**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Black Circles
time_limit_ms: 2000
memory_limit_mb: 256
Description: There are $n$ circles on a two-dimensional plane. The $i$-th circle is centered at $(x_i,y_i)$. Initially, all circles have a radius of $0$.

The circles' radii increase at a rate of $1$ unit per second.

You are currently at $(x_s,y_s)$; your goal is to reach $(x_t,y_t)$ without touching the circumference of any circle (including the moment you reach $(x_t,y_t)$). You can move in any direction you want. However, your speed is limited to $1$ unit per second.

Please determine whether this is possible.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1\le n\le10^5$) — the number of circles.

The next $n$ lines each contain two integers $x_i$, $y_i$ ($1\le x_i,y_i\le10^9$) — the center of each circle.

The final line contains four integers $x_s$, $y_s$, $x_t$, $y_t$ ($1\le x_s,y_s,x_t,y_t\le10^9$) — the coordinates of the starting point and the goal, respectively.

It is guaranteed that these $n+2$ points are distinct.

It is guaranteed that the sum of $n$ over all testcases does not exceed $10^5$.

Output: For each test case, output $\texttt{YES}$ if it is possible to reach the goal without touching the circle boundaries, and output $\texttt{NO}$ otherwise.

You can output $\texttt{Yes}$ and $\texttt{No}$ in any case (for example, strings $\texttt{yEs}$, $\texttt{yes}$, $\texttt{Yes}$, and $\texttt{YES}$ will be recognized as a positive response).

Examples:
input:
7
3
2 5
2 14
10 13
4 9 9 7
3
10 11
6 9
12 12
14 13 4 8
1
5 7
12 6 11 13
2
1000000000 2
2 1000000000
1 1 2 2
1
999999998 1000000000
999999999 999999999 1 1
1
1000000000 1
1 1000000000 1 1
10
989237121 2397081
206669655 527238537
522705783 380636165
532545346 320061691
207818728 199485303
884520552 315781807
992311437 802563521
205138355 324818663
223575704 395073023
281560523 236279118
216941610 572010615 323956540 794523071
output:
YES
NO
YES
YES
YES
NO
YES

Note: In the first test case, a feasible way of movement is as follows.

![](CDN_BASE_URL/235d765e20897c623b3ac974eceac134)

Buggy Code:
// . . . Bismillahir Rahmanir Rahim . . .

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

typedef long long ll;
#ifndef ONLINE_JUDGE
#define dbg_out cout
#define debug(...) dbg_out << "DBG )> "; __f(#__VA_ARGS__, __VA_ARGS__);
template<typename T1, typename T2> ostream& operator<<(ostream& out, pair<T1, T2> pr) { out << "{ " << pr.first << ", " << pr.second << " }"; return out; }
template<typename T1> ostream& operator<<(ostream& out, vector<T1> vec) { out << "{ "; for (auto &x: vec) out << x << ", "; out << "}"; return out; }
template<typename T1, size_t size> ostream& operator<<(ostream& out, array<T1, size> arr) { out << "{ "; for (auto &x: arr) out << x << ", "; out << "}"; return out; }
template<typename T1, typename T2> ostream& operator<<(ostream& out, map<T1, T2> mp) { out << "{ ";for (auto &x: mp) out << x.first << ": " << x.second <<  ", "; out << "}"; return out; }
template <typename Arg1> void __f(const char* name, Arg1&& arg1) { while (isspace(name[0])) name++; (isalpha(name[0]) || name[0] == '_') ? dbg_out << name << ": " << arg1 << "\n" : dbg_out << arg1 << "\n"; dbg_out.flush();}
template <typename Arg1, typename... Args> void __f (const char* names, Arg1&& arg1, Args&&... args) { const char *comma = strchr(names + 1, ','); while (isspace(names[0])) names++; (isalpha(names[0]) || names[0] == '_') ? dbg_out.write(names, comma - names) << ": " << arg1 << " | " : dbg_out << arg1 << " | "; __f(comma + 1, args...);}
#else
#define debug(...)
#endif

struct hash_pair {
    template <class T1, class T2>
    size_t operator()(const pair<T1, T2> &p) const {
        auto hash1 = hash<T1>{}(p.first);
        auto hash2 = hash<T2>{}(p.second);
        return hash1 ^ hash2;
    }
};

