**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
# Read the input
n = int(input().strip())

# Calculate the minimum number of planks needed
min_planks_needed = 2 * n

# Print the result
print(min_planks_needed)

# Wxa421jrJqGTBLCfM8H

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to consider the scenario where the code's assumption of needing `2 * n` planks is incorrect. The code assumes that each plank of 60 units can only be used to cover one bridge, which is not necessarily true. By cutting the planks efficiently, we can often use fewer planks.

### Step-by-Step Analysis:
1. **Understanding the Code's Assumption:**
   - The code assumes that each plank of 60 units can only cover one bridge, leading to `2 * n` planks being needed.
   - This is incorrect because a single plank can be cut to cover parts of multiple bridges.

2. **Finding an Edge Case:**
   - We need a test case where the efficient use of planks can significantly reduce the number of planks needed.
   - For example, if `n = 2`, we can cut one 60-unit plank into two 25-unit and one 10-unit plank, and another 60-unit plank into one 21-unit and one 39-unit plank. This way, we can cover all three bridges with just 2 planks.

3. **Constructing the Test Case:**
   - We need a small `n` that clearly shows the inefficiency of the provided code.
   - Let's use `n = 2` as it is small and easy to verify manually.

### Test Input:
```plaintext
2
```

### Explanation:
- For `n = 2`, the correct minimum number of planks needed is 2, not 4 as the code would output.
- This test case clearly exposes the bug in the provided code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.Scanner;

public class Main{
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int n = scanner.nextInt();

        // 2 * n is derived based on observation that combining planks optimally requires twice the base planks for minimal waste.
        int requiredPlanks = 2 * n;
        System.out.println(requiredPlanks);
    }
}

// Dl4FPoy6WSIMAkNYOL1

**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
12
```

**Explanation:**

The buggy code assumes that the minimum number of planks required is always `2 * n`. However, this is not always correct. The optimal solution involves cutting the planks in a way that minimizes waste.

For `n = 12`, let's analyze the actual minimum number of planks required:

- For the bridge with width 18: 12 planks of 18 units each require \(12 \times 18 = 216\) units.
- For the bridge with width 21: 12 planks of 21 units each require \(12 \times 21 = 252\) units.
- For the bridge with width 25: 12 planks of 25 units each require \(12 \times 25 = 300\) units.

Total required length:
\[ 216 + 252 + 300 = 768 \]

Since each standard plank is 60 units long, the minimum number of planks required is:
\[ \lceil \frac{768}{60} \rceil = \lceil 12.8 \rceil = 13 \]

The buggy code would output:
\[ 2 * 12 = 24 \]

This test case clearly shows that the buggy code is overestimating the number of planks required, thus exposing the bug.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.Scanner;

public class Main{
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int n = scanner.nextInt();
        int result = (int) Math.ceil(2 * (n / 3.0) + (n % 3));
        System.out.println(result);
    }
}

// p1I7wXhoUOTn3SHqkiW

**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
9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

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

int main()
{
    std::ios_base::sync_with_stdio(0);
    cout.tie(0);
    cin.tie(0);

    long long n;
    cin >> n;
    long long suma = n*(18+21+25);

    int x = (suma%60 == 0) ? suma/60 : (suma/60)+1;
    cout << x;

    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
40
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.Scanner;

public class BridgeRenovation {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        sc.close();

        int[] plankLengths = {18, 21, 25};

        int[] totalPlanksNeeded = {n, n, n};

        int result = calculateMinPlanks(plankLengths, totalPlanksNeeded);
        System.out.println(result);
    }

    private static int calculateMinPlanks(int[] plankLengths, int[] totalPlanksNeeded) {
        int maxPlankLength = 60;
        int totalPlanks = 0;

        while (totalPlanksNeeded[0] > 0 || totalPlanksNeeded[1] > 0 || totalPlanksNeeded[2] > 0) {
            int remainingLength = maxPlankLength;

            for (int i = 0; i < 3; i++) {
                while (totalPlanksNeeded[i] > 0 && remainingLength >= plankLengths[i]) {
                    remainingLength -= plankLengths[i];
                    totalPlanksNeeded[i]--;
                }
            }

            totalPlanks++;
        }

        return totalPlanks;
    }
}

**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 focus on edge cases where the algorithm might not efficiently use the available plank lengths. The code provided has a greedy approach where it tries to use the remaining length of the plank to fit the required planks. However, this approach might not always yield the optimal solution.

### Edge Case Analysis:
1. **High Demand for the Longest Plank:**
   - If the demand for the longest plank (25 units) is high, the algorithm might not efficiently use the remaining length of the plank for the shorter planks (18 and 21 units).

2. **Balanced Demand:**
   - If the demand is balanced across all three plank lengths, the algorithm might still not efficiently use the remaining length of the plank, especially when the remaining length is not enough to fit any of the required planks.

3. **Low Demand for the Shortest Plank:**
   - If the demand for the shortest plank (18 units) is low, the algorithm might not use the remaining length efficiently for the longer planks.

### Chosen Test Input:
To maximize the failure probability, we will choose a test input where the demand for the longest plank (25 units) is high, and the remaining lengths are not efficiently used.

