \bmod 6 = 3$$$
- $$$10 \bmod 4 = 2$$$ but $$$10 \bmod 6 = 4$$$
- $$$11 \bmod 4 = 3$$$ but $$$11 \bmod 6 = 5$$$

Buggy Code:
n=int(input())
for _ in range(n):
    a,b=map(int,input().split())
    m=min(a,b)
    while(m%a!=m%b):
        m=m+1
    print(m)

**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 3
1000 1000
```