ll gcd(ll a, ll b){ while (b){ a %= b; swap(a, b);} return a;}
ll lcm(ll a, ll b){ return (a/gcd(a, b)*b);}
ll ncr(ll a, ll b){ ll x = max(a-b, b), ans=1; for(ll K=a, L=1; K>=x+1; K--, L++){ ans = ans * K; ans /= L;} return ans;}
ll bigmod(ll a,ll b,ll mod){
	b %= (mod-1);
	a %= mod;

	if(b==0){
	 return 1;
	}
	ll tm=bigmod(a,b/2,mod);
	tm=(tm*tm)%mod;
	if(b%2==1) tm=(tm*a)%mod;
	return tm;
}
ll egcd(ll a,ll b,ll &x,ll &y){ if(a==0){ x=0; y=1; return b;} ll x1,y1; ll d=egcd(b%a,a,x1,y1); x=y1-(b/a)*x1; y=x1; return d;}
ll modpow(ll a,ll p,ll mod) {ll ans=1;while(p){if(p%2)ans=(ans*a)%mod;a=(a*a)%mod;p/=2;} return ans;}
ll inverse_mod(ll n,ll mod) {return modpow(n,mod-2,mod);}

#define int long long
const long long eps = 1e8;
bool check(pair<int, int> x, pair<int, int> y, pair<int, int> a)
{
	//bool ok = 1;
	int dist = (a.first-y.first)*(a.first-y.first) + (a.second-y.second)*(a.second-y.second);
	//int mid = sqrt(dist);
	dist = (ll)eps*sqrt(dist);
	//dist *= eps;

	int ans = (x.first-y.first)*(x.first-y.first) + (x.second-y.second)*(x.second-y.second);
	ans = (ll)eps*sqrt(ans);
	//if(mid*mid < ans) mid++;

	//cout << ans << ' ' << dist << ' ';

	//if(ans == dist){
		//if(y.first >= a.first){
			//if(x.first >= y.first)
		//}
	//}

	int l = ans/eps;
	if(l*eps != ans) l++;
	int r = dist/eps;
	if(r*eps != ans) r++;

	if(l == r){
		if(l*eps != ans){
			if(r*eps != dist) return true;
			return false;
		}
		return false;
	}

	if(l < r) return false;
	return true;
}
signed main()
{
	ios_base::sync_with_stdio(false);
	cin.tie(0);
	cout.tie(0);

	int t = 1;
	cin >> t;
	for(int cs = 1; cs <= t; cs++){
		// cout << "Case " << cs << ": ";

		int n; cin >> n;
		vector<pair<int, int> > ara(n);
		for(int K = 0; K < n; K++) cin >> ara[K].first >> ara[K].second;

		pair<int, int> a, b;
		cin >> a.first >> a.second;
		cin >> b.first >> b.second;

		bool ok = 1;

		for(int K = 0; K < n; K++){
			if(!check(ara[K], b, a)){
				ok = 0;
				//cout << K << ' ';
				break;
			}
		}

		//sort(ara.begin(), ara.end());
		//if(a.first > b.first) swap(a, b);

		//int i = max(a.second, b.second);
		//int j = min(a.second, b.second);
		//for(int K = 0; K < n; K++){
			//if(ara[K].first < a.first || ara[K].first > b.first) continue;

			//if(ara[K].second >= a.second){
				//int mid = ara[K].second-a.second - (ara[K].first-a.first);
				//if(mid < 0){
					//ok = 0;
					//break;
				//}
				//i = min(i, ara[K].second-(ara[K].first-a.first)-mid/2-1);
			//}
			//else{
				//int mid = a.second-ara[K].second - (ara[K].first-a.first);
				//if(mid < 0){
					//ok = 0;
					//break;
				//}
				//j = max(j, ara[K].second+(ara[K].first-a.first)-mid/2+1);
			//}
		//}

		//cout << ok << ' ' << i << ' ' << j << "\n";
		if(ok ) cout << "YES\n";
		else cout << "NO\n";
	}

	return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that maximize the failure probability. Here are the steps to think through this:

1. **Understanding the Bug**:
   - The buggy code uses a function `check` to determine if the starting point can reach the goal without touching the circumference of any circle.
   - The function `check` calculates distances and uses a threshold `eps` to compare these distances.
   - The bug might be in the way distances are calculated or compared, especially with the use of `eps`.