```plaintext
5
```

### Explanation:
- For \( n = 5 \):
  - The total number of planks needed for each bridge is 5.
  - The algorithm will try to use the 60-unit planks to fit the 25-unit planks first, which might leave inefficiently small remaining lengths for the 18 and 21-unit planks.

This test input will challenge the algorithm to efficiently use the remaining lengths of the 60-unit planks, especially when the demand for the longest plank is high.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.*;
public class Planks {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        try {
        int n=sc.nextInt();
        if (n < 1 || n > 1000) {
            System.out.println("Invalid input: n must be between 1 and 1000.");
            return;
        }
        int totalLength = n * 18 + n * 21 + n * 25;
        int Planksneed = (totalLength + 59) / 60;
        System.out.println(Planksneed);
    } catch (Exception e) {
        System.out.println("Runtime error occured: " + e.getMessage());
    } finally {
        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
4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
#include<iostream>

using namespace std;

int main() {
	int n;
	cin >> n;
	int n2 = n;
	int cnt = 0;
	if (n % 3 == 0) {
		cnt += 2 * n / 3;
	}
	if (n % 3 == 1) {
		cnt += 2 * n / 3 +1;
	}
	if (n % 3 == 2) {
		cnt += 2 * n / 3 + 2;
		n2--;
	}
	if (n2 % 2 == 1) {
		cnt += n2 / 2 + 1;
	}
	else cnt += n2 / 2;
	cout << cnt;
	return 0;
}

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

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

Think step by step.

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

### Analysis of the Buggy Code
1. **Input Handling**: The code reads an integer `n` from the input.
2. **Main Logic**:
   - It calculates the number of planks needed based on the value of `n % 3`.
   - It then calculates the number of planks needed based on the value of `n2 % 2`, where `n2` is initially equal to `n`.

### Potential Issues
1. **Modulo Operations**:
   - The code uses `n % 3` to determine the number of planks needed. However, the logic for handling `n % 3` and `n2 % 2` seems to be flawed.
   - The code might not correctly handle cases where `n` is small or where the distribution of planks is not evenly divisible.

2. **Edge Cases**:
   - Small values of `n` (e.g., 1, 2, 3) might not be handled correctly.
   - Values of `n` that are multiples of 3 or 2 might also cause issues.

### Challenging Test Input
To expose the bug, we should test with values of `n` that are small and might cause the modulo operations to behave unexpectedly. Let's test with `n = 2`.

### Test Input
```plaintext
2
```

### Explanation
- For `n = 2`, the code should ideally calculate the number of planks required to cover the three bridges.
- The correct output should be `3` because:
  - For the first bridge: 2 planks of length 18 (1 plank of 60 can be split into 3 planks of 18).
  - For the second bridge: 2 planks of length 21 (1 plank of 60 can be split into 2 planks of 21 and 18).
  - For the third bridge: 2 planks of length 25 (2 planks of 60 can be split into 2 planks of 25 and 10 remaining).

The buggy code might not handle this correctly due to its flawed logic in the modulo operations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
n = int(input())
a = 0
from math import ceil
a+=ceil(n/2)

f = 0
if n%2!=0 and n!=1:
   b = (n-1)
f = (n-3)/6
p = ceil(n/2)
'''if n%2==0:
    f = ceil(n / 6)'''
print(ceil(a+f+p))

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

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

Think step by step.

Response:
```plaintext
5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import math
n = int(input())
a1 = math.ceil(n/2)
if n/2 != math.ceil(n/2):
    a2 = math.ceil((n-a1-1)/3)
    a3 = 0
    if (n-a1-1) % 3 == 1:
        a3 = 1
else:
    a2 = math.ceil((n-a1)/3)
    a3 = 0
    if (n-a1) % 3 == 1:
        a3 = 1
a3 += math.ceil(n/2)
print(a1+a2+a3)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases where the distribution of planks might not be handled correctly. The code seems to have issues with how it calculates the number of planks needed for each bridge, especially when dealing with remainders and rounding.

Let's consider the following scenarios:
1. **Small values of \( n \)**: These can help identify issues with basic calculations.
2. **Values of \( n \) that are multiples of 2 or 3**: These can help identify issues with ceiling and floor functions.
3. **Values of \( n \) that are not multiples of 2 or 3**: These can help identify issues with remainders and edge cases.

Given the nature of the problem, a small but challenging input is \( n = 7 \). This value is not a multiple of 2 or 3, and it should expose any issues with the distribution of planks.

Let's break down why \( n = 7 \) is a good test case:
- For the first bridge, \( n = 7 \) planks of length 18 are needed.
- For the second bridge, \( n = 7 \) planks of length 21 are needed.
- For the third bridge, \( n = 7 \) planks of length 25 are needed.

The total length required for each bridge is:
- First bridge: \( 7 \times 18 = 126 \) units
- Second bridge: \( 7 \times 21 = 147 \) units
- Third bridge: \( 7 \times 25 = 175 \) units

The total length required for all three bridges is \( 126 + 147 + 175 = 448 \) units.

Since each plank is 60 units long, the minimum number of planks needed is:
- \( \lceil \frac{448}{60} \rceil = 8 \) planks

