**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
def find_remaining_restaurant(visited):
    # Create a set of visited restaurants
    visited_set = set(visited)

    # Iterate through all possible restaurant numbers
    for i in range(1, 6):
        if i not in visited_set:
            return i

# Example usage:
visited_restaurants = [1, 3, 2, 5]
remaining_restaurant = find_remaining_restaurant(visited_restaurants)
print(remaining_restaurant)

# FrZsSNcjL46COMK5Dwq

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
public class Main {
    public static void main(String[] args) {
        int numeroDiaLunes = 1;
        int numeroDiaMartes = 3;
        int numeroDiaMiercoles = 2;
        int numeroDiaJueves = 5;

        /*calculamos la suma de los restaurantes visitados de lunes a jueves.
           Dependiendo de la suma (14, 13, 12, o 11) determinamos qué restaurante
           falta y lo imprimimos como resultado */

        int sumaTotal = numeroDiaLunes + numeroDiaMartes + numeroDiaMiercoles + numeroDiaJueves;

        if (sumaTotal == 14) {
            System.out.println(1);
        } else if (sumaTotal == 13) {
            System.out.println(2);
        } else if (sumaTotal == 12) {
            System.out.println(3);
        } else if (sumaTotal == 11) {
            System.out.println(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
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
import java.util.*;

public class Boring {
    public static int MEX(int[] stats, int[] nums){
        HashMap<Integer,Integer> freq = new HashMap<>();
        int idx =0;
        for (Integer i:nums){
            if (freq.containsKey(i)){
                freq.put(i,freq.get(i)+1);
            }
            else{
                freq.put(i,1);
            }
        }
        while(freq.containsKey(idx)){
            if (freq.get(idx)>1){
                if (freq.containsKey(idx+stats[1])){
                    freq.put(idx+stats[1],freq.get(idx+stats[1])+(freq.get(idx)-1));
                }
                else{
                    freq.put(idx+stats[1],freq.get(idx)-1);
                }
            }
            idx++;
        }
        return idx;
    }
    public static int[] beautifulArray(int[] stats){
        int[] beau = new int[3];
        beau[0] = stats[1];
        beau[1] = stats[1];
        beau[2] = 3*stats[0]-2*stats[1];
        return beau;
    }
    public static int Bowling(int[] stats){
        int total = stats[0]+stats[1];
        boolean over = false;
        for (int i = 1; i<=stats[0]+stats[1]; i++){
            total-=i;
            if (total<0){
                return i-1;
            }
            else if (total == 0){
                return i;
            }
        }
        return 0;
    }
    public static String DIY(int[] stats, int points){
        HashMap<Integer,Integer> freq = new HashMap<>();
        Arrays.sort(stats);
        int count = 0;
        int[] use = new int[4];
        for (Integer i:stats){
            if (freq.containsKey(i)){
                freq.put(i,freq.get(i)+1);
            }
            else{
                freq.put(i,1);
            }
        }
        for(int i = 0; i<points;i++){
            int a = stats[i];
            if (freq.get(a)>=2){
                freq.put(a,freq.get(a)-2);
                use[count] = a;
                count++;
                i--;
            }
            if (count == 2){
                break;
            }
        }
        for (int i = points-1;i>=0;i--){
            int a = stats[i];
            if (freq.get(a)>=2){
                freq.put(a,freq.get(a)-2);
                use[count] = a;
                count++;
                i--;
            }
            if (count == 4){
                break;
            }
        }
        if (count != 4){
            return "NO";
        }
        else{
            System.out.println("YES");
            return use[0]+" "+use[1]+" "+use[0]+" "+use[2]+ " "+use[3]+" "+use[1]+" "+use[3]+" "+use[2];
        }
    }
    public static long actionfigure(int stats, String open){
        int[] figures = new int[stats];
        for(int i = 1; i<=stats; i++){
            figures[i-1] = i;
        }
        int r = stats-1;
        int l = 0;
        int ridx = stats-1;
        long price = 0;
        int a = r;
        while(r>l){
            if ('1' == (open.charAt(r))){
                if (a>r){
                    a = r;
                }
                while(a>=0){
                    if ('0' == open.charAt(a)&&figures[a]!=0){
                        price+=figures[a];
                        figures[a] = 0;
                        figures[r] = 0;
                        ridx = r-1;
                        break;
                    }
                    a--;
                }
            }
            r--;
        }
        while (ridx>l){
            if ('1' == open.charAt(ridx)){
                for(int i = l; i<ridx; i++){
                    if (figures[i] != 0){
                        price+=figures[i];
                        figures[i] = 0;
                        figures[ridx] = 0;
                        l++;
                        break;
                    }
                }
            }
            ridx--;
        }
        for(int i =0; i<stats; i++){
            price+=figures[i];
        }
        return price;
    }
    public static int alternatingStrings(int len, String s){
        int min = len+1;
        if (len%2==0){
            return altStringEven(len, s);
        }
        else{
            for(int i = 0; i<len; i++){
                StringBuilder newS = new StringBuilder(s);
                int cont = altStringEven(len-1, ""+newS.deleteCharAt(i));
                if (cont<min){
                    min = cont;
                }
            }
            return min+1;
        }
    }
    public static int altStringEven(int len, String s){
        HashMap<Character,Integer> even = new HashMap<>();
        HashMap<Character,Integer> odd = new HashMap<>();
        int Emax = 0;
        int Omax = 0;
        for (int i = 0; i<len; i++){
            if (i%2 == 1){
                if (odd.containsKey(s.charAt(i))){
                    odd.put(s.charAt(i),odd.get(s.charAt(i))+1);
                }
                else{
                    odd.put(s.charAt(i),1);
                }
            }
            else{
                if (even.containsKey(s.charAt(i))){
                    even.put(s.charAt(i),even.get(s.charAt(i))+1);
                }
                else{
                    even.put(s.charAt(i),1);
                }
            }
        }
        for (char i:even.keySet()){
            if (even.get(i)>Emax){
                Emax = even.get(i);
            }
        }
        for(char i:odd.keySet()){
            if (odd.get(i)>Omax){
                Omax = odd.get(i);
            }
        }
        return len-(Omax+Emax);
    }
    public static String IntrestingString(String a, String b){
        HashMap<String,Integer> memo = new HashMap<>();
        int min = a.length()+b.length();
        String minS = "";
        for(int i = 0; i<a.length(); i++){
            for(int n = 0; n<b.length(); n++){
                String check = a.substring(0,i+1)+b.substring(n,b.length());
                if (memo.containsKey(check)){
                    memo.put(check,memo.get(check)+1);
                    if (check.length()<min){
                        min = check.length();
                        minS = check;
                    }
                }
                else{
                    memo.put(check,1);
                }
            }
        }
        return minS;
    }
    public static long StarGazer(long len, long boring, long left, long right) {
        long points = len+1;
        int layer = 1;
        long sum = 0;
        while(len>=boring){
            if (len%2 == 1){
                sum+=layer;
            }
            layer*=2;
            len = (long) Math.floor(len/2);
        }
        return points*sum/2;
    }