2. **Edge Cases to Consider**:
   - **Circles with very close centers**: This can cause precision issues in distance calculations.
   - **Circles that are very large or very small**: This can test the limits of the distance comparison.
   - **Starting point and goal very close to a circle's boundary**: This can test the precision of the distance threshold.
   - **Starting point and goal on the same line as a circle's center**: This can test the linearity of the movement and the distance calculation.

3. **Constructing the Test Input**:
   - We will create a test case with a few circles that are close to each other and the starting and goal points.
   - The circles will be positioned in such a way that the distance calculations are very close to the threshold, which can expose the bug.

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

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

### Explanation:
- **Number of test cases**: 1
- **Number of circles**: 3
- **Circle centers**:
  - (1, 1)
  - (2, 2)
  - (3, 3)
- **Starting point**: (0, 0)
- **Goal point**: (4, 4)

This test case is designed to:
- Place circles in a straight line from the starting point to the goal.
- Ensure that the distances are very close to the threshold, which can cause precision issues in the `check` function.
- Test the ability of the algorithm to handle close distances and linearity.

This input should expose the bug in the distance comparison and threshold handling in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Black Circles
time_limit_ms: 2000
memory_limit_mb: 256
Description: There are $n$ circles on a two-dimensional plane. The $i$-th circle is centered at $(x_i,y_i)$. Initially, all circles have a radius of $0$.

The circles' radii increase at a rate of $1$ unit per second.

You are currently at $(x_s,y_s)$; your goal is to reach $(x_t,y_t)$ without touching the circumference of any circle (including the moment you reach $(x_t,y_t)$). You can move in any direction you want. However, your speed is limited to $1$ unit per second.

Please determine whether this is possible.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1\le n\le10^5$) — the number of circles.

The next $n$ lines each contain two integers $x_i$, $y_i$ ($1\le x_i,y_i\le10^9$) — the center of each circle.

The final line contains four integers $x_s$, $y_s$, $x_t$, $y_t$ ($1\le x_s,y_s,x_t,y_t\le10^9$) — the coordinates of the starting point and the goal, respectively.

It is guaranteed that these $n+2$ points are distinct.

It is guaranteed that the sum of $n$ over all testcases does not exceed $10^5$.

Output: For each test case, output $\texttt{YES}$ if it is possible to reach the goal without touching the circle boundaries, and output $\texttt{NO}$ otherwise.

You can output $\texttt{Yes}$ and $\texttt{No}$ in any case (for example, strings $\texttt{yEs}$, $\texttt{yes}$, $\texttt{Yes}$, and $\texttt{YES}$ will be recognized as a positive response).

Examples:
input:
7
3
2 5
2 14
10 13
4 9 9 7
3
10 11
6 9
12 12
14 13 4 8
1
5 7
12 6 11 13
2
1000000000 2
2 1000000000
1 1 2 2
1
999999998 1000000000
999999999 999999999 1 1
1
1000000000 1
1 1000000000 1 1
10
989237121 2397081
206669655 527238537
522705783 380636165
532545346 320061691
207818728 199485303
884520552 315781807
992311437 802563521
205138355 324818663
223575704 395073023
281560523 236279118
216941610 572010615 323956540 794523071
output:
YES
NO
YES
YES
YES
NO
YES

Note: In the first test case, a feasible way of movement is as follows.

![](CDN_BASE_URL/235d765e20897c623b3ac974eceac134)

Buggy Code:
import java.util.Scanner;

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

        while (t-- > 0) {
            int n = scanner.nextInt();
            int count = 0;
            double[] a = new double[2 * n];

            for (int i = 0; i < 2 * n; i++) {
                a[i] = scanner.nextDouble() / 1000.0;
            }

            double x = scanner.nextDouble();
            double y = scanner.nextDouble();
            double z = scanner.nextDouble();
            double k = scanner.nextDouble();

            double d = Math.pow(x - z, 2) + Math.pow(y - k, 2);

            for (int i = 0; i < 2 * n - 1; i += 2) {
                if (Math.pow(x/1000.0 - z/1000.0, 2) + Math.pow(y/1000.0 - k/1000.0, 2) -
                    Math.pow(a[i] - z/1000.0, 2) - Math.pow(a[i+1] - k/1000.0, 2) < 0) {
                    count++;
                }
            }

            if (count == n) {
                System.out.println("YES");
            } else {
                System.out.println("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
1
3
1 1
2 2
3 3
1 1 4 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Black Circles
time_limit_ms: 2000
memory_limit_mb: 256
Description: There are $n$ circles on a two-dimensional plane. The $i$-th circle is centered at $(x_i,y_i)$. Initially, all circles have a radius of $0$.