Now, let's see if the buggy code handles this correctly.

```plaintext
7
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

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

#define int long long
typedef long long ll;
typedef vector<long long> vll;
typedef vector<int> vi;
typedef vector<char> vc;
typedef vector<bool> vb;
typedef vector<vector<int> > vvi;
typedef vector<vector<char> > vvc;
typedef pair<int, int> pi;
typedef pair<ll, ll> pll;
typedef stack<int> sti;
typedef queue<int> qi;
typedef vector<pair<long, long> > vpll;

#define endl "\n"
#define Ft front
#define Bk back
#define F first
#define S second
#define PB push_back
#define MP make_pair
#define rep(i,a,b) for (int i = a; i <= b; i++)

const ll MOD = 1e9 + 7;

#define fastio() ios_base::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL)
#define MOD 1000000007
#define MOD1 998244353
#define INF 1e18
#define nline "\n"
#define pb push_back
#define ppb pop_back
#define mp make_pair
#define ff first
#define ss second
#define PI 3.141592653589793238462
#define set_bits __builtin_popcountll
#define sz(x) ((int)(x).size())
#define all(x) (x).begin(), (x).end()

typedef long long ll;
typedef unsigned long long ull;
typedef long double lld;
// typedef tree<pair<int, int>, null_type, less<pair<int, int>>, rb_tree_tag, tree_order_statistics_node_update > pbds; // find_by_order, order_of_key

#ifndef ONLINE_JUDGE
#define debug(x) cerr << #x <<" "; _print(x); cerr << endl;
#else
#define debug(x)
#endif

void _print(ll t) {cerr << t;}
// void _print(int t) {cerr << t;}
void _print(string t) {cerr << t;}
void _print(char t) {cerr << t;}
void _print(lld t) {cerr << t;}
void _print(double t) {cerr << t;}
void _print(ull t) {cerr << t;}

template <class T, class V> void _print(pair <T, V> p);
template <class T> void _print(vector <T> v);
template <class T> void _print(set <T> v);
template <class T, class V> void _print(map <T, V> v);
template <class T> void _print(multiset <T> v);
template <class T, class V> void _print(pair <T, V> p) {cerr << "{"; _print(p.ff); cerr << ","; _print(p.ss); cerr << "}";}
template <class T> void _print(vector <T> v) {cerr << "[ "; for (T i : v) {_print(i); cerr << " ";} cerr << "]";}
template <class T> void _print(set <T> v) {cerr << "[ "; for (T i : v) {_print(i); cerr << " ";} cerr << "]";}
template <class T> void _print(multiset <T> v) {cerr << "[ "; for (T i : v) {_print(i); cerr << " ";} cerr << "]";}
template <class T, class V> void _print(map <T, V> v) {cerr << "[ "; for (auto i : v) {_print(i); cerr << " ";} cerr << "]";}

void solve() {
    int count = 0;
    int n;
    cin>>n;
    if(n & 1){
        cout<<n / 2 + 1 + ceil((double(n) / 3) * 2)<<endl;
    }
    else{
        cout<<n / 2 + ceil((double(n) / 3) * 2)<<endl;
    }
//    cout<<count<<endl;

}

signed main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

#ifndef ONLINE_JUDGE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    freopen("debug.txt", "w", stderr);
#endif

    int t = 1;
    // cin >> t;ṇ
    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 expose the bug in the provided buggy code, we need to focus on edge cases where the logic of the code might fail. The code is supposed to calculate the minimum number of planks of standard length (60 units) required to cover three bridges of different widths (18, 21, and 25 units) for a given number of planks \( n \).

### Step-by-Step Analysis:

1. **Understanding the Code Logic:**
   - The code uses a simple formula to calculate the number of planks needed based on the parity (odd or even) of \( n \).
   - For odd \( n \), it calculates \( n / 2 + 1 + \lceil (n / 3) * 2 \rceil \).
   - For even \( n \), it calculates \( n / 2 + \lceil (n / 3) * 2 \rceil \).

2. **Potential Issues:**
   - The code does not account for the optimal way to cut planks to minimize waste.
   - The formula might not always yield the correct result, especially for small values of \( n \).

3. **Edge Cases to Consider:**
   - **Small values of \( n \):** These can often expose issues in the logic.
   - **Specific values of \( n \) that are multiples or near multiples of 3 or 2.**

4. **Test Input:**
   - Let's try \( n = 2 \). This is a small value and should be easy to verify manually.

### Manual Verification:
- For \( n = 2 \):
  - We need 2 planks of 18 units, 2 planks of 21 units, and 2 planks of 25 units.
  - The total length required is \( 2 \times 18 + 2 \times 21 + 2 \times 25 = 36 + 42 + 50 = 128 \) units.
  - The standard planks are 60 units each.
  - We need at least 3 planks to cover 128 units (since \( 60 \times 2 = 120 \) and \( 60 \times 3 = 180 \)).

### Expected Output:
- The correct output should be 3.

### Buggy Code Output:
- The buggy code for \( n = 2 \):
  - \( n \) is even, so the formula is \( n / 2 + \lceil (n / 3) * 2 \rceil \).
  - \( n / 2 = 1 \).
  - \( \lceil (2 / 3) * 2 \rceil = \lceil 1.33 \rceil = 2 \).
  - Total = \( 1 + 2 = 3 \).

The output is correct for \( n = 2 \), so let's try another edge case.

### Test Input:
- Let's try \( n = 3 \).

### Manual Verification:
- For \( n = 3 \):
  - We need 3 planks of 18 units, 3 planks of 21 units, and 3 planks of 25 units.
  - The total length required is \( 3 \times 18 + 3 \times 21 + 3 \times 25 = 54 + 63 + 75 = 192 \) units.
  - The standard planks are 60 units each.
  - We need at least 4 planks to cover 192 units (since \( 60 \times 3 = 180 \) and \( 60 \times 4 = 240 \)).

### Expected Output:
- The correct output should be 4.

### Buggy Code Output:
- The buggy code for \( n = 3 \):
  - \( n \) is odd, so the formula is \( n / 2 + 1 + \lceil (n / 3) * 2 \rceil \).
  - \( n / 2 = 1.5 \) (integer division gives 1).
  - \( \lceil (3 / 3) * 2 \rceil = \lceil 2 \rceil = 2 \).
  - Total = \( 1 + 1 + 2 = 4 \).

The output is correct for \( n = 3 \), so let's try another edge case.

### Test Input:
- Let's try \( n = 4 \).

### Manual Verification:
- For \( n = 4 \):
  - We need 4 planks of 18 units, 4 planks of 21 units, and 4 planks of 25 units.
  - The total length required is \( 4 \times 18 + 4 \times 21 + 4 \times 25 = 72 + 84 + 100 = 256 \) units.
  - The standard planks are 60 units each.
  - We need at least 5 planks to cover 256 units (since \( 60 \times 4 = 240 \) and \( 60 \times 5 = 300 \)).

### Expected Output:
- The correct output should be 5.

### Buggy Code Output:
- The buggy code for \( n = 4 \):
  - \( n \) is even, so the formula is \( n / 2 + \lceil (n / 3) * 2 \rceil \).
  - \( n / 2 = 2 \).
  - \( \lceil (4 / 3) * 2 \rceil = \lceil 2.67 \rceil = 3 \).
  - Total = \( 2 + 3 = 5 \).

The output is correct for \( n = 4 \), so let's try another edge case.

### Test Input:
- Let's try \( n = 5 \).

### Manual Verification:
- For \( n = 5 \):
  - We need 5 planks of 18 units, 5 planks of 21 units, and 5 planks of 25 units.
  - The total length required is \( 5 \times 18 + 5 \times 21 + 5 \times 25 = 90 + 105 + 125 = 320 \) units.
  - The standard planks are 60 units each.
  - We need at least 6 planks to cover 320 units (since \( 60 \times 5 = 300 \) and \( 60 \times 6 = 360 \)).

### Expected Output:
- The correct output should be 6.

### Buggy Code Output:
- The buggy code for \( n = 5 \):
  - \( n \) is odd, so the formula is \( n / 2 + 1 + \lceil (n / 3) * 2 \rceil \).
  - \( n / 2 = 2.5 \) (integer division gives 2).
  - \( \lceil (5 / 3) * 2 \rceil = \lceil 3.33 \rceil = 4 \).
  - Total = \( 2 + 1 + 4 = 7 \).

The output is incorrect for \( n = 5 \). The correct output should be 6, but the buggy code outputs 7.

### Final Test Input:
```plaintext
5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

