**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Easy Problem
time_limit_ms: 1000
memory_limit_mb: 256
Description: Cube is given an integer $$$n$$$. She wants to know how many ordered pairs of positive integers $$$(a,b)$$$ there are such that $$$a=n-b$$$. Since Cube is not very good at math, please help her!

Input: The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 99$$$) — the number of test cases.

The only line of each test case contains an integer $$$n$$$ ($$$2 \leq n \leq 100$$$).

Output: For each test case, output the number of ordered pairs $$$(a, b)$$$ on a new line.

Examples:
input:
3
2
4
6
output:
1
3
5

Note: In the first test case, the only ordered pair that works is $$$(a,b)=(1,1)$$$.

In the second test case, the three ordered pairs of $$$(a,b)$$$ that work are $$$(3,1), (2,2), (1,3)$$$.

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

public class Main {

    private static PrintWriter out = new PrintWriter(new BufferedWriter((new OutputStreamWriter(System.out))));

    public static void main(String[] args) {
        int tcSize = Reader.nextInt();
        for (int idx = 0; idx < tcSize; ++idx) {
            int n = Reader.nextInt();
            out.println(n - (n % 2 == 0 ? 1 : 0));
        }
        out.flush();
    }

    public static class Reader {
        static BufferedReader reader = new BufferedReader(new InputStreamReader(System.in));
        static StringTokenizer tokenizer = new StringTokenizer("");

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

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

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

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

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

Think step by step.

Response:
```plaintext
3
2
3
100
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

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

int main()
{
 ios_base::sync_with_stdio(false); cin.tie(NULL);
 ll times;
 cin>>times;
 while(times--)
 {
 ll l,r;
 cin>>l>>r;
 if(l==1)
 {
     if(r<3)
     {
         cout<<0<<endl;
     }
     else if(r==3)
     {
         cout<<1<<endl;
     }
     else
     {
         if(r%2==1)
         {
            cout<<1+((r-(4+3))/4)+1<<endl;
         }
         else
         {
             cout<<1+((r-(4+2))/4)+1<<endl;
         }
     }

 }
 else
 {
     if(l%2==1&&r%2==1)
     {
         cout<<(r-l)/3<<endl;
     }
     else if(l%2==0&&r%2==1)
     {
         cout<<((r-(l+3))/4)+1<<endl;
     }
     else if(l%2==1&&r%2==0)
     {
          cout<<((r-(l+2))/4)+1<<endl;
     }
     else
     {
         cout<<(((r-1)-(l+3))/4)+1<<endl;
     }
 }

}
    return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's consider the logic and edge cases in the code. The code handles different ranges of integers based on the parity (odd or even) of `l` and `r`. The key is to find cases where the code's assumptions might be incorrect or where the logic might break down.

### Steps to Identify Challenging Test Cases

1. **Small Range with Special Properties:**
   - Test cases where the range is small and the numbers have specific properties (e.g., all odd, all even, or a mix).

2. **Edge Cases for `l == 1`:**
   - Test cases where `l == 1` and `r` is small or just above a multiple of 4.

3. **Mixed Parity Cases:**
   - Test cases where `l` and `r` have different parities (one odd, one even).

4. **Large Range with Specific Properties:**
   - Test cases where the range is large and the numbers have specific properties (e.g., all odd, all even, or a mix).

### Chosen Test Cases

1. **Small Range with All Odd Numbers:**
   - `l = 3, r = 9` (all odd numbers)

2. **Small Range with All Even Numbers:**
   - `l = 4, r = 10` (all even numbers)

3. **Small Range with Mixed Parity:**
   - `l = 5, r = 12` (mixed parity)

4. **Edge Case for `l == 1`:**
   - `l = 1, r = 10` (includes 1 and a mix of odd and even numbers)

5. **Large Range with Specific Properties:**
   - `l = 100, r = 200` (mixed parity)

### Test Input

```plaintext
5
3 9
4 10
5 12
1 10
100 200
```

This test input includes a variety of ranges with different properties, which should help expose any logical 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: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
#include<bits/stdc++.h>
#include<numeric>
using namespace std;
int main(){
    int t;
    cin>>t;
    while(t--){
        int l,r;
        cin>>l>>r;
        int arr[(r-l)+1];
        arr[0]=l;
        for(int i=1;i<=(r-l);i++){
            arr[i]=arr[i-1]+1;
        }
        int flag=0;
        int count=0;
        for(int i=0;i<=(r-l)-2;i++){
            if(arr[i]==0){
                continue;
            }
            for(int j=i+1;j<=(r-l)-1;j++){
                if(gcd(arr[i],arr[j])==1){
                    for(int k=j+1;k<=(r-l);k++){
                        if(gcd(arr[j],arr[k])==1 && gcd(arr[k],arr[i])==1){
                            count++;
                            arr[i]=0;
                            arr[j]=0;
                            arr[k]=0;
                            flag=1;
                            break;
                    }
                    if(flag){
                        break;
                    }
            }
        }
    }
}
cout<<count<<endl;
    }
    return 0;
}

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

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

Think step by step.

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

1. **Edge Cases**: Test the boundaries of the input range, such as when `l` and `r` are very close to each other.
2. **Special Cases**: Test cases where the set contains specific integers that might cause the algorithm to fail, such as consecutive integers or sets with a small number of elements.
3. **Large Range**: Test a range where the number of elements is large enough to stress the algorithm but not so large that it exceeds the input length limit.

Let's consider a few specific scenarios:

1. **Small Range with Consecutive Integers**:
   - This can test the algorithm's ability to handle small sets and ensure it correctly identifies and removes valid triplets.
   - Example: `l = 1, r = 3`

2. **Range with a Small Number of Elements**:
   - This can test the algorithm's ability to handle cases where the set has only a few elements.
   - Example: `l = 10, r = 12`

3. **Range with a Moderate Number of Elements**:
   - This can test the algorithm's ability to handle a larger set and ensure it correctly identifies and removes valid triplets.
   - Example: `l = 10, r = 20`

4. **Range with a Large Number of Elements**:
   - This can test the algorithm's ability to handle a large set and ensure it correctly identifies and removes valid triplets.
   - Example: `l = 1, r = 100`

5. **Range with Specific Integers**:
   - This can test the algorithm's ability to handle specific integers that might cause issues, such as sets with many even or odd numbers.
   - Example: `l = 2, r = 8`

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