    public static int BentoBox(int[] stats){
        int[] freq = new int[5];
        Arrays.fill(freq,1);
        for(int i = 0; i<4;i++){
            freq[i] = 0;
        }
        for(int i = 0; i<5; i++){
            if (freq[i] == 1){
                return i;
            }
        }
        return 0;
    }

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int[] stats  = new int[4];
        for(int i = 0; i<4; i++){
            stats[i] = scanner.nextInt();
        }
        System.out.println(BentoBox(stats));
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define rep(i,a,b)for(int i=a;i<=b;i++)
int main(){

  vector<int>a(5);
  rep(i,1,4)cin>>a[i];
  sort(a.begin(),a.end());
  rep(i,1,4)if(a[i]!=i){cout<<i<<endl;break;}

}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
#include<iostream>
#include<string>
#include<map>
#include<vector>
#include<unordered_map>
#include <algorithm>
using namespace std;
void solve(){
    int n,a,b;
    cin>>n>>a>>b;
    string s;
    cin>>s;
    int x=0;
    int y=0;
    for(int i=0;i<s.size();i++){
        if(s[i]=='N')y++;
        else if(s[i] =='E')x++;
        else if(s[i] == 'S')y--;
        else if(s[i] == 'W')x--;
    }
    cout<<"X: "<<x<<"Y: "<<y<<endl;
    if(a/x == b/y)cout<<"YES"<<endl;
    else cout<<"NO"<<endl;
}
int main(){
    // int t;
    // cin>>t;
    // while(t--){
    //     solve();
    // }
    vector<int>v;
    map<int,bool>mp;
    for(int i=0;i<4;i++){
        int a;
        cin>>a;
        mp[a]=true;
    }
    for(int i=1;i<5;i++){
        if(!mp[i])cout<<i<<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 focus on edge cases or scenarios that maximize the failure probability. The buggy code is a mix of two different problems, but the main issue we need to address is the correct identification of the missing restaurant number.

### Analysis of the Buggy Code:
1. The code reads four integers representing the restaurants visited from Monday to Thursday.
2. It uses a `map` to track which restaurants have been visited.
3. It iterates through the numbers 1 to 5 and prints the one that has not been visited.

### Potential Bugs:
- The code does not handle the case where the input numbers are not distinct.
- The code does not handle the case where the input numbers are out of the valid range (1 to 5).

### Challenging Test Input:
To expose the bug, we need to ensure that the input numbers are distinct and within the valid range. However, we can still create a test input that maximizes the failure probability by ensuring the input is as complex as possible within the constraints.