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

int n, k;
int a[10000], b[10000];
int c[10000], d[10000];

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

    cin >> n;

    if (n <= 3)
    {
        cout << n + 1 << endl;
        return 0;
    }

    if (n % 2 == 0)
    {
        int i = n / 2;
        cout << 2 * i + i / 3 + 1 << endl;
    }
    else
    {
        int ans = 0;
        ans += 2 * n / 3 + n / 2 + 2;
        if (2 * n % 3 < 2)
        {
            ans += n / 2 + 1;
        }
        else if (2 * n % 3 == 2)
        {
            ans += n / 2 + 2;
        }

        cout << ans << endl;
    }

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import math
import sys
input = sys.stdin.readline
n = int(input())
ans = math.ceil((2 * n) / 3) + math.ceil(n / 2)
print(ans)

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

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

Think step by step.

Response:
```plaintext
4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
# لا تَجلِس فارِغًا [ حرك لسانك ]
# اسْتَغفِر ، سَبّح ، هَلّل ، كَبّر ، حَولق

import sys
from random import randint, shuffle, choice
from math import gcd , sqrt, isqrt, perm, comb, factorial, log2, ceil, floor
from collections import Counter, defaultdict, deque
from functools import lru_cache, reduce, cmp_to_key
from itertools import accumulate, combinations, permutations , product , repeat, takewhile , starmap , dropwhile , cycle
from heapq import nsmallest, nlargest, heappushpop, heapify, heappop, heappush , _heapify_max
from bisect import bisect_left, bisect_right , insort_left , insort_right
#from operator import sub , mul,pow  , truediv , floordiv
input = lambda: sys.stdin.buffer.readline().decode().rstrip()
OneByOne = lambda: sys.stdin.read(1)
def lcm(a , b) :
    return a * b // gcd(a ,b)
def W(i) :
    return (i , str(i))

def MakeFile(s : str) :
    import sys
    sys.stdin = open(f'{s}.in', 'r')
    sys.stdout = open(f'{s}.out', 'w')

I = lambda: input()
II = lambda: int(input())
MII = lambda: map(int, input().split())
LI = lambda: list(input().split())
LII = lambda: list(map(int, input().split()))
GMI = lambda: map(lambda x: int(x) - 1, input().split())
LGMI = lambda: list(map(lambda x: int(x) - 1, input().split()))
yes = lambda: print('YES')
no = lambda: print('NO')
DIR4 = ((-1, 0), (0, 1), (1, 0), (0, -1))
DIR8 = ((-1, 0), (-1, 1), (0, 1), (1, 1), (1, 0), (1, -1), (0, -1), (-1, -1))
MOD = 10 ** 9 + 7
inf = float('inf')

#for _ in range(II()) :

n = II()

print(2 * (n + 1 ) // 2 + (n + 1 ) // 6)

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases and scenarios that maximize the failure probability. The buggy code calculates the number of planks required using a formula that might not always be optimal. Let's analyze the formula used in the code:

```python
print(2 * (n + 1 ) // 2 + (n + 1 ) // 6)
```

This formula seems to be an attempt to calculate the minimum number of planks needed, but it might not be accurate for all values of `n`. To find a test case that exposes the bug, we should look for values of `n` where the formula might not yield the correct result.

### Step-by-Step Analysis:
1. **Basic Case**: For small values of `n`, the formula might work correctly. For example, when `n = 1`, the formula gives `2`, which is correct.
2. **Edge Case**: For larger values of `n`, we need to check if the formula still holds. Specifically, we should check values of `n` where the formula might not distribute the planks optimally.

### Test Case Generation:
Let's consider `n = 5`. This is a small value that is not trivial and can help us see if the formula fails.

### Calculation:
For `n = 5`:
- The required planks for each bridge are:
  - 5 planks of length 18
  - 5 planks of length 21
  - 5 planks of length 25

We need to check if the formula `2 * (n + 1 ) // 2 + (n + 1 ) // 6` gives the correct result.

### Formula Calculation:
- `2 * (5 + 1) // 2 + (5 + 1) // 6`
- `2 * 6 // 2 + 6 // 6`
- `2 * 3 + 1`
- `6 + 1`
- `7`

### Optimal Solution:
- We need to check if 7 planks are indeed the minimum required. Let's manually calculate:
  - 1 plank can be cut into 18, 21, and 21 (18 + 21 + 21 = 60)
  - 1 plank can be cut into 25, 18, and 17 (25 + 18 + 17 = 60)
  - 1 plank can be cut into 25, 21, and 14 (25 + 21 + 14 = 60)
  - 1 plank can be cut into 18, 18, and 24 (18 + 18 + 24 = 60)
  - 1 plank can be cut into 25, 18, and 17 (25 + 18 + 17 = 60)
  - 1 plank can be cut into 25, 21, and 14 (25 + 21 + 14 = 60)
  - 1 plank can be cut into 21, 18, and 21 (21 + 18 + 21 = 60)

So, 7 planks are indeed required, which means the formula is correct for `n = 5`.

### Edge Case:
Let's try `n = 6` to see if the formula fails.

### Formula Calculation for `n = 6`:
- `2 * (6 + 1) // 2 + (6 + 1) // 6`
- `2 * 7 // 2 + 7 // 6`
- `2 * 3 + 1`
- `6 + 1`
- `7`

### Optimal Solution for `n = 6`:
- We need to check if 7 planks are indeed the minimum required. Let's manually calculate:
  - 1 plank can be cut into 18, 21, and 21 (18 + 21 + 21 = 60)
  - 1 plank can be cut into 25, 18, and 17 (25 + 18 + 17 = 60)
  - 1 plank can be cut into 25, 21, and 14 (25 + 21 + 14 = 60)
  - 1 plank can be cut into 18, 18, and 24 (18 + 18 + 24 = 60)
  - 1 plank can be cut into 25, 18, and 17 (25 + 18 + 17 = 60)
  - 1 plank can be cut into 25, 21, and 14 (25 + 21 + 14 = 60)
  - 1 plank can be cut into 21, 18, and 21 (21 + 18 + 21 = 60)
  - 1 plank can be cut into 25, 18, and 17 (25 + 18 + 17 = 60)

So, 8 planks are required, which means the formula fails for `n = 6`.

### Test Input:
```plaintext
6
```

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.Scanner;

public class codeforces_2038L {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int res = (2*n + 2)/3 + (n+1)/2;
        System.out.println(res);
    }
}

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

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

Think step by step.

Response:
To identify a test input that exposes the bug in the provided code, we need to analyze the logic and identify where it might fail. The given code calculates the number of planks required using the formula:

```java
int res = (2*n + 2)/3 + (n+1)/2;
```

This formula is intended to calculate the minimum number of planks needed to cover all three bridges. However, it does not correctly account for the optimal way to cut the planks to minimize the total number of planks used.

### Analysis:
1. **Understanding the Formula:**
   - `(2*n + 2)/3` is an attempt to calculate the number of planks needed for the first and second bridges.
   - `(n+1)/2` is an attempt to calculate the number of planks needed for the third bridge.

2. **Potential Issues:**
   - The formula might not correctly account for the optimal cutting of planks.
   - It might not handle edge cases where the number of planks needed for each bridge is small or specific.

### Edge Cases to Consider:
1. **Small Values of `n`:**
   - When `n` is small, the formula might not correctly calculate the minimum number of planks.
   - For example, when `n = 1`, the formula should return `2`, but we need to verify if it does so correctly.

2. **Specific Values of `n`:**
   - Values like `n = 2` or `n = 3` might also reveal issues in the formula.

### Test Input:
Let's test with `n = 4` to see if the formula fails to correctly calculate the minimum number of planks.

### Calculation:
- For `n = 4`:
  - The first bridge needs 4 planks of 18 units each.
  - The second bridge needs 4 planks of 21 units each.
  - The third bridge needs 4 planks of 25 units each.

  - **Optimal Cutting:**
    - One plank of 60 units can be cut into:
      - 1 plank of 25 units and 1 plank of 18 units (remaining 17 units).
      - Another plank of 60 units can be cut into:
        - 1 plank of 21 units and 1 plank of 39 units (remaining 0 units).
      - Another plank of 60 units can be cut into:
        - 1 plank of 25 units and 1 plank of 18 units (remaining 17 units).
      - Another plank of 60 units can be cut into:
        - 1 plank of 21 units and 1 plank of 39 units (remaining 0 units).

  - Total planks needed: 4 planks of 60 units.

### Expected Output:
- The expected output for `n = 4` should be `4`.

### Test Input:
```plaintext
4
```

This input should expose the bug in the provided code if the formula does not correctly calculate the minimum number of planks.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bridge Renovation
time_limit_ms: 2000
memory_limit_mb: 512
Description: Recently, Monocarp started working as a director of a park located near his house. The park is quite large, so it even has a small river splitting it into several zones. Several bridges are built across this river. Three of these bridges are especially old and need to be repaired.

All three bridges have the same length but differ in width. Their widths are $$$18$$$, $$$21$$$ and $$$25$$$ units, respectively.

During the park renovation process, Monocarp has to replace the old planks that served as the surface of the bridges with the new ones.

Planks are sold with a standard length of $$$60$$$ units. Monocarp already knows that he needs $$$n$$$ planks for each bridge. But since the bridges have different widths, he needs $$$n$$$ planks of length $$$18$$$ for the first bridge, $$$n$$$ planks of length $$$21$$$ for the second one, and $$$n$$$ planks of length $$$25$$$ for the last one.

Workers in charge of renovation have no problem with cutting planks into parts but refuse to join planks, since it creates weak spots and looks ugly.

Monocarp wants to buy as few planks as possible but struggles to calculate the required number of planks. Can you help him?

Input: The first and only line contains a single integer $$$n$$$ ($$$1 \le n \le 1000$$$) — the number of planks required for each of the three bridges.

Output: Print a single integer — the minimum number of planks of standard length ($$$60$$$ units) Monocarp needs to cover all three bridges if the planks can be cut into parts.

Examples:
input:
1
output:
2

input:
3
output:
4

input:
1000
output:
1167

Note: In the first example, it is possible to cut one plank of length $$$60$$$ into three planks with lengths $$$25$$$, $$$18$$$ and $$$17$$$, and cut another plank of length $$$60$$$ into two planks with lengths $$$39$$$ and $$$21$$$. That way, Monocarp will have all the required planks.

Buggy Code:
import java.util.Scanner;
public class Problem {
	public static void main(String[] args) {
		Scanner hi = new Scanner(System.in);
		int t = hi.nextInt();
		System.out.println(t+1 + t/6) ;
	}
}

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

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

Think step by step.

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

### Buggy Code Analysis
The buggy code reads an integer `t` and prints `t + 1 + t / 6`. This logic is clearly incorrect because it does not consider the different widths of the planks and how they can be optimally cut from the standard 60-unit planks.

### Correct Approach
The correct approach involves calculating the minimum number of 60-unit planks required to cover the total length of planks needed for all three bridges. The total length of planks needed for each bridge is `n * width` where the widths are 18, 21, and 25 units.

### Challenging Test Input
To expose the bug, we need a test input where the incorrect logic in the buggy code produces a significantly different result from the correct logic. Let's consider a small `n` that is not trivial and can highlight the issue.

#### Example Calculation
For `n = 4`:
- Total length of planks needed for the first bridge: `4 * 18 = 72`
- Total length of planks needed for the second bridge: `4 * 21 = 84`
- Total length of planks needed for the third bridge: `4 * 25 = 100`

Total length required: `72 + 84 + 100 = 256`

To find the minimum number of 60-unit planks:
- `256 / 60 = 4.2667` (which means we need 5 planks)

### Buggy Code Output
For `n = 4`:
- Buggy code output: `4 + 1 + 4 / 6 = 5` (which is incorrect)

### Correct Output
- Correct output: `5` (as calculated above)

### Test Input
```plaintext
4
```

This test input should clearly expose the bug in the buggy code because it will produce an incorrect result. The correct output should be 5, but the buggy code will also output 5, which is coincidentally correct for this specific input. However, the logic in the buggy code is flawed and will produce incorrect results for other inputs.

To ensure the bug is exposed, let's consider a slightly larger `n`:

#### Example Calculation for `n = 6`
- Total length of planks needed for the first bridge: `6 * 18 = 108`
- Total length of planks needed for the second bridge: `6 * 21 = 126`
- Total length of planks needed for the third bridge: `6 * 25 = 150`

Total length required: `108 + 126 + 150 = 384`

To find the minimum number of 60-unit planks:
- `384 / 60 = 6.4` (which means we need 7 planks)

### Buggy Code Output
For `n = 6`:
- Buggy code output: `6 + 1 + 6 / 6 = 8` (which is incorrect)

### Correct Output
- Correct output: `7` (as calculated above)

### Test Input
```plaintext
6
```

This test input should clearly expose the bug in the buggy code because the correct output is 7, but the buggy code will output 8.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Rectangle Filling
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$3\cdot 10^5$$$.

Output: For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Examples:
input:
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
output:
NO
YES
YES
YES
YES
NO
YES
NO

Note: In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

Buggy Code:
#include<iostream>
#include<algorithm>
#include<map>
#include<cstring>
const int N=10000;

using namespace std;

map<int,int> mp;
typedef long long ll;
char a[N][N];
int main(){
	int t;
	cin>>t;
	while(t--){
		int flag=0;
		int n,m;
		cin>>n>>m;
		for(int i=1;i<=n;i++){
			for(int j=1;j<=m;j++){
				cin>>a[i][j];
			}
		}

		if(n==m&&n==1){
			cout<<"YES"<<endl;
			continue;
		}
//		for(int i=1;i<=n;i++){
//			for(int j=1;j<=m;j++){
//				cout<<a[i][j];
//			}
//			cout<<endl;
//		}
//		cout<<endl;
		if(a[1][1]==a[n][m]||a[1][m]==a[n][1]){
			cout<<"YES"<<endl;
			continue;
		}
		if(n==1&&a[1][1]!=a[1][m]){
			cout<<"NO"<<endl;
			continue;
		}
		int ans1=0,ans2=0;
		for(int i=1;i<m;i++){
			if(a[1][i]!=a[1][i+1]){ //两种都出现过
				ans1=1;

			}
		}
		for(int i=1;i<m;i++){
			if(a[n][i]!=a[n][i+1]){
				ans2=1;
			}
		}
		if(ans1==1||ans2==1){
//			cout<<ans1<<endl;
			cout<<"YES"<<endl;

		}
		else{
			cout<<"NO"<<endl;
		}
	}
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases and scenarios that the code might handle incorrectly. Here are the key points to consider:

1. **Single Row or Column Cases:**
   - The code has special handling for single row or column cases, so we should test these scenarios.
   - For a single row or column, the code checks if all elements are the same. If they are not, it outputs "NO".

2. **Symmetrical and Asymmetrical Grids:**
   - The code checks if the corners of the grid are the same. If they are, it outputs "YES".
   - We should test grids where the corners are the same but the rest of the grid is not uniform.

3. **Mixed Color Patterns:**
   - The code checks if there are any changes in the first and last rows. If there are, it outputs "YES".
   - We should test grids where the first and last rows have mixed colors but the grid cannot be made uniform.

4. **Special Cases:**
   - The code has a special case for a single cell grid, which is always "YES".
   - We should test a single cell grid to ensure it works correctly.

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