```plaintext
5
1 3
10 12
10 20
1 100
2 8
```

This test input includes a variety of ranges to ensure the algorithm is tested under different conditions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

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

int main() {
    int t;
    cin >> t;
    while (t--) {
        int l, r;
        cin >> l >> r;
        // 计算初始集合的大小
        int range_size = r - l + 1;
        if (range_size != 3) {
        // 计算可以执行的最大操作数
            int mp = range_size * 1 / 4;
            printf("%d\n", mp);
        } else {
            if ((l & 1) == 0) printf("0\n");
            else printf("1\n");
        }
    }
    return 0;
}

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

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

Think step by step.

Response:
```plaintext
1
1 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
from math import gcd

if __name__ == '__main__':
    t = int(input())

    for _ in range(t):
        l, r = map(int, input().split())

        ans = 0
        if (r - l + 1) < 3:
            ans = 0
        elif (r - l + 1) == 3:
            ans = (1 if gcd(l,l+1) == gcd(l+1,l+2) == gcd(l, l+2) == 1 else 0)
        else:
            ans = (r - l + 1) // 4
        print(ans)

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

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

Think step by step.

Response:
To 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's logic might fail. The buggy code has a few issues:

1. It incorrectly handles the case where the range length is exactly 3.
2. It uses a fixed formula \((r - l + 1) // 4\) for ranges longer than 3, which might not always yield the correct maximum number of operations.

Let's consider the following scenarios:

1. **Range length exactly 3**: The code checks if \(\gcd(l, l+1) = \gcd(l+1, l+2) = \gcd(l, l+2) = 1\). This is generally true for any three consecutive integers, so the code should always output 1 for such ranges. However, we should still verify this.

2. **Range length greater than 3**: The code uses \((r - l + 1) // 4\). This formula might not always be optimal. For example, if the range has a length of 5, the correct answer should be 1 (since 5 cannot be divided into 4 groups of 3), but the code will output 1, which is correct. However, for a range of 7, the correct answer should be 2, but the code will output 1.

3. **Range length with specific patterns**: We should test ranges where the number of integers is not a multiple of 4 but can still be divided into multiple valid operations.

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

```plaintext
5
1 3
3 7
10 12
2 8
51 60
```

Explanation:
- **Test Case 1 (1 3)**: The range has exactly 3 integers. The correct answer is 1.
- **Test Case 2 (3 7)**: The range has 5 integers. The correct answer is 1.
- **Test Case 3 (10 12)**: The range has 3 integers. The correct answer is 1.
- **Test Case 4 (2 8)**: The range has 7 integers. The correct answer is 2.
- **Test Case 5 (51 60)**: The range has 10 integers. The correct answer is 3.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import math as m
v=int(input())
n=[]
for i in range(v):
    a,b=map(int,input().split())
    l=[]
    c=0
    for o in range(a,b+1):
        l.append(o)
    for q in l:
        h=q
        g=0
        for w in l:
            j=w
            if w!=q:
                for e in l:
                    k=e
                    if k!=h and k!=j:
                        if m.gcd(h,j)==m.gcd(j,k)==m.gcd(h,k)==1:
                            l.pop(l.index(h))
                            l.pop(l.index(j))
                            l.pop(l.index(k))
                            c+=1
                            g=1
                            break
            if g==1:
                break
    n.append(c)
for ele in n:
    print(ele,sep='\n')

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the buggy code, we need to consider scenarios where the code might fail due to inefficiency or incorrect logic. Here are the steps to think through this:

1. **Edge Cases:**
   - Small ranges where the set has very few elements.
   - Ranges with consecutive prime numbers.
   - Ranges where the set has a specific pattern of coprime numbers.

2. **Large Ranges:**
   - Ranges where the set has a large number of elements, which can expose inefficiencies in the code.

3. **Specific Patterns:**
   - Ranges where the set has a specific pattern of numbers that are coprime in groups of three.

Let's consider a few specific test cases:

### Test Case 1: Small Range with Few Elements
- **Range:** [1, 3]
- **Expected Output:** 1
- **Explanation:** The set is {1, 2, 3}. The only possible operation is to remove 1, 2, and 3.

### Test Case 2: Range with Consecutive Primes
- **Range:** [5, 7]
- **Expected Output:** 1
- **Explanation:** The set is {5, 6, 7}. The only possible operation is to remove 5, 6, and 7.

### Test Case 3: Range with Specific Coprime Pattern
- **Range:** [10, 15]
- **Expected Output:** 3
- **Explanation:** The set is {10, 11, 12, 13, 14, 15}. Possible operations include:
  - Remove 10, 11, 13
  - Remove 12, 14, 15
  - Remove 10, 13, 14

### Test Case 4: Large Range
- **Range:** [1, 10]
- **Expected Output:** 3
- **Explanation:** The set is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Possible operations include:
  - Remove 1, 2, 3
  - Remove 4, 5, 6
  - Remove 7, 8, 9

### Test Case 5: Range with No Possible Operations
- **Range:** [1, 2]
- **Expected Output:** 0
- **Explanation:** The set is {1, 2}. There are not enough elements to perform any operations.

### Test Case 6: Range with a Large Number of Elements
- **Range:** [1, 20]
- **Expected Output:** 6
- **Explanation:** The set is {1, 2, 3, ..., 20}. Possible operations include:
  - Remove 1, 2, 3
  - Remove 4, 5, 6
  - Remove 7, 8, 9
  - Remove 10, 11, 13
  - Remove 12, 14, 15
  - Remove 16, 17, 19

### Test Case 7: Range with a Specific Coprime Pattern
- **Range:** [1, 12]
- **Expected Output:** 4
- **Explanation:** The set is {1, 2, 3, ..., 12}. Possible operations include:
  - Remove 1, 2, 3
  - Remove 4, 5, 6
  - Remove 7, 8, 9
  - Remove 10, 11, 12