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

### Explanation:
- This input is valid and follows the constraints.
- The numbers 1, 2, 3, and 4 are all distinct and within the range 1 to 5.
- The missing restaurant number should be 5.

### Expected Output:
```plaintext
5
```

This test input should correctly identify the missing restaurant number as 5, and if the code fails to produce this output, it will expose the bug.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
L=list(map(int,input().split()))
L1=[1,2,3,4,5]
i=0
while(i<len(L)):
    if L1[i] in L :
        i+=1
    else :
        print(L1[i])
        break

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
a=list(map(int,input().split()))
a.sort()

b=[]
for i in range(0,len(a)):
   if i+1!=a[i]:
      b.append(i)
print((b[0])+1)

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
import java.util.*;
public class alpha{
    public static void main(String args[]){
        Scanner sc=new Scanner(System.in);
        int a[]=new int[4];
        for(int i=0;i<4;i++){
            a[i]=sc.nextInt();
        }
        Arrays.sort(a);
        if(a[0]!=1){
                System.out.println(1);
            }
            else{
        for(int i=0;i<3;i++){
            if(a[i]!=(a[i+1]-1))
            {
                System.out.println(a[i]+1);
            }
        }
            }
        sc.close();
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
import java.util.Arrays;
import java.util.Scanner;

public class BentoBoxAdventure {
  public static void main(String[] args){
    Scanner scanner = new Scanner(System.in);
    String[] arr = scanner.nextLine().split(" ");
    int[] intArr = new int[arr.length];
    for(int i = 0; i < arr.length; i++){
      intArr[i] = Integer.parseInt(arr[i]);
    }
    Arrays.sort(intArr);
    int[] newint = new int[] {1,2,3,4,5};
    for(int i = 0; i < intArr.length; i++){
      if (intArr[i] != newint[i]){
        System.out.println(newint[i]);
        break;
      }
    }
  }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
#include<bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;
template<class T> using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;
template<class T> using ordered_multiset = tree<T, null_type, less_equal<T>, rb_tree_tag, tree_order_statistics_node_update>;
#define all(x) (x).begin(), (x).end()
#define pb push_back
#define pob pop_back
#define F first
#define S second
#define nl '\n'
#define lb lower_bound
#define ub upper_bound
#define read(a)			\
	for(auto &i : a)	\
		cin>>i;
#define ot(a)			\
	for(auto &i : a)    \
		cout<<i<<" ";   \
	cout << "\n";
typedef long long ll;
#define int ll
typedef vector<int> vi;
typedef pair<int,int> pii;
typedef vector<pair<int,int>> vpll;
const int mod = 1000000007;
const int N = 200001;
#define inf (int)LLONG_MAX

void solve(){
	vi v(4);
	for(int i=0; i<4; i++){
		cin >> v[i];
	}
	sort(all(v));
	for(int i=0; i<4; i++){
		if(v[i]!=i+1){
			cout << i+1 << nl;
			return;
		}
	}

	// cerr<<"worked in "<<1000*((double)clock())/(double)CLOCKS_PER_SEC<<"ms\n";
}

int32_t main(){
    ios_base::sync_with_stdio(false);
    cin.tie(0);

    // int testCases;
    // cin >> testCases;
    // while(testCases--)
    	solve();
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider the logic of the code and identify potential edge cases where it might fail. The code sorts the input array and then checks if each element is equal to its index plus one. If it finds an element that does not match, it outputs the index plus one as the missing restaurant.

### Analysis of the Code:
1. **Sorting the Input**: The code sorts the input array.
2. **Checking for Missing Element**: It checks if each element in the sorted array is equal to its index plus one.

### Potential Bugs:
- **Edge Case 1**: The input contains the numbers 1, 2, 3, and 5. The code will sort the array to [1, 2, 3, 5]. It will then check if each element is equal to its index plus one. Since 5 is not equal to 4, it will output 4, which is correct.
- **Edge Case 2**: The input contains the numbers 1, 2, 4, and 5. The code will sort the array to [1, 2, 4, 5]. It will then check if each element is equal to its index plus one. Since 4 is not equal to 3, it will output 3, which is correct.
- **Edge Case 3**: The input contains the numbers 1, 3, 4, and 5. The code will sort the array to [1, 3, 4, 5]. It will then check if each element is equal to its index plus one. Since 3 is not equal to 2, it will output 2, which is correct.
- **Edge Case 4**: The input contains the numbers 2, 3, 4, and 5. The code will sort the array to [2, 3, 4, 5]. It will then check if each element is equal to its index plus one. Since 2 is not equal to 1, it will output 1, which is correct.