The circles' radii increase at a rate of $1$ unit per second.

You are currently at $(x_s,y_s)$; your goal is to reach $(x_t,y_t)$ without touching the circumference of any circle (including the moment you reach $(x_t,y_t)$). You can move in any direction you want. However, your speed is limited to $1$ unit per second.

Please determine whether this is possible.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1\le t\le10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1\le n\le10^5$) — the number of circles.

The next $n$ lines each contain two integers $x_i$, $y_i$ ($1\le x_i,y_i\le10^9$) — the center of each circle.

The final line contains four integers $x_s$, $y_s$, $x_t$, $y_t$ ($1\le x_s,y_s,x_t,y_t\le10^9$) — the coordinates of the starting point and the goal, respectively.

It is guaranteed that these $n+2$ points are distinct.

It is guaranteed that the sum of $n$ over all testcases does not exceed $10^5$.

Output: For each test case, output $\texttt{YES}$ if it is possible to reach the goal without touching the circle boundaries, and output $\texttt{NO}$ otherwise.

You can output $\texttt{Yes}$ and $\texttt{No}$ in any case (for example, strings $\texttt{yEs}$, $\texttt{yes}$, $\texttt{Yes}$, and $\texttt{YES}$ will be recognized as a positive response).

Examples:
input:
7
3
2 5
2 14
10 13
4 9 9 7
3
10 11
6 9
12 12
14 13 4 8
1
5 7
12 6 11 13
2
1000000000 2
2 1000000000
1 1 2 2
1
999999998 1000000000
999999999 999999999 1 1
1
1000000000 1
1 1000000000 1 1
10
989237121 2397081
206669655 527238537
522705783 380636165
532545346 320061691
207818728 199485303
884520552 315781807
992311437 802563521
205138355 324818663
223575704 395073023
281560523 236279118
216941610 572010615 323956540 794523071
output:
YES
NO
YES
YES
YES
NO
YES

Note: In the first test case, a feasible way of movement is as follows.

![](CDN_BASE_URL/235d765e20897c623b3ac974eceac134)

Buggy Code:
import java.util.*;

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

        while (t-- > 0) {
            int n = scanner.nextInt();
            int count = 0;
            double[] a = new double[n + n];

            for (int i = 0; i < n + n; i++) {
                a[i] = scanner.nextDouble() / 1000;
            }

            double x = scanner.nextDouble();
            double y = scanner.nextDouble();
            double z = scanner.nextDouble();
            double k = scanner.nextDouble();

            double d = Math.pow(x - z, 2) + Math.pow(y - k, 2);

            for (int i = 0; i + 1 < n + n; i += 2) {
                if (Math.pow(x/1000 - z/1000, 2) + Math.pow(y/1000 - k/1000, 2) -
                    Math.pow(a[i] - z/1000, 2) - Math.pow(a[i+1] - k/1000, 2) < 0) {
                    count++;
                }
            }

            if (count == n) {
                System.out.println("YES");
            } else {
                System.out.println("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
1
1
1 1
2 2 1 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
#include <stdio.h>
int main(){
	int n,k,s=0;
	scanf("%d %d",&n,&k);
	long long a[n];
	int b[n];
	for(int m=0;m<n;m++)
	{
		scanf("%lld",&a[m]);
	}
	for(int m=0;m<n;m++)
	{
		scanf("%d",&b[m]);
	}
	int c[n];
	for(int m=0;m<n;m++)
    c[m]=0;
	for(int m=n-1;m>=0;m--)
	{
		c[m]=a[m]/b[m];
		s=s+c[m];
		if(s>=k)
		{
			c[m]=k-s+c[m];
			break;
		}
	}
	for(int m=0;m<n;m++)
	printf("%d ",c[m]);
	return 0;
}

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

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

Think step by step.