```plaintext
4
1 5
WBBWB
2 2
BB
BB
3 3
WBB
BBW
BBB
4 4
WBBW
BBWB
WBBW
BBWB
```

### Explanation:
1. **Single Row (1 5):**
   - The first row has mixed colors: "WBBWB". Since not all elements are the same, the code should output "NO".

2. **2x2 Grid (2 2):**
   - The grid is already uniform: "BB" and "BB". The code should output "YES".

3. **3x3 Grid (3 3):**
   - The corners are 'W' and 'B', but the rest of the grid is not uniform. The code should output "NO" because it cannot be made uniform with the given operations.

4. **4x4 Grid (4 4):**
   - The corners are 'W' and 'B', and the rest of the grid has a pattern that cannot be made uniform. The code should output "NO" because it cannot be made uniform with the given operations.

This test input should help expose the bug in the given buggy code by covering various edge cases and mixed color patterns.

4
1 5
WBBWB
2 2
BB
BB
3 3
WBB
BBW
BBB
4 4
WBBW
BBWB
WBBW
BBWB


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Rectangle Filling
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$3\cdot 10^5$$$.

Output: For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Examples:
input:
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
output:
NO
YES
YES
YES
YES
NO
YES
NO

Note: In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

Buggy Code:
#include<iostream>
using namespace std;
int a[501][501];
int main(){
	int t;
	cin>>t;
	while(t--)
	{
		int n,m;
		cin>>n>>m;
		for(int i=1;i<=n;i++)
		{
			for(int j=1;j<=m;j++)
			{

			char c;
			cin>>c;
			if(c=='W') a[i][j]=1;
			else a[i][j]=0;
		    }
		}

		////
		if(n==1&&a[n][1]==a[n][m])
		{
			//cout<<1<<endl;
			cout<<"YES"<<endl;
			continue;
		}
		else if(n==1&&a[n][1]!=a[n][m]){
			//cout<<1<<endl;
			cout<<"NO"<<endl;
			continue;
		}

		if(m==1&&a[1][m]==a[n][m])
		{
			//cout<<2<<endl;
			cout<<"YES"<<endl;
			continue;
		}
		else if(m==1&&a[1][m]!=a[n][m]){
			//cout<<2<<endl;
			cout<<"NO"<<endl;
			continue;
		}
		////
		if(a[1][1]==a[n][m])
		{
			//cout<<3<<endl;
			cout<<"YES"<<endl;
			continue;
		}
		if(a[1][m]==a[n][1])
		{
		//	cout<<3<<endl;
			cout<<"YES"<<endl;
			continue;
		}
		////
		bool flag=false;
		if(a[1][1]==a[1][m])
		{

			for(int i=1;i<=m;i++)
			{
				if(a[n][i]==a[1][1])
				{
					flag=true;
					break;
				}
			}
		}
		 if(a[1][1]==a[n][1])
		{
			for(int i=1;i<=n;i++)
			{
				if(a[i][m]==a[1][1])
				{
					flag=true;
					break;
				}
			}
		}
	     if(a[n][1]==a[n][m])
		{
			for(int i=1;i<=m;i++)
			{
				if(a[1][i]==a[n][1])
				{
					flag=true;
					break;
				}
			}
		}
	    if(a[1][m]==a[n][m])
		{
			for(int i=1;i<=n;i++)
			{
				if(a[1][i]==a[1][m])
				{
					flag=true;
					break;
				}
			}
		}
		if(flag)
		cout<<"YES"<<endl;
		else cout<<"NO"<<endl;

	}
	return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
2 2
WB
BW
3 3
WBB
BWB
BBW
4 4
WBWB
BWBW
WBWB
BWBW
```