### Challenging Test Input:
The code will fail if the input is such that the missing number is the last one in the sequence. For example, if the input is 1, 2, 3, 4, the code will sort the array to [1, 2, 3, 4] and then check if each element is equal to its index plus one. Since all elements match, it will not output anything, which is incorrect.

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

This input will expose the bug in the code because the missing number is 5, but the code will not output anything.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int main()
{
    int a,b,c,d;
    cin >> a >> b >> c >> d;
    vector <int> v;
    v.push_back(a);
    v.push_back(b);
    v.push_back(c);
    v.push_back(d);
    sort(v.begin(),v.end());
    for(int i=0; i<4; i++){
        if(v[i]!=i+1){
            cout << i+1 << "\n";
            break;
        }
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider the logic of the code and identify scenarios where it might fail. The code sorts the input numbers and then checks for the first missing number in the sorted sequence. The bug lies in the assumption that the missing number will always be the first one that doesn't match its position in the sorted list.

Let's break down the steps:

1. **Understanding the Code Logic:**
   - The code reads four integers and stores them in a vector.
   - It sorts the vector.
   - It then checks each element to see if it matches its position (i.e., `v[i] == i + 1`).
   - The first position where this condition fails is assumed to be the missing number.

2. **Identifying the Bug:**
   - The code assumes that the missing number will always be the first one that doesn't match its position in the sorted list.
   - This assumption fails if the missing number is the largest number (5) because the sorted list will always be in the order of the numbers that are present, and the code will not correctly identify the missing number.

3. **Creating the Test Input:**
   - To expose this bug, we need a test case where the missing number is 5.
   - The input should be a permutation of the numbers 1, 2, 3, and 4.

Given these considerations, the following test input will expose the bug:

```plaintext
1 2 3 4
```

This input will cause the code to output `1` instead of `5`, which is incorrect. The correct output should be `5` because the missing restaurant number is 5.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
l = list(map(int,input().split()))
for i in range(1,len(l)+1):
    if i not in l:
        print(i)
        break

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
import java.util.*;
public class remaining{
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int arr[] = new int[4];
        for(int i =0;i<4;i++){
             arr[i]= sc.nextInt();

        }
        Arrays.sort(arr);
        for(int j =0;j<4;j++){
            if(arr[j]!=(j+1)){
                System.out.println(j+1);
                break;

            }
        }

    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
import java.util.HashMap;
import java.util.HashSet;
import java.util.Scanner;

public class CF2041 {
    public static void main(String[] args) {
        Scanner scan = new Scanner(System.in);
        HashSet<Integer> penis = new HashSet<Integer>();

        for (int i = 0; i < 4; i++) {
            penis.add(scan.nextInt());
        }

        int[] penis2 = {1, 2, 3, 4, 5};

        for (int i = 0; i < 4; i++) {
            if (!penis.contains(penis2[i])) {
                System.out.println(penis2[i]);
            }
        }
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Bento Box Adventure
time_limit_ms: 1000
memory_limit_mb: 1024
Description: Image generated by ChatGPT 4o.

Boxed meals in Taiwan are very common, offering convenient and affordable nutrition-balanced choices for students and office workers. These meals typically include a variety of vegetables, protein, and rice, providing a well-rounded diet. With numerous options available at local self-service restaurants, they are a popular choice for those looking for a quick, healthy lunch.

There are five Taiwanese self-service restaurants numbered from 1 to 5. You plan to visit a different restaurant each day from Monday to Friday to pack a boxed meal for lunch. You've already visited one restaurant from Monday to Thursday, each time visiting a different one. Now, it's Friday, and you want to visit the last remaining restaurant that you haven't been to yet this week.

Write a program that takes as input the four restaurants you've already visited (one for each day from Monday to Thursday) and outputs the restaurant you should visit on Friday.

Input: A single line of input containing four integers $$$a, b, c, d$$$, each between 1 and 5 (inclusive), representing the restaurant numbers you visited from Monday to Thursday, in order.

- $$$1\le a, b, c, d\le 5$$$
- All four numbers will be different.

Output: Output the restaurant number you should visit on Friday.

Examples:
input:
1 3 2 5
output:
4

input:
2 5 4 3
output:
1

Note:

Buggy Code:
/******************************************************************************

                            Online Java Compiler.
                Code, Compile, Run and Debug java program online.
Write your code in this editor and press "Run" button to execute it.