Response:
```plaintext
3 10
10 10 10
1 2 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

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

int sum(int n, vector<int> a, vector<int> b, int j){
	int sum = 0;
	for (int i = j; i < n; i++) {
		sum = sum + int(a[i] / b[i]);
	}
	return sum;
}

int main() {
	int n, k;
	cin >> n;
	cin >> k;
	vector <int> a(n);
	for (int j = 0; j < n; j++) {
		cin >> a[j];
	}
	vector <int> b(n);
	for (int j = 0; j < n; j++){
		cin >> b[j];
	}
	vector <int> c(n);

	for (int j = 0; j < n - 1;j++) {
		if (k - sum(n, a, b, j + 1) < 0) {
			c[j] = 0;

		}
		else {
			c[j] = k - sum(n, a, b, j + 1);
			k = sum(n, a, b, j + 1);
		}

	}
	c[n - 1] = k;

	for (int i = 0; i < n; i++) {
		cout << c[i] << endl;
	}

	return 0;
}

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

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

Think step by step.

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

### Explanation:
This test case is designed to expose the bug in the given buggy code. Here’s why it’s challenging:

1. **Total Work Units (k) is 7**: The project requires 7 units of work.
2. **Bonuses (a)**:
   - Engineer 1: 6 burles
   - Engineer 2: 7 burles
   - Engineer 3: 8 burles
3. **Work Unit Costs (b)**:
   - Engineer 1: 1 burle per unit
   - Engineer 2: 2 burles per unit
   - Engineer 3: 3 burles per unit

### Expected Behavior:
- **Engineer 1** can complete up to 6 units of work (since \(a_1 / b_1 = 6 / 1 = 6\)).
- **Engineer 2** can complete up to 3 units of work (since \(a_2 / b_2 = 7 / 2 = 3.5\), but we take the integer part).
- **Engineer 3** can complete up to 2 units of work (since \(a_3 / b_3 = 8 / 3 = 2.67\), but we take the integer part).

### Correct Solution:
- **Engineer 1** should complete 6 units of work.
- **Engineer 2** should complete 1 unit of work.
- **Engineer 3** should complete 0 units of work.

### Why This Test Case is Challenging:
- The buggy code calculates the sum of work units that can be completed by the remaining engineers and subtracts it from the total work required. This approach can lead to incorrect assignments of work units.
- In this test case, the sum of work units that can be completed by Engineers 2 and 3 is \(3 + 2 = 5\). If Engineer 1 completes 6 units of work, the project will be completed, but the buggy code might incorrectly assign 0 units to Engineer 1 and then fail to distribute the remaining work correctly.

This test case should help in identifying and fixing the bug in the given code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define pb(x) push_back(x)
#define F first
#define S second

void solve (ll t) {
    ll n, k;
    cin >> n >> k;
    vector<ll> a(n);
    for (ll i = 0; i < n; i++) {
        cin >> a[i];
    }
    vector<ll> b(n);
    for (ll i = 0; i < n; i++) {
        cin >> b[i];
    }
    vector<ll> ans;
    ll i = 0;
    for(i = n - 1; i >= 0; i--) {
        ll tot = a[i]/b[i];
        if(k > tot) {
            k -= tot;
            ans.pb(tot);
        } else {
            ans.pb(k);
            k = 0;
            i--;
            break;
        }

    }
    while(i >= 0) {
        ans.pb(0);
        i--;
    }
    reverse(ans.begin(), ans.end());
    for(auto x: ans) {
        cout << x << " ";
    }
    cout << endl;
}

int main () {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);
    ll t = 1;
    while(t)
    solve(t--);
}

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

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

Think step by step.

Response:
```plaintext
3 10
10 20 30
3 2 1
```

### Explanation:
This test case is designed to expose the bug in the buggy code. Here’s how it works:

1. **Engineers and Work Requirements:**
   - There are 3 engineers, and the project requires 10 units of work.

2. **Bonuses and Work Values:**
   - The bonuses are 10, 20, and 30 burles for engineers 1, 2, and 3, respectively.
   - The work values are 3, 2, and 1 burles per unit of work for engineers 1, 2, and 3, respectively.

3. **Optimal Work Distribution:**
   - The optimal distribution should maximize the benefit for each engineer while ensuring the project is completed.
   - Engineer 3 has the lowest work value (1 burle per unit), so they should do as much work as possible to maximize their benefit.
   - Engineer 2 has the next lowest work value (2 burles per unit), so they should do the next amount of work.
   - Engineer 1 has the highest work value (3 burles per unit), so they should do the least amount of work.

4. **Expected Output:**
   - Engineer 3 should do 3 units of work (since \(30 - 3 \cdot 1 = 27\)).
   - Engineer 2 should do 4 units of work (since \(20 - 4 \cdot 2 = 12\)).
   - Engineer 1 should do 3 units of work (since \(10 - 3 \cdot 3 = 1\)).
   - The total work is \(3 + 4 + 3 = 10\), which meets the project requirement.

5. **Bug Exposure:**
   - The buggy code starts from the last engineer and assigns work based on the ratio of bonus to work value. However, it does not consider the optimal distribution of work across all engineers.
   - The correct output should be `3 4 3`, but the buggy code will likely produce a different and incorrect result due to its flawed logic.

This test case should effectively highlight the issues with the buggy code's approach to distributing the work.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
n,k = map(int,input().split())
a = list(map(int,input().split()))
b = list(map(int,input().split()))
result = []
for i in range(n):
    result = [min(k,a[n-1-i]//b[n-1-i])] + result
    k-=result[0]
if sum(result) < k:
    for c in result:
        print(0,end=' ')
else:
    for c in result:
        print(c,end=' ')

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

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

Think step by step.

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

1. **Edge Cases**: Small values of `n` and `k` to ensure the test is small-scale.
2. **Negative Benefit**: Ensure that some engineers have a negative benefit to test how the code handles this.
3. **Infeasible Completion**: Ensure that the total work required (`k`) is such that it cannot be completed by the engineers, leading to a scenario where all engineers should do zero work.
4. **Optimal Distribution**: Ensure that the work is distributed in a way that maximizes the benefit for each engineer.

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

- **n = 3**: Three engineers.
- **k = 7**: The project requires 7 units of work.
- **a = [10, 20, 30]**: Bonuses for the engineers.
- **b = [3, 2, 1]**: Work unit costs for the engineers.

This test input is designed to:
- Ensure that the first engineer (with the highest cost per unit of work) does not do any work because it would result in a negative benefit.
- Ensure that the second and third engineers can complete the project optimally.