### Test Case 8: Range with a Large Number of Elements
- **Range:** [1, 30]
- **Expected Output:** 10
- **Explanation:** The set is {1, 2, 3, ..., 30}. Possible operations include:
  - Remove 1, 2, 3
  - Remove 4, 5, 6
  - Remove 7, 8, 9
  - Remove 10, 11, 13
  - Remove 12, 14, 15
  - Remove 16, 17, 19
  - Remove 18, 20, 21
  - Remove 22, 23, 25
  - Remove 24, 26, 27
  - Remove 28, 29, 30

### Final Test Input
```plaintext
8
1 3
5 7
10 15
1 10
1 2
1 20
1 12
1 30
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
t=int(input())
for i in range(0,t):
    l,r=map(int,input().split())
    x=r-l+1
    if x>3:
        print(x//4)
    else:
        print(1)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on scenarios where the code's logic fails. The buggy code calculates the maximum number of operations as `x // 4` for `x > 3`, where `x` is the number of integers in the range `[l, r]`. This logic is incorrect because it does not consider the actual constraints of the problem, particularly the requirement that the chosen integers must have a GCD of 1.

### Key Points to Consider:
1. **Small Range with Many Coprime Numbers**: A range with many numbers that are coprime to each other can expose the bug.
2. **Ranges with Special Properties**: Ranges where the number of coprime numbers is not a simple multiple of 4 can also be challenging.
3. **Edge Cases**: Ranges with specific values of `l` and `r` that might cause the code to miscalculate the number of operations.

### Challenging Test Input:
Let's consider a small range where the number of integers is just above 3 and contains several coprime numbers. For example, the range `[1, 7]`:

- The range `[1, 7]` contains 7 integers: 1, 2, 3, 4, 5, 6, 7.
- The maximum number of operations can be 2, as we can choose:
  - First operation: 1, 2, 3 (gcd(1, 2) = 1, gcd(2, 3) = 1, gcd(1, 3) = 1)
  - Second operation: 4, 5, 6 (gcd(4, 5) = 1, gcd(5, 6) = 1, gcd(4, 6) = 2, but 4 and 6 are not coprime, so we need to choose 4, 5, 7 instead)
  - Third operation: 4, 5, 7 (gcd(4, 5) = 1, gcd(5, 7) = 1, gcd(4, 7) = 1)

The buggy code would output `x // 4 = 7 // 4 = 1`, which is incorrect.

### Test Input:
```plaintext
1
1 7
```

This test input will help expose the bug in the provided buggy code by showing that it does not correctly handle the maximum number of operations for a range with more than 3 integers.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import java.util.*;
public class Find{
    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();
            int num=b-a+1;
            int maxop=num/4;
            System.out.println(maxop==0?1:maxop);
        }
    }
}

**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 4
2 6
1 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import java.io.*;
import java.util.*;
import static java.lang.System.*;
import static java.lang.Math.*;
public class A {
    void run() {
        int tc=ni();
        while(tc-->0) {
        	int l=ni(), r=ni();
        	int diff = r-l+1;
        	ap((diff == 3 ? diff + l%2 : diff )>> 2);
            ap("\n");
        }
        out.println(sb);
        out.flush();
        out.close();
    }

    public static void main(String[] args)throws Exception {
        try {
            // new Thread(null, new A()::run, "1", 1 << 26).start();
            new Thread(null, new A("ONLINE_JUDGE")::run, "1", 1 << 26).start();
        } catch(Exception e) {}
    }

    FastReader sc=null;PrintWriter out = null;
    public A()throws Exception {
        sc = new FastReader(new FileInputStream("../input.txt"));
        out = new PrintWriter(new BufferedWriter(new FileWriter("../output.txt")));
    }
    public A(String JUDGE) {
        sc = new FastReader(System.in);
        out = new PrintWriter(System.out);
    }

    long ceil(long a, long b) {
        return (a + b - 1) / b;
    }

    StringBuilder sb = new StringBuilder();
    final int IMAX = Integer.MAX_VALUE;
    final int IMIN = Integer.MIN_VALUE;
    final long LMAX = Long.MAX_VALUE;
    final long LMIN = Long.MIN_VALUE;
    final int MOD = (int)1e9+7;

    void ap(String... str) {
        for(String s: str) sb.append(s);
    }
    void ap(Integer... arr) {
        for(Integer a: arr) sb.append(a);
    }
    void ap(Long... arr) {
        for(Long a: arr) sb.append(a);
    }
    void debug(String... str) {
        for(String s: str) System.out.print(s+" - ");
        System.out.println();
    }
    void debug(Integer... a) {
        for(Integer v: a) System.out.print(v+" - ");
        System.out.println();
    }
    void debug(Long... a) {
        for(long v: a) System.out.print(v+" - ");
        System.out.println();
    }
    void debug(int a[], Integer... b) {
        System.out.println(Arrays.toString(a));
        debug(b);
    }
    void debug(int a[][], Integer... b) {
        System.out.println(Arrays.deepToString(a));
        debug(b);
    }
    void debug(long a[], Long... b) {
        System.out.println(Arrays.toString(a));
        debug(b);
    }
    void debug(long a[][], Long... b) {
        System.out.println(Arrays.deepToString(a));
        debug(b);
    }
    String s(Long n) {
        return String.valueOf(n);
    }
    String s(Integer n) {
        return String.valueOf(n);
    }

    String ns() { return sc.next(); }
    int ni() { return sc.nextInt(); }
    long nl() { return sc.nextLong(); }
    char[] nc() {
        return ns().toCharArray();
    }
    char[][] nc(int n) {
        char c[][] = new char[n][];
        for(int i=0;i<n;i++) {
            c[i] = ns().toCharArray();
        }
        return c;
    }
    int[][] _ni(int n) {
        int a[][] = new int[n][];
        for(int i=0;i<n;a[i]=new int[]{i, ni()});
        return a;
    }
    int[] ni(int n) {
        int a[]=new int[n];
        for(int i=0;i<n;a[i++]=ni());
        return a;
    }
    long[] nl(int n) {
        long a[]=new long[n];
        for(int i=0;i<n;a[i++]=nl());
        return a;
    }

    int[][] ni(int n,int m) {
        int a[][]=new int[n][m];
        for(int i=0;i<n;i++)
            for(int j=0;j<m;j++)
                a[i][j]=ni();
        return a;
    }
    long[][] nl(int n,int m) {
        long a[][]=new long[n][m];
        for(int i=0;i<n;i++)
            for(int j=0;j<m;j++)
                a[i][j]=nl();
        return a;
    }
    int gcd(int a, int b) {
        return b==0?a:gcd(b,a%b);
    }
    static class FastReader {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int curChar;
        private int numChars;
        private FastReader.SpaceCharFilter filter;
        public FastReader(InputStream stream) {
            this.stream = stream;
        }

        public int read() {
            if (numChars == -1) throw new InputMismatchException();
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
                if (numChars <= 0) return -1;
            }
            return buf[curChar++];
        }

        public int nextInt() {
            int c = read();
            while (isSpaceChar(c)) c = read();
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            int res = 0;
            do {
                if (c < '0' || c > '9') throw new InputMismatchException();
                res *= 10;
                res += c - '0';
                c = read();
            }
            while (!isSpaceChar(c));
            return res * sgn;
        }

        public long nextLong() {
            int c = read();
            while (isSpaceChar(c)) c = read();
            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = read();
            }
            long res = 0;
            do {
                if (c < '0' || c > '9') throw new InputMismatchException();
                res = res*1L*10;
                res += c - '0';
                c = read();
            }
            while (!isSpaceChar(c));
            return res *1L* sgn;
        }

        public String next() {
            int c = read();
            while (isSpaceChar(c)) c = read();
            StringBuilder res = new StringBuilder();
            do {
                res.appendCodePoint(c);
                c = read();
            } while (!isSpaceChar(c));
            return res.toString();
        }

        public boolean isSpaceChar(int c) {
            if (filter != null) return filter.isSpaceChar(c);
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        public interface SpaceCharFilter {
            public boolean isSpaceChar(int ch);

        }
    }
}

**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 faulty logic in the code can be easily identified. The buggy code has a flaw in the way it calculates the number of operations. Specifically, it uses a simple division by 4 to determine the number of operations, which does not correctly account for the gcd conditions.

Here are the steps to think through the test input:

1. **Identify the Bug**: The bug in the code is that it does not correctly handle the gcd conditions. It simply divides the number of elements by 4, which is not always correct.

2. **Edge Cases**: We need to consider edge cases where the range of numbers has specific properties that make it difficult to find three numbers with gcd 1.

3. **Small Scale**: Since we need a small-scale test input, we should choose a range that is manageable and can be easily verified.

4. **Example Test Case**: A good example is a range where the numbers are such that it is difficult to find three numbers with gcd 1. For instance, a range where most numbers are even or have common factors.

Given these considerations, let's choose the range `[2, 4]` as a test case. In this range, the numbers are 2, 3, and 4. The only valid operation is to choose 2, 3, and 4 because gcd(2, 3) = gcd(3, 4) = gcd(2, 4) = 1. After this operation, the set is empty, and no more operations can be performed.

Here is the test input:

```plaintext
1
2 4
```

This test case is small and valid, and it should expose the bug in the code because the buggy code will incorrectly calculate the number of operations as 1, which is correct, but it does not handle the gcd conditions correctly in more complex scenarios.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
#include <iostream>
#include <vector>
using namespace std;

int gcd(int a, int b) {
    if (a == 0) {
        return b;
    }
    return gcd(b % a, a);
}

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

    while (testCases--) {
        int l, r;
        cin >> l >> r;

        int op = 0;
        vector<int> vec;

        for (int i = l; i <= r; i++) {
            vec.push_back(i);
        }

        for (int i = 0; i <= vec.size() - 3; i++) {
            if (vec[i] != -1 && vec[i+1] != -1 && vec[i+2] != -1) {
                int a = vec[i], b = vec[i+1], c = vec[i+2];

                if (gcd(a, b) == 1 && gcd(b, c) == 1 && gcd(a, c) == 1) {

                    vec[i] = vec[i+1] = vec[i+2] = -1;
                    op++;
                }
            }
        }

        cout << op << endl;
    }

    return 0;
}

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

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

Think step by step.

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

### Explanation:
This test case is designed to expose the bug in the buggy code by focusing on a small range of numbers where the algorithm can be easily analyzed.

- The range [1, 5] contains the numbers 1, 2, 3, 4, and 5.
- The correct solution should be able to perform 1 operation by selecting the numbers 1, 2, and 3, as \(\gcd(1, 2) = 1\), \(\gcd(2, 3) = 1\), and \(\gcd(1, 3) = 1\).
- After removing 1, 2, and 3, the set will contain 4 and 5, and no more operations can be performed.

The buggy code, however, may fail to correctly identify and remove the numbers 1, 2, and 3 in a single operation, or it might incorrectly leave some numbers unprocessed, leading to an incorrect output. This test case will help in identifying such issues.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define cel(n,k) ((n-1)/k+1)
#define lli long long int
#define ci(k) cin>>k
#define cii(a,b) cin>>a>>b
#define vi vector<int>
#define ll long long
#define fr(n) for(int i=0;i<n;i++)
#define fr1(n) for(int i=1;i<=n;i++)
#define frij(a,b) for(int i=a;i<=b;i++)
#define frj(n) for(int j=0;j<n;j++)
#define frj1(n) for(int j=1;j<=n;j++)
 #define pb push_back
#define all(v) v.begin(),v.end()
#define min3(a,b,c) min(a,min(b,c)
#define max3(a,b,c) max(a,max(b,c)
#define ff first
#define ss second
#define MOD 10000007
#define Assamualaikum int main()
#define AllahHafiz return 0;
typedef long long i64;
typedef unsigned long long ull;
typedef pair<int,int> pii;
typedef pair<long long,long long> pll;
ll gcd(ll a, ll b){
    if(b == 0) return a;
    return gcd(b,a%b);
}
ll power(ll a, ll b){
    ll res = 1;
    while(b){
        if(b%2) {
            res = (res*a);
            b--;
        }
        b/=2;
        a=(a*a);
    }
    return res;
}
void solve(){
    //write your cod here
    int l,r;cii(l,r);
    set<int> s;
    for(int i=l;i<=r;i++) s.insert(i);
   // cout<<s.size()/3<<endl;
    ll ans=0;
    map<int,bool> mp;
    for(int i=l;i+2<=r;i++){
        if(mp[i] == 1) continue;
        if(mp[i+1] == 1 ) {i++;continue;}
        for(int j=i+2;j<=r;j++){
            if(__gcd(j,i) == 1 && __gcd(i+1,j) == 1){
                mp[j]=1;
                ans++;
            }
        }
        i++;
    }
    cout<<(ans+1)/2<<endl;
}
Assamualaikum
{
    cin.tie(nullptr)->sync_with_stdio(false);
    int t=1;
    ci(t);
    while(t--){
        solve();
    }
    AllahHafiz
}

**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 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import sys
input = sys.stdin.readline
from math import ceil
def in1():
    return int(input())

def in2():
    return [*map(int, input().split())]

t = in1()
for _ in range(t):
    l, r = in2()
    d = r-l+1
    if d == 3:
        if l%2 == 0:
            print(0)
        else:
            print(1)
    else:
        print((r-l+1)//4)

**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 4
2 5
1 6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import math

t = int(input())
for _ in range(t):
    l, r = map(int, input().split())
    dist = r - l + 1
    if dist == 3 and math.gcd(l, r) == 1:
        print(1)
    else:
        print(dist // 4)

**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 and scenarios where the code's logic might fail. The buggy code incorrectly assumes that the maximum number of operations can be determined by simply dividing the number of elements in the set by 4. However, this approach does not consider the constraints of the problem, particularly the requirement that the selected integers must be pairwise coprime (i.e., their greatest common divisor (gcd) must be 1).

### Step-by-Step Analysis:
1. **Case with 3 Elements:**
   - The code correctly handles the case where the set has exactly 3 elements and all are coprime.
   - Example: `l = 1, r = 3` (Output: 1)

2. **Case with 4 Elements:**
   - The code incorrectly assumes that 4 elements can always be divided into 1 operation.
   - Example: `l = 2, r = 5` (Set: {2, 3, 4, 5})
   - Possible operation: `a = 2, b = 3, c = 5` (gcd conditions met)
   - Remaining set: {4} (No more operations can be performed)
   - Correct output: 1
   - Buggy output: 1 (Correct, but let's verify other cases)

3. **Case with 5 Elements:**
   - The code incorrectly assumes that 5 elements can be divided into 1 operation.
   - Example: `l = 1, r = 5` (Set: {1, 2, 3, 4, 5})
   - Possible operation: `a = 1, b = 2, c = 3` (gcd conditions met)
   - Remaining set: {4, 5}
   - Possible operation: `a = 4, b = 5` (No more operations can be performed)
   - Correct output: 1
   - Buggy output: 1 (Correct, but let's verify other cases)

4. **Case with 6 Elements:**
   - The code incorrectly assumes that 6 elements can be divided into 1 operation.
   - Example: `l = 1, r = 6` (Set: {1, 2, 3, 4, 5, 6})
   - Possible operation: `a = 1, b = 2, c = 3` (gcd conditions met)
   - Remaining set: {4, 5, 6}
   - Possible operation: `a = 4, b = 5, c = 6` (gcd conditions met)
   - Remaining set: {}
   - Correct output: 2
   - Buggy output: 1 (Incorrect)

5. **Case with 7 Elements:**
   - The code incorrectly assumes that 7 elements can be divided into 1 operation.
   - Example: `l = 1, r = 7` (Set: {1, 2, 3, 4, 5, 6, 7})
   - Possible operation: `a = 1, b = 2, c = 3` (gcd conditions met)
   - Remaining set: {4, 5, 6, 7}
   - Possible operation: `a = 4, b = 5, c = 6` (gcd conditions met)
   - Remaining set: {7}
   - Possible operation: `a = 7` (No more operations can be performed)
   - Correct output: 2
   - Buggy output: 1 (Incorrect)

### Challenging Test Input:
Based on the above analysis, the following test input will expose the bug in the buggy code:

```plaintext
3
1 6
1 7
2 8
```

This test input includes cases with 6, 7, and 7 elements, respectively, which should correctly output 2, 2, and 1, respectively, but the buggy code will output 1, 1, and 1.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
//import java.io.File;
// import java.io.FileOutputStream;
// import java.io.PrintStream;

import java.util.*;

public class A {

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);

//---------Code starts here---------
        int t = sc.nextInt();

        while (t-- > 0) {
            int l = sc.nextInt();
            int r = sc.nextInt();
            if(r-l < 4) System.out.println(1);
            else System.out.println(((r-l+1)/2)/2);

            // AHelper obj = new AHelper();
            // System.out.println(obj.aHelperMethod(strip, n));
        }

        sc.close();
    }

    private static int gcd(int a, int b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a % b);
    }

}

// class AHelper{
//     public int aHelperMethod(String strip, int n) {
//     }
// }

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

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

Think step by step.

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

### Analysis of the Buggy Code:
1. **Bug 1: Incorrect Condition for Small Ranges:**
   - The code checks if `r - l < 4` and prints `1` if true. This is incorrect because for ranges where `r - l < 3`, no operations can be performed.
   - For example, if `l = 1` and `r = 3`, the set is `{1, 2, 3}` and one operation can be performed, but for `l = 1` and `r = 2`, the set is `{1, 2}` and no operation can be performed.

2. **Bug 2: Incorrect Calculation for Larger Ranges:**
   - The code calculates the maximum number of operations as `((r - l + 1) / 2) / 2`. This is incorrect because it does not consider the actual number of coprime triplets that can be formed.
   - For example, for `l = 10` and `r = 21`, the set is `{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}`. The code outputs `3`, but the correct answer is `4` (as shown in the example).

### Challenging Test Input:
To expose these bugs, we need to create test cases that:
1. Test the condition for small ranges where `r - l < 3`.
2. Test the condition for small ranges where `r - l = 3`.
3. Test the condition for larger ranges where the correct number of operations is more than the buggy code's output.

### Test Input:
```plaintext
5
1 2
1 3
10 21
2 8
51 60
```

### Explanation:
1. **Test Case 1: `1 2`**
   - The set is `{1, 2}`. No operations can be performed.
   - Expected output: `0`
   - Buggy output: `1` (Incorrect)

2. **Test Case 2: `1 3`**
   - The set is `{1, 2, 3}`. One operation can be performed.
   - Expected output: `1`
   - Buggy output: `1` (Correct)

3. **Test Case 3: `10 21`**
   - The set is `{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}`. Four operations can be performed.
   - Expected output: `4`
   - Buggy output: `3` (Incorrect)

4. **Test Case 4: `2 8`**
   - The set is `{2, 3, 4, 5, 6, 7, 8}`. One operation can be performed.
   - Expected output: `1`
   - Buggy output: `1` (Correct)

5. **Test Case 5: `51 60`**
   - The set is `{51, 52, 53, 54, 55, 56, 57, 58, 59, 60}`. Two operations can be performed.
   - Expected output: `2`
   - Buggy output: `1` (Incorrect)

This test input will expose the bugs in the provided code by testing both small and large ranges, ensuring that the code's incorrect logic is clearly demonstrated.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

public class MaxOperations {

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int t = scanner.nextInt();
        List<Integer> results = new ArrayList<>();

        for (int i = 0; i < t; i++) {
            int l = scanner.nextInt();
            int r = scanner.nextInt();
            if (l == 1 && r == 1000) {
                results.add(250);
            } else {
                results.add(countOperations(l, r));
            }
        }

        for (int result : results) {
            System.out.println(result);
        }
        scanner.close();
    }

    public static int countOperations(int l, int r) {
        List<Integer> numbers = new ArrayList<>();
        for (int i = l; i <= r; i++) {
            numbers.add(i);
        }
        int n = numbers.size();
        int operations = 0;
        int x = n - 1;

        outerLoop: for (int i = x; i >= 0; i--) {
            if (numbers.get(i) == -1)
                continue;
            for (int j = i - 1; j >= 0; j--) {
                if (numbers.get(j) == -1)
                    continue;
                for (int k = j - 1; k >= 0; k--) {
                    if (numbers.get(k) == -1)
                        continue;

                    int a = numbers.get(i);
                    int b = numbers.get(j);
                    int c = numbers.get(k);

                    if (gcd(a, b) == 1 && gcd(b, c) == 1 && gcd(a, c) == 1) {
                        operations++;
                        numbers.set(i, -1);
                        numbers.set(j, -1);
                        numbers.set(k, -1);
                        x = i;
                        continue outerLoop;
                    }
                }
            }
        }
        return operations;
    }

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

**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 3
```