*******************************************************************************/
import java.util.*;
public class Main
{
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		HashSet<Integer> set = new HashSet<>();
		for(int i = 0; i < 4; i++){
		    int a = sc.nextInt();
		    set.add(a);
		}
		int o = 1;
		for(int i : set){
		    if(!set.contains(o)){
		        System.out.println(o);
		        break;
		    }
		    o++;
		}
	}
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
#include<stdio.h>

int main()
{
	int t;
	scanf("%d", &t);
	long long x,y,k;
	while(t--){
		int sum=0;
		scanf("%lld %lld %lld",&x,&y,&k);
		if(x>y){
			if(x%k==0)
				sum=2*(x/k)-1;
			else
				sum=2*(x/k)+1;
		}
		else{
			if(y%k==0)
				sum=2*y/k;
			else
				sum=2*(y/k+1);
		}
		printf("%d\n",sum);
	}
	return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
1 1 1
10 10 10
0 0 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
#include<iostream>
#include<string>
using namespace std;
int main()
{
	int t, i;
	long long x, y, k;
	cin >> t;
	for (i = 0; i < t; i++)
	{
		cin >> x >> y >> k;
		long long x1 = (x + k - 1) / k;
		long long y1 = (y + k - 1) / k;
		if (x1 >= y1&&x1!=0)
			cout << 2 * x1 - 1 << endl;
		else
			cout << 2 * y1 << endl;
	}
}

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

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

Think step by step.

Response:
```plaintext
3
0 0 1
1 1 1
1000000000 1000000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
#include<stdio.h>
int main()
{
	int t;
	scanf("%d",&t);
	while(t--){
		long long x,y,k;
		long long sum=0;
		scanf("%lld %lld %lld",&x,&y,&k);
		long long a,b;
		a=(x-1)/k+1;
		if(x==0)
			a=0;
		b=(y-1)/k+1;
		if(y==0)
			b=0;
		if(x>y)
			sum=2*a-1;
		else if(x==y)
			sum=a+b;
		else
			sum=2*b;
		printf("%lld\n",sum);
	}
	return 0;
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
t = int(input())
for _ in range(t):
    x,y,k = map(int,input().split())
    if y>=x:
        if y%k==0:
            print(int(y/k*2))
        else:
            print((int(y/k)+1)*2)
    else:
        if x%k==0:
            print(int(x/k)*2-1)
        else:
            print(((int(x/k)+1)*2)-1)

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

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

Think step by step.

Response:
```plaintext
3
0 0 1
1 0 1
0 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import math

t = int(input())

vec = []
for _ in range(t):
    x, y, k = map(int, input().split())
    vec.append((x, y, k))

for x, y, k in vec:
    stepX = math.ceil(x / k)
    stepY = math.ceil(y / k)
    flag = 0
    if stepX == y or stepY == x:
        flag = 1
    res = 2 * max(stepX, stepY)
    print(res - flag)

**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
0 0 1
1 1 1
10 10 10
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
#strategy
#divide the maximum of a or b by d
#ceil it
#add the same amount to it

import math

t = int(input())
r = []

for i in range(t):
    x, y, k = map(int, input().split())
    minimum_moves = 0

    if y/k >= x/k:
        minimum_moves = 2 * math.ceil(y/k)
    elif x/k > y/k:
        minimum_moves = 2 * math.ceil(x/k) - 1

    r.append(minimum_moves)

for i in range(len(r)):
    print(r[i])

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

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

Think step by step.

Response:
```plaintext
3
0 0 1
1 1 1
1000000000 1000000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import java.util.Scanner;

public class Testing3 {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int t = sc.nextInt();
		while(t>0){
			int x = sc.nextInt();
			int y = sc.nextInt();
			int k = sc.nextInt();
			if(x>y) System.out.println((int)(Math.ceil(x/(k + 0.0d))) * 2 - 1);
			else System.out.println((int)(Math.ceil(y/(k + 0.0d))) * 2);

			t--;
		}
	}
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import java.util.*;
import java.io.*;
//(n(n-1))/2 combinations to select 2 items from n nC2(n!/(n-2)!2!)
// count number of odds between start and end, (end+1)/2-(start)/2
public class codeforces{
    public static void main(String[] args) {
        FastReader sc = new FastReader();
        int t = sc.nextInt();
        for(int i=0;i<t;i++){
             int x = sc.nextInt();
                     int y= sc.nextInt();
                             int z = sc.nextInt();
                             int sp = 0;
                             while(x!=0 || y!=0){
                                 if(x>=z){
                                     x-=z;
                                     sp++;
                                 }
                                 else if(x>0 && x<z){
                                     x-=x;
                                     sp++;
                                 }
                                 else if(x==0 && y>0){
                                     sp++;
                                 }
                                 if(x==0 && y==0){
                                     break;
                                 }
                                 if(y>=z){
                                     y-=z;
                                     sp++;
                                 }
                                 else if(y>0 && y<z){
                                     y-=y;
                                     sp++;
                                 }
                                 else if(y==0 && x>0){
                                     sp++;
                                 }
                                 if(x==0 && y==0){
                                     break;
                                 }
                             }

            System.out.println(sp);

        }
        //System.out.println();
    }

    public static class FastReader {
        BufferedReader br;
        StringTokenizer st;

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

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

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

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

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

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

Think step by step.

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

1. **Edge cases where `x` or `y` is zero**: The code should handle these cases correctly.
2. **Cases where `x` and `y` are both non-zero but less than `k`**: The code should be able to handle these cases efficiently.
3. **Cases where `x` and `y` are both large and greater than `k`**: The code should be able to handle these cases without getting stuck in an infinite loop or taking too long.

Let's create a test input that includes these scenarios:

1. **Case 1**: `x = 0`, `y = 0`, `k = 1` (Both coordinates are zero)
2. **Case 2**: `x = 1`, `y = 0`, `k = 2` (One coordinate is zero)
3. **Case 3**: `x = 1`, `y = 1`, `k = 2` (Both coordinates are non-zero but less than `k`)
4. **Case 4**: `x = 10`, `y = 10`, `k = 3` (Both coordinates are large and greater than `k`)

Here is the test input:

```plaintext
4
0 0 1
1 0 2
1 1 2
10 10 3
```

This test input will help in identifying any issues in the code, especially in handling edge cases and large values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

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

int main()
{
       ios_base::sync_with_stdio(false);
       cin.tie(NULL);
       cout.tie(NULL);

       int t; cin>>t;
       while(t--)
       {

          long long x,y,k; cin >> x>>y >>k;
          long long res;

          if(x>y)
          {
                    if(x%k==0)
                    res=((x/k)*2)-1;
                    else
                            res= (( (x/k)+1)*2)-1;

          }
          else if(y>=x)
          {
                  if(y%k==0)
                              res=(y/k)*2;
                  else
                    res=((y/k)+1)*2;
          }

          cout<<res<<endl;

       }

           return 0;

}

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

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

Think step by step.

Response:
```plaintext
1
5 5 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
/*
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠘⠣⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢋⣩⣥⣬⣍⣭⣤⣬⣍⡈⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡔⣦⠱⡀
⠀⠀⠀⠀⠀⠀⢠⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠰⡀
⠀⠀⠀⠀⠀⠀⡆⣼⢿⣿⣿⣿⣿⠏⠀⠈⣿⣿⣿⡟⠀⠈⢻⣿⣿⣿⣿⡿⡇⢃
⠀⠀⠀⠀⠀⠀⡇⠁⣿⣿⣿⡟⠁⠠⣀⠀⡼⠿⠿⡇⠀⡠⠀⠉⢻⣿⣿⡇⡁⢸
⠀⠀⠀⠀⠀⠀⣧⡇⢻⢿⣿⣧⡀⠀⠀⡰⠁⣿⡇⡘⣄⠀⠀⢀⣾⣿⡿⡇⡗⡌
⠀⠀⠀⠀⠀⠀⠁⢡⠘⠈⠀⢲⣦⣭⣋⡅⠀⠿⡇⠈⢨⣙⣭⣴⠖⠈⠁⢁⠃⠁
⠀⠀⠀⠀⠀⠀⡀⠔⢥⣶⣶⣦⣀⠈⠻⣷⡐⠀⠀⢆⣿⠋⠀⣠⣶⣶⣶⡜⠤⡀
⠀⠀⠀⠀⢠⢊⡔⠒⠢⡈⠻⣿⣿⣂⠀⣿⠁⠀⠀⠸⡇⠀⢸⣿⣿⠋⠁⠐⠒⢮⢢
⠀⠀⠀⠀⢸⢸⡀⠀⠀⠈⢂⠈⠉⢀⢀⠘⠦⢀⡀⠔⠁⡸⠀⠙⠁⠀⠀⠀⠀⢸⢇⠂
⠀⠀⠀⠀⠈⢎⢷⡀⠀⠀⢸⠀⢄⠠⠃⠈⠁⠒⠒⠉⠁⠑⢔⠀⠸⠀⠀⠀⣠⠏⠜
⠀⠀⠀⠀⠀⠀⠑⠍⣓⣒⡡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢕⣒⡛⠅⠊
*/

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define endl "\n"
signed main()
{
	ios_base::sync_with_stdio(false);
	cin.tie(NULL);
	cout.tie(NULL);

	int t;
	cin >> t;

	while (t--)
	{
		ll x, y, k;
		cin >> x >> y >> k;

		ll ans = 0;

		ll first = (x + k - 1) / k;
		ll second = (y + k - 1) / k;

		if (second > first)
		{
			ans += first * 2;

			ans = ans + (second - first) * 2;
		}
		else if (second == first)
		{
			ans += (second * 2) - 1;
		}
		else
		{
			if (x % y == 0 && y % k == 0)
			{
				ans += (second * 2) - 1;
				ans = ans + (first - second) * 2;
			}
			else
			{
				ans += second * 2;
				ans = ans + (first - second) * 2;
			}
		}
		cout << ans << endl;
	}
}

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

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

Think step by step.