3
2 2
WB
BW
3 3
WBB
BWB
BBW
4 4
WBWB
BWBW
WBWB
BWBW


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Rectangle Filling
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$3\cdot 10^5$$$.

Output: For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Examples:
input:
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
output:
NO
YES
YES
YES
YES
NO
YES
NO

Note: In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

Buggy Code:
//#include<bits/stdc++.h>
#define _CRT_SECURE_NO_WARNINGS
#include<iostream>
#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<algorithm>
//#include<map>
//#include<stack>
using namespace std;
int a[520][520] = { 0 };
int main()
{
    int t;
    cin >> t;
    while (t--) {
        int n, m;
        int num = 0;
        memset(a, 0, sizeof a);
        scanf("%d%d\n", &n, &m);
        for (int i = 1; i <= n; i++) {
            string x;
            cin >> x;
            for (int j = 1; j <= m; j++) {
                if (x[j-1] == 'W')
                    a[i][j] = 0;
                else a[i][j] = 1;
            }
        }
        if (n == 1 && m == 1) {
            num = 1;
        }
        else if (n == 1 || m == 1)
        {
            if (a[1][1] == a[n][m])num = 1;
            else num = 0;
        }
        else {
            if (a[1][1] == a[n][m] || a[1][m] == a[n][1])
                num = 1;
            else {
                if (a[1][1] == 0) {
                    for (int i = 1; i <= n; i++) {
                        for (int j = 1; j <= m; j++) {
                            if (a[i][1] == 1 && a[1][j] == 1) {
                                num = 1;
                                break;
                            }
                        }
                    }
                }
                else if(a[1][1] == 1) {
                    for (int i = 1; i <= n; i++) {
                        for (int j = 1; j <= m; j++) {
                            if (a[i][1] == 0 && a[1][j] == 0) {
                                num = 1;
                                break;
                            }
                        }
                    }
                }
                if (num == 0) {
                    if (a[1][m] == 0) {
                        for (int i = 1; i <= n; i++) {
                            for (int j = 1; j <= m; j++) {
                                if (a[i][m] == 1 && a[1][j] == 1) {
                                    num = 1;
                                    break;
                                }
                            }
                        }
                    }
                    else if (a[1][m] == 1) {
                        for (int i = 1; i <= n; i++) {
                            for (int j = 1; j <= m; j++) {
                                if (a[i][m] == 0 && a[1][j] == 0) {
                                    num = 1;
                                    break;
                                }
                            }
                        }
                    }
                }
            }
        }
        if (num == 1)cout << "YES" << endl;
        else cout << "NO" << 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
2 2
WB
BW
```

### Explanation:
This test case is designed to expose the bug in the buggy code. The grid is a 2x2 grid with alternating colors. The correct answer should be "NO" because it is impossible to make all squares the same color with the given operations. However, the buggy code might incorrectly output "YES" due to its flawed logic.

The key points to consider:
- The grid is small (2x2), making it easy to visualize and understand.
- The grid has alternating colors, which should make it impossible to unify all squares to a single color.
- The buggy code might incorrectly check the corners or edges, leading to an incorrect result.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Rectangle Filling
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$3\cdot 10^5$$$.

Output: For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Examples:
input:
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
output:
NO
YES
YES
YES
YES
NO
YES
NO

Note: In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

Buggy Code:
# def solve(r, n, m):
#     if n == 1:
#         for i in range(1, m):
#             if r[0][i] != r[0][i - 1]:
#                 return False

#         return True

#     if m == 1:
#         for i in range(1, n):
#             if r[i][0] != r[i - 1][0]:
#                 return False

#         return True

#     possible = False
#     if r[0][0] == 'B':
#         last_row_black = False
#         for i in range(m):
#             if r[n - 1][i] == 'B':
#                 last_row_black = True
#                 break
#         last_column_black = False
#         for i in range(n):
#             if r[i][m - 1] == 'B':
#                 last_column_black = True
#                 break

#         possible |= (last_row_black and last_column_black)

#     if r[0][0] == 'W':
#         last_row_white = False
#         for i in range(m):
#             if r[n - 1][i] == 'W':
#                 last_row_white = True
#                 break
#         last_column_white = False
#         for i in range(n):
#             if r[i][m - 1] == 'W':
#                 last_column_white = True
#                 break

#         possible |= (last_row_white and last_column_white)

#     if r[n - 1][m - 1] == 'B':
#         first_row_black = False
#         for i in range(m):
#             if r[0][i] == 'B':
#                 first_row_black = True
#                 break
#         first_column_black = False
#         for i in range(n):
#             if r[i][0] == 'B':
#                 first_column_black = True
#                 break

#         possible |= (first_row_black and first_column_black)

#     if r[n - 1][m - 1] == 'W':
#         first_row_white = False
#         for i in range(m):
#             if r[0][i] == 'W':
#                 first_row_white = True
#                 break
#         first_column_white = False
#         for i in range(n):
#             if r[i][0] == 'W':
#                 first_column_white = True
#                 break

#         possible |= (first_row_white and first_column_white)

#     if r[0][m - 1] == 'B':
#         last_row_black = False
#         for i in range(m):
#             if r[n - 1][i] == 'B':
#                 last_row_black = True
#                 break
#         first_column_black = False
#         for i in range(n):
#             if r[i][0] == 'B':
#                 first_column_black = True
#                 break

#         possible |= (last_row_black and first_column_black)

#     if r[0][m - 1] == 'W':
#         last_row_white = False
#         for i in range(m):
#             if r[n - 1][i] == 'W':
#                 last_row_white = True
#                 break
#         first_column_white = False
#         for i in range(n):
#             if r[i][0] == 'W':
#                 first_column_white = True
#                 break

#         possible |= (last_row_white and first_column_white)

#     if r[n - 1][0] == 'B':
#         first_row_black = False
#         for i in range(m):
#             if r[0][i] == 'B':
#                 first_row_black = True
#                 break
#         last_column_black = False
#         for i in range(n):
#             if r[i][m - 1] == 'B':
#                 last_column_black = True
#                 break

#         possible |= (first_row_black and last_column_black)

#     if r[n - 1][0] == 'W':
#         first_row_white = False
#         for i in range(m):
#             if r[0][i] == 'W':
#                 first_row_white = True
#                 break
#         last_column_white = False
#         for i in range(n):
#             if r[i][m - 1] == 'W':
#                 last_column_white = True
#                 break

#         possible |= (first_row_white and last_column_white)

#     return possible

def can_be_toggled(r, i, j, n, m):
    if r[i][j] == 'B':
        need = 'W'
    else:
        need = 'B'

    if i == 0 and j == 0:
        first_row_has = False
        for k in range(0, m):
            if r[0][k] == need:
                first_row_has = True
                break
        first_col_has = False
        for k in range(0, n):
            if r[k][0] == need:
                first_col_has = True

        return first_row_has and first_col_has

    if i == 0 and j == m - 1:
        first_row_has = False
        for k in range(0, m):
            if r[0][k] == need:
                first_row_has = True
                break
        last_col_has = False
        for k in range(0, n):
            if r[k][m-1] == need:
                last_col_has = True

        return first_row_has and last_col_has

    if i == n - 1 and j == 0:
        last_row_has = False
        for k in range(0, m):
            if r[n-1][k] == need:
                last_row_has = True
                break
        first_col_has = False
        for k in range(0, n):
            if r[k][0] == need:
                first_col_has = True

        return last_row_has and first_col_has

    if i == n - 1 and j == m - 1:
        last_row_has = False
        for k in range(0, m):
            if r[n-1][k] == need:
                last_row_has = True
                break
        last_col_has = False
        for k in range(0, n):
            if r[k][m-1] == need:
                last_col_has = True

        return last_row_has and last_col_has

    return False

def solve(r, n, m):
    if n == 1:
        for i in range(1, m):
            if r[0][i] != r[0][i - 1]:
                return False

        return True

    if m == 1:
        for i in range(1, n):
            if r[i][0] != r[i - 1][0]:
                return False

        return True

    if r[0][0] == r[n-1][m-1]:
        return True

    if can_be_toggled(r, 0, 0, n, m) or can_be_toggled(r, n-1, m-1, n, m) \
            or can_be_toggled(r, 0, m-1, n, m) or can_be_toggled(r, n-1, 0, n, m):
        return True

    return False

t = int(input())
for _ in range(t):
    n, m = map(int, input().split())
    r = []

    for i in range(n):
        r.append(input())

    if solve(r, n, m):
        print("YES")
    else:
        print("NO")

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

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

Think step by step.

Response:
```plaintext
3
2 2
BW
WB
2 2
WB
BW
1 5
WBBWB
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Rectangle Filling
time_limit_ms: 1000
memory_limit_mb: 256
Description: There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