This test case is designed to expose potential issues in the buggy code. Here's the reasoning:

1. **Small Range**: The range `[1, 3]` is small and contains only three integers: 1, 2, and 3.
2. **GCD Condition**: All three integers (1, 2, and 3) are coprime with each other, meaning \(\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1\).
3. **Expected Output**: The expected output is 1, as you can perform one operation by selecting 1, 2, and 3, and then the set becomes empty.
4. **Buggy Code Behavior**: The buggy code should correctly identify this case and output 1. However, if there is any issue in the logic for identifying and removing elements, this small and straightforward case can help expose it.

By using this test case, you can verify if the code correctly handles the basic scenario and if it can properly manage the set and perform the operations as expected.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

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

        using ll = long long;
        using vi = vector<long long>;
        // using vi = vector<int>;
        using vpi = vector<pair<ll, ll>>;
        typedef long double ld;
        #define all(x) (x).begin(), (x).end()
        #define rall(a) (a).rbegin(), (a).rend()
        #define lb lower_bound
        #define ps(x, y) fixed << setprecision(y) << x
        #define int long long
        #define pb push_back
        #define ff first
        #define ss second
        #define endl "\n"

        template<class T> bool ckmin(T&a, T&b) { bool B = a > b; a = min(a,b); return B; }
        bool ckmin(int& a, int b){ return b < a ? a = b, true : false; }
        bool ckmax(int& a, int b){ return b > a ? a = b, true : false; }

        ll INF = 1e18;
        ll MOD = 1e9 + 7;
        // ll MOD = 998244353;
        // ll MOD = 1e18 + 7;

        int dx[] = {0, 0, -1, 1};
        int dy[] = {-1, 1, 0, 0}; // useful when dealing with points
        ll maxx(vi &a)
        {
            return (*max_element(a.begin(), a.end()));
        }

        ll minn(vi &a)
        {
            return (*min_element(a.begin(), a.end()));
        }

        bool is_set(ll number, ll x) {
            return (number >> x) & 1;
        }

        long long gcd(ll a, ll b)
        {
            if (b == 0)
                return a;
            return gcd(b, a % b);
        }

        // Function to return LCM of two numbers
        long long lcm(ll a, ll b)
        {
            return (a / gcd(a, b)) * b;
        }

        bool isPrime(ll n)
        {
            if (n <= 1)
            {
                return false;
            }
            for (ll i = 2; i * i <= n; i++)
            {
                if (n % i == 0)
                {
                    return false;
                }
            }
            return true;
        }

        // mod stuff starts here
        long long mod(long long x)
        {
            return ((x % MOD + MOD) % MOD);
        }
        long long add(long long a, long long b)
        {
            return mod(mod(a) + mod(b));
        }
        long long mul(long long a, long long b)
        {
            return mod(mod(a) * mod(b));
        }
        // this is mod stuff

        void input(vi &a)
        {
            int n = a.size();
            for (int i = 0; i < n; i++)
            {
                cin >> a[i];
            }
        }

        void print(vi &a)
        {
            for (auto x : a)
            {
                cout << x << " ";
            }
            cout << endl;
        }

        int stringToInt(string s)
        {
            stringstream geek(s);
            int x = 0;
            geek >> x;
            return x;
        }

        bool isPowerOfTwo(unsigned int n) {
            return n && !(n & (n - 1));
        }

        bool isPalindrome(string s)
        {
            int n = s.size();
            for (int i = 0; i < n; i++)
            {
                if (s[i] != s[n - i - 1])
                {
                    return false;
                }
            }
            return true;
        }
        string binaryTransformation(long long x)
        {
            if (x == 0)
                return "";
            else
            {
                string s = binaryTransformation(x / 2);
                s.push_back(char('0' + x % 2));
                return s;
            }
        }
        ll power(ll a, ll b, ll mod)
        {
            if (b == 0)
            {
                return 1;
            }
            ll ans = power(a, b / 2, mod);
            ans *= ans;
            ans %= mod;
            if (b % 2)
            {
                ans *= a;
            }
            return ans % mod;
        }
        double power(double a, int b)
        {
            if (b == 0)
            {
                return 1;
            }
            double t = power(a, b / 2);
            if (b & 1)
            {
                return t * t * a;
            }
            else
            {
                return t * t;
            }
        }
        int modularInverse(int number, int mod)
        {
            return power(number, mod - 2, mod);
        }
        // it should be 1LL and not 1
        // write more tests
        // google if stuck
        // take a walk if stuck

        bool isPerfectSquare(ll x)
        {
            if (x >= 0) {

                long long sr = sqrt(x);
                return (sr * sr == x);
            }

            return false;
        }

        void solve()
        {
            int l,r; cin>>l>>r;
            if(r-l+1<3)
            {
                cout<<0<<endl;
            }

            int ans=0;

            if(l%2==0) l++;
            for(auto i=l;i<=r;i+=4)
            {
                if(r-i+1>=3) ++ans;
            }

            cout<<ans<<endl;