Response:
```plaintext
3
0 0 1
1 1 1
1000000000 1000000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import math
for _ in range(int(input())):
    x, y, k = map(int, input().split())
    xstep = math.ceil(x / k)
    ystep = math.ceil(y / k)
    if x == 0 and y == 0:
        print(0)
    elif xstep >= ystep:
        print(2 * xstep - 1)
    else:
        print(2 * ystep)

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

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

Think step by step.

Response:
```plaintext
3
1 1 1
10 10 10
1000000000 1000000000 1000000000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
t = int(input())
for _ in range(t):
    x, y, k = map(int, input().split())
    xtime , ytime = x//k , y//k
    xless, yless = x%k , y%k
    if xtime > ytime:
        if xless == 0:
            print(2*xtime-1)
        else:
            print(2*xtime+1)
    elif xtime < ytime:
        if yless == 0:
            print(2*ytime)
        else:
            print(2*ytime+2)
    else:
        if xless == 0 and yless == 0:
            print(2*xtime)
        elif xless != 0 and yless == 0:
            print(2*xtime+1)
        else:
            print(2*xtime+2)

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

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

Think step by step.

Response:
```plaintext
3
1 1 1
0 0 1
1000000000 1000000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import java.util.*;
public class LegendFrog {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t=sc.nextInt();
        while(t-->0)
        {
            int x=sc.nextInt();
            int y=sc.nextInt();
            int d=sc.nextInt();

            int move=0,curx=0,cury=0;
            boolean dir=true;

            while(curx!=x || cury!=y)
            {
                if(dir)
                {
                    int dis=Math.abs(x-curx);

                    if(dis>=d)
                    {
                        curx += (x>curx)?d:-d;
                    }
                    else
                    {
                        curx=x;
                    }
                }
                else
                {
                    int dis = Math.abs(y-cury);

                    if(dis>=d)
                    {
                        cury +=(y>cury)?d:-d;
                    }
                    else
                    {
                        cury=y;
                    }
                }
                move++;
                dir=!dir;
            }
            System.out.println(move);
        }
    }
}

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

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

Think step by step.

Response:
```plaintext
3
1 1 1
1000000000 1000000000 1
0 0 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
import java.util.*;
import java.io.*;
public class akarsh
{
    public static boolean even(int a)//even odd check
    {
        if((a&1)==0)
        return true;
        return false;
    }
    public static void printarray(int[] a)//printing arrray
    {
        for(int k : a)
        System.out.print(a+" ");
        System.out.println();
    }
    public static int[] prefix(int[] a)//prefix sum
    {
        int n = a.length;
        int[] ans = new int[n];
        int sum = 0;
        for(int i=0;i<n;i++)
        {
            sum += a[i];
            ans[i] = sum;
        }
        return ans;
    }
    public static void steve()//steve for prime
    {
        int k = 100000+1;
        boolean[] prime = new boolean[k];
        Arrays.fill(prime,true);
        prime[0] = false;
        prime[1] = false;
        for(int i=2;i*i<k;i++)
        {
            if(prime[i]==true)
            {
                for(int j=i*i;j<k;j+=i)
                prime[j] = false;
            }
        }
    }
    public static void main(String args[])//main functions
    {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while((t--)>0)
        {
            int x = sc.nextInt();
            int y = sc.nextInt();
            int k = sc.nextInt();
            int ans = 0;
            boolean u = true, m = true;
            while(m)
            {
                if(u)
                {
                    x = x-Math.min(x,k);
                    u = false;
                }
                else
                {
                    y = y-Math.min(y,k);
                    u = true;
                }
                if(x==0&&y==0)
                m = false;
                ans++;
            }
            System.out.println(ans);
        }
    }
    public static boolean checkprime(int N)
    {
        int i;
        if(N==1)
        return false;
        if((N&1)==0&&N!=2)
        return false ;
        else if(N%3==0&&N!=3)
        return false ;
        else if(N%11==0&&N!=11)
        return false ;
        else if(N%13==0&&N!=13)
        return false ;
        else if(N%17==0&&N!=17)
        return false ;
        else
        {
            for(i=3;i<=Math.sqrt(N);i+=2)
            {
                if(N%i==0)
                return false ;
            }
        }
        return true ;
    }
    public static boolean checksquare(int n)
    {
        int sqrt = (int)(Math.sqrt(n));
        if((sqrt*sqrt)==n)
        return true;
        return false;
    }
    public static boolean checkfactor(int n,int x)
    {
        for(int i=2;i<=n;i++)
        {
            if(n%i==0&&i<x)
            return false;
        }
        return true;
    }
}
class pair//for pairs
{
    int a, b;
    pair(int a,int b)
    {
        this.a = a;
        this.b = b;
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#define int long long
void solve()
{
	int x,y,k;
	cin>>x>>y>>k;
	if(x%k==0&&y%k==0)
	cout<<max(x/k,y/k)*2-1<<endl;
	else if(x%k==0&&y%k!=0)
	cout<<max(x/k,y/k+1)*2<<endl;
	else if(x%k!=0&&y%k==0)
	cout<<max(x/k+1,y/k)*2<<endl;
	else
	cout<<max(x/k+1,y/k+1)*2<<endl;
}
signed main()
{
	int t;
	cin>>t;
	while(t--)
	{
		solve();
	}
	return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given code, we need to consider edge cases and scenarios that might not be handled correctly by the code. Here are some steps to think through:

1. **Edge Cases for x and y:**
   - When `x` or `y` is 0.
   - When `x` and `y` are very close to each other.
   - When `x` and `y` are multiples of `k` but not exactly divisible.
   - When `x` and `y` are just slightly more than multiples of `k`.

2. **Edge Cases for k:**
   - When `k` is 1.
   - When `k` is very large, close to the upper limit.

3. **Combination of Edge Cases:**
   - When `x` and `y` are both 0.
   - When `x` and `y` are both very large and `k` is small.
   - When `x` and `y` are both very small and `k` is large.

Given these considerations, a challenging test input could be:

```plaintext
4
0 0 1
1 1 1
1000000000 1000000000 1
1 1 1000000000
```

Explanation:
1. **Test Case 1: (0, 0, 1)**
   - Freya is already at the target, so the number of jumps should be 0.
   - The code might not handle this correctly if it doesn't check for the initial position.

2. **Test Case 2: (1, 1, 1)**
   - Both `x` and `y` are 1, and `k` is 1.
   - The optimal solution is 2 jumps: (0,0) -> (1,0) -> (1,1).
   - The code should handle this correctly but is a good check for basic functionality.

3. **Test Case 3: (1000000000, 1000000000, 1)**
   - Both `x` and `y` are very large, and `k` is 1.
   - The optimal solution is 2,000,000,000 jumps, which is the maximum number of jumps possible.
   - This tests the code's ability to handle large inputs.

4. **Test Case 4: (1, 1, 1000000000)**
   - Both `x` and `y` are 1, but `k` is very large.
   - The optimal solution is 2 jumps: (0,0) -> (1,0) -> (1,1).
   - This tests the code's ability to handle large `k` values.

These test cases cover a range of edge cases and should help expose any bugs in the given code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: The Legend of Freya the Frog
time_limit_ms: 2000
memory_limit_mb: 256
Description: Freya the Frog is traveling on the 2D coordinate plane. She is currently at point $$$(0,0)$$$ and wants to go to point $$$(x,y)$$$. In one move, she chooses an integer $$$d$$$ such that $$$0 \leq d \leq k$$$ and jumps $$$d$$$ spots forward in the direction she is facing.

Initially, she is facing the positive $$$x$$$ direction. After every move, she will alternate between facing the positive $$$x$$$ direction and the positive $$$y$$$ direction (i.e., she will face the positive $$$y$$$ direction on her second move, the positive $$$x$$$ direction on her third move, and so on).

What is the minimum amount of moves she must perform to land on point $$$(x,y)$$$?

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

Each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$0 \leq x, y \leq 10^9, 1 \leq k \leq 10^9$$$).

Output: For each test case, output the number of jumps Freya needs to make on a new line.

Examples:
input:
3
9 11 3
0 10 8
1000000 100000 10
output:
8
4
199999

Note: In the first sample, one optimal set of moves is if Freya jumps in the following way: ($$$0,0$$$) $$$\rightarrow$$$ ($$$2,0$$$) $$$\rightarrow$$$ ($$$2,2$$$) $$$\rightarrow$$$ ($$$3,2$$$) $$$\rightarrow$$$ ($$$3,5$$$) $$$\rightarrow$$$ ($$$6,5$$$) $$$\rightarrow$$$ ($$$6,8$$$) $$$\rightarrow$$$ ($$$9,8$$$) $$$\rightarrow$$$ ($$$9,11$$$). This takes 8 jumps.

Buggy Code:
/*Problem: Virtual_Round_971 C	Author: pkr1012
Url: */

# include<bits/stdc++.h>

using namespace std;

typedef long long ll;

void solve(){
    ll x, y, k;
    cin >> x >> y >> k;
    ll ret = 0;
    if (x > y){
        if (x > k)
            ret = 2 * (x / k) - 1;
        else ret = 2;
    }else if (y > x){
        ret = 2 * (y / k + 1);
    }else{
        if (x > k)
            ret = 2 * (x / k);
        else ret = 2;
    }
    cout << ret << endl;
}

int main(){
    // Online Judge
    #ifndef ONLINE_JUDGE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    #endif
    ll T;
    cin >> T;
    while(T--){
        solve();
    }

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
10 10 10
1 1 1
1000000000 1000000000 1000000000
```