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

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

It is guaranteed that the sum of $$$n\cdot m$$$ over all test cases does not exceed $$$3\cdot 10^5$$$.

Output: For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

Examples:
input:
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
output:
NO
YES
YES
YES
YES
NO
YES
NO

Note: In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

Buggy Code:
def main():
    t = int(input())

    for _ in range(t):
        n, m = map(int, input().split())
        square = []
        for i in range(n):
            square.append(input())

        r1 = m>=2 and n>=2 and (square[0][0]==square[n-1][m-2] and square[0][0]==square[n-2][m-1])
        r2 = m>=2 and n>=2 and (square[n-1][m-1]==square[1][0] and square[n-1][m-1]==square[0][1])
        r3 = m>=2 and n>=2 and (square[n-1][0]==square[0][m-2] and square[n-1][0]==square[1][m-1])
        r4 = n>=2 and m>=2 and (square[0][m-1]==square[n-2][0] and square[0][m-1]==square[n-1][1])
        if square[0][0] == square[n-1][m-1] or square[0][m-1] == square[n-1][0] or r1 or r2 or r3 or r4:
            print("YES")
        else:
            print("NO")

# runner.py
import sys
from io import StringIO

def run_with_file(input_file='./practice/A.txt'):
    # Save the original stdin
    original_stdin = sys.stdin

    # Replace sys.stdin with file contents
    with open(input_file, 'r') as f:

        sys.stdin = StringIO(f.read())

    try:
        # Run the main solution
        main()
    finally:
        # Restore original stdin
        sys.stdin = original_stdin

if __name__ == "__main__":
    if len(sys.argv) > 1 and sys.argv[1] == '--test':
        run_with_file()
    else:
        main()  # Regular execution for submission

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

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

Think step by step.

Response:
```plaintext
1
2 2
WB
BW
```

This test case is designed to expose the bug in the given buggy code. The grid is a 2x2 matrix with alternating colors. The buggy code checks specific corner cases but fails to consider the possibility that the entire grid can be made uniform by selecting appropriate squares. In this case, it is possible to make all squares the same color by selecting the top-left and bottom-right squares (both 'W') and then the top-right and bottom-left squares (both 'B'). However, the buggy code will incorrectly output "NO" because it does not handle this scenario correctly.