        }

        int32_t main()
        {
            ios_base::sync_with_stdio(false);
            cin.tie(NULL);
            // Sieve();
            // precompute();
            // precalc();
            int t = 1;
            cin >> t;
            while(t--)
            {
                solve();
            }
            return 0;
        }

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

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

Think step by step.

Response:
```plaintext
1
1 4
```

### Explanation:
The provided test input is designed to expose the bug in the given buggy code. Here's why this test case is challenging:

1. **Range Selection**: The range `[1, 4]` contains exactly four integers: 1, 2, 3, and 4.
2. **GCD Condition**: The only valid triplet of integers from this range that satisfies the GCD condition is (1, 2, 3). The GCD of (1, 2), (2, 3), and (1, 3) is 1.
3. **Expected Output**: The correct output for this test case should be 1, as only one valid triplet can be removed.

### Analysis of the Buggy Code:
- The buggy code checks if the range length is less than 3 and immediately returns 0 if true.
- It then initializes `ans` to 0 and starts a loop from the first odd number in the range, incrementing by 4 each time.
- The loop condition `if(r-i+1>=3) ++ans;` is intended to count the number of valid triplets, but it fails to consider the correct logic for finding valid triplets.

### Why This Test Case Exposes the Bug:
- The loop in the buggy code starts from the first odd number and increments by 4. For the range `[1, 4]`, the loop will start at 1 and check if `4 - 1 + 1 >= 3`, which is true. However, it will only increment `ans` once and then exit the loop.
- The correct logic should consider all possible triplets in the range, not just those starting from the first odd number and incrementing by 4.

This test case will help identify the issue with the loop logic and the incorrect increment step 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: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

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

bool check_palindrome(vector<int> &temp,int start,int end){
    while(start < end){
        if(temp[start] != temp[end])return false;
        start++;
        end--;
    }
    return true;
}

bool bSearch_on_answer(int x,vector<int> &arr){
    vector<int> temp;
    for(int i = 0;i<arr.size();i++){
        temp.push_back(arr[i]%x);
    }

    return check_palindrome(temp,0,arr.size()-1) ? true : false;
}

int bSearch(int low,int high,vector<int> &arr,int target){
    while(low <= high){
        int mid = low + (high - low)/2;
        if(bSearch_on_answer(mid,arr)){
            low = mid+1;
        }
        else{
            high = mid - 1;
        }
    }

    return low-1;
}

int Prefix_Sum(int n,vector<int> &nums,vector<int> &P_sum){
    P_sum[0] = nums[0] < 0 ? 0 : nums[0];
    int maxi = nums[0];
    for(int i = 1;i<n;i++){
        if(nums[i] < 0){
            P_sum[i] = P_sum[i-1];
        }
        else{
            P_sum[i] = P_sum[i-1] + nums[i];
        }
        maxi = max(maxi,P_sum[i]);
    }
    return maxi;
}

void solve(){
    int l,r;
    cin>>l>>r;

    if(l == r){
        cout<<0<<"\n";
    }

    if(l % 2 == 0)l++;
    if(r % 2 == 0)r--;

    int x = (r-l)/2 + 1;
    cout<<x/2<<"\n";
}

signed main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    cout.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:
```plaintext
5
1 2
2 3
3 4
4 5
5 6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import math

def max_operations(l, r):
    n = r - l + 1
    if n == 3:
        return 1
    elif (n-3)/4+1 == int((n-3)/4+1):
        return (n-3)//4
    return (n-3)//4+1

# Đọc số lượng test case
t = int(input())

# Xử lý từng test case
results = []
for _ in range(t):
    l, r = map(int, input().split())
    operations = max_operations(l, r)
    results.append(operations)

# Xuất kết quả
for result in results:
    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:
```plaintext
5
1 3
4 6
10 12
1 10
1 100
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
for _ in range(int(input())):
    l, r = map(int, input().split())
    if l == 1 and r == 3: print(1)
    else:
        print(((r-l+1)//4))

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

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

Think step by step.

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

### Explanation:
1. **Test Case 1: 1 3**
   - The set is {1, 2, 3}. The only possible operation is to remove 1, 2, and 3. The expected output is 1.

2. **Test Case 2: 1 4**
   - The set is {1, 2, 3, 4}. The possible operations are:
     - Remove 1, 2, 3.
   - The expected output is 1.

3. **Test Case 3: 1 5**
   - The set is {1, 2, 3, 4, 5}. The possible operations are:
     - Remove 1, 2, 3.
     - Remove 4, 5, 6 (if 6 was in the set, but it isn't).
   - The expected output is 1.

4. **Test Case 4: 1 6**
   - The set is {1, 2, 3, 4, 5, 6}. The possible operations are:
     - Remove 1, 2, 3.
     - Remove 4, 5, 6.
   - The expected output is 2.

5. **Test Case 5: 1 7**
   - The set is {1, 2, 3, 4, 5, 6, 7}. The possible operations are:
     - Remove 1, 2, 3.
     - Remove 4, 5, 6.
   - The expected output is 2.

### Why These Test Cases?
- **Edge Cases:**
  - **1 3:** This is the smallest range that allows one operation.
  - **1 4:** This is the smallest range that does not allow more than one operation.
  - **1 5:** This is the smallest range that does not allow more than one operation, but it's not immediately obvious.
  - **1 6:** This is the smallest range that allows two operations.
  - **1 7:** This is another range that allows two operations but is slightly larger than 1 6.

- **Buggy Code Analysis:**
  - The buggy code uses the formula `((r-l+1)//4)`, which fails for ranges where the number of elements is not a multiple of 4. For example, for the range 1 to 5, the correct answer is 1, but the buggy code outputs 1, which is correct by coincidence. For the range 1 to 6, the correct answer is 2, but the buggy code outputs 1, which is incorrect. These test cases will expose the bug in the faulty solution.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Dora's Set
time_limit_ms: 1000
memory_limit_mb: 256
Description: Dora has a set $s$ containing integers. In the beginning, she will put all integers in $[l, r]$ into the set $s$. That is, an integer $x$ is initially contained in the set if and only if $l \leq x \leq r$. Then she allows you to perform the following operations:

  * Select three distinct integers $a$, $b$, and $c$ from the set $s$, such that $\gcd(a, b) = \gcd(b, c) = \gcd(a, c) = 1^\dagger$.    * Then, remove these three integers from the set $s$.

What is the maximum number of operations you can perform?

$^\dagger$Recall that $\gcd(x, y)$ means the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor) of integers $x$ and $y$.

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 500$) — the number of test cases. The description of the test cases follows.

The only line of each test case contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 1000$) — the range of integers in the initial set.

Output: For each test case, output a single integer — the maximum number of operations you can perform.

Examples:
input:
8
1 3
3 7
10 21
2 8
51 60
2 15
10 26
1 1000
output:
1
1
3
1
2
3
4
250

Note: In the first test case, you can choose $a = 1$, $b = 2$, $c = 3$ in the only operation, since $\gcd(1, 2) = \gcd(2, 3) = \gcd(1, 3) = 1$, and then there are no more integers in the set, so no more operations can be performed.

In the second test case, you can choose $a = 3$, $b = 5$, $c = 7$ in the only operation.

In the third test case, you can choose $a = 11$, $b = 19$, $c = 20$ in the first operation, $a = 13$, $b = 14$, $c = 15$ in the second operation, and $a = 10$, $b = 17$, $c = 21$ in the third operation. After the three operations, the set $s$ contains the following integers: $12$, $16$, $18$. It can be proven that it's impossible to perform more than $3$ operations.

Buggy Code:
import java.io.*;
import java.lang.reflect.Array;
import java.util.*;

public class CodeForces {
    public static void main(String[] args) throws IOException {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        StringBuilder result = new StringBuilder();
        FastReader fr = new FastReader();
        int t = fr.nextInt();
        while (t-- > 0) {
            int l = fr.nextInt();
            int r = fr.nextInt();

            // Calculate the number of integers in the range [l, r]
            int count = r - l + 1;
              int maxOperations=0;
            // The maximum number of operations is the number of groups of 3 that can be formed
             if(count<4) maxOperations = count / 3;
              else maxOperations = count/4;
            System.out.println(maxOperations);
        }

        }
    }

class FastReader {
    BufferedReader br;
    StringTokenizer st;

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

    String next() {
        while (st == null || !st.hasMoreElements()) {
            try {
                // Read the next line
                String line = br.readLine();

                // Check if the line is null
                if (line == null) {
                    return null;  // Handle end of input or unexpected input issues
                }

                // Initialize the StringTokenizer with the read line
                st = new StringTokenizer(line);

            } catch (IOException e) {
                e.printStackTrace();
            }
        }
        return st.nextToken();
    }

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

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

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

    String nextLine() {
        String str = "";
        try {
            if (st.hasMoreTokens()) {
                str = st.nextToken("\n");
            } else {
                str = br.readLine();
            }
        } catch (IOException e) {
            e.printStackTrace();
        }
        return str;
    }
}

//        // Method to calculate GCD using Euclidean algorithm
//        public static int gcd(int a, int b) {
//            if (b == 0) {
//                return a;
//            }
//            return gcd(b, a % b);
//        }
// Custom implementation of upper_bound to mimic C++ behavior
//    private static int upperBound(List<Integer> list, int key) {
//        int low = 0, high = list.size();
//        while (low < high) {
//            int mid = (low + high) / 2;
//            if (list.get(mid) <= key) {
//                low = mid + 1;
//            } else {
//                high = mid;
//            }
//        }
//        return low;
//    }
//static long pow(long x, long q) {
//    if (q == 0) {
//        return 1;
//    }
//    long i = pow(x, q / 2);
//    if (q % 2 == 0) {
//        return (i * i) % p;
//    }
//    return (i * i * x) % p;
//}

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

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

Think step by step.

Response:
To generate a test input that exposes the bug in the given buggy code, we need to focus on the logic that calculates the maximum number of operations. The buggy code incorrectly calculates the maximum number of operations as `count / 4` for `count >= 4`, which is incorrect. The correct calculation should be `count / 3` for any valid `count`.

### Steps to Generate the Test Input:
1. **Identify the Bug**: The bug is in the calculation of the maximum number of operations. The code uses `count / 4` for `count >= 4`, which is incorrect. It should be `count / 3`.
2. **Edge Cases**:
   - Small ranges where `count < 4` (e.g., `1 2`, `1 3`).
   - Ranges where `count` is exactly 4 (e.g., `1 4`).
   - Ranges where `count` is a multiple of 3 (e.g., `1 6`).
   - Ranges where `count` is not a multiple of 3 (e.g., `1 7`).

### Test Input:
```plaintext
4
1 2
1 3
1 4
1 6
```

### Explanation:
1. **Test Case 1: `1 2`**
   - `count = 2`
   - Correct output: `0` (since `2 / 3 = 0`)
   - Buggy output: `0` (correct, but this doesn't expose the bug)

2. **Test Case 2: `1 3`**
   - `count = 3`
   - Correct output: `1` (since `3 / 3 = 1`)
   - Buggy output: `1` (correct, but this doesn't expose the bug)

3. **Test Case 3: `1 4`**
   - `count = 4`
   - Correct output: `1` (since `4 / 3 = 1`)
   - Buggy output: `1` (incorrect, but this exposes the bug)

4. **Test Case 4: `1 6`**
   - `count = 6`
   - Correct output: `2` (since `6 / 3 = 2`)
   - Buggy output: `1` (incorrect, but this exposes the bug)

### Final Test Input:
```plaintext
4
1 2
1 3
1 4
1 6
```