```plaintext
3 7
10 20 30
3 2 1
```

Explanation:
- The first engineer (with `a_1 = 10` and `b_1 = 3`) would have a negative benefit if they do any work, so they should do zero work.
- The second engineer (with `a_2 = 20` and `b_2 = 2`) can do up to 10 units of work, but they will do 3 units to maximize their benefit.
- The third engineer (with `a_3 = 30` and `b_3 = 1`) can do up to 30 units of work, but they will do 4 units to complete the remaining work.

The expected output should be:
```plaintext
0 3 4
```

This test input will expose the bug in the buggy code, as the code incorrectly calculates the work distribution and does not handle the negative benefit scenario correctly.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
def solve():
    n, k = map(int, input().split())
    a = list(map(int, input().split()))
    b = list(map(int, input().split()))
    limits = [a[i] // b[i] for i in range(n)]
    s = sum(limits)
    for i in range(n):
        otherswork = s - limits[i]
        work = max(0, min(limits[i], k - otherswork))
        print(work, end=' ')
        k -= work
        s -= limits[i]
    print()

#a1 - c1.b1 + a2 - c2.b2 + ...+ an - cn.bn
#sum(a) - (c1.b1 + c2.b2 + ....+ cn.bn)

t = 1
for _ in range(t):
    solve()

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to consider edge cases and scenarios that maximize the failure probability. The code has a logic that calculates the maximum work each engineer can do based on their limits and the remaining work required. However, it might fail in scenarios where the distribution of work is not optimal or where the limits are not correctly handled.

### Step-by-Step Analysis:
1. **Equal Limits**: If all engineers have the same work unit cost, the code might not distribute the work optimally.
2. **High Work Requirements**: If the total required work is very high compared to the limits, the code might incorrectly set some engineers to do zero work.
3. **Low Work Requirements**: If the total required work is very low, the code might distribute work in a suboptimal way.
4. **Mixed Limits**: If the limits are mixed (some high, some low), the code might not handle the distribution correctly.

### Chosen Test Input:
We will use a scenario where the total required work is just slightly more than the sum of the limits, and the limits are mixed. This will test the code's ability to handle a situation where the work distribution is critical.

### Test Input:
```plaintext
4 13
10 15 20 25
1 2 3 4
```

### Explanation:
- **n = 4**: There are 4 engineers.
- **k = 13**: The project requires 13 units of work.
- **a = [10, 15, 20, 25]**: The bonuses for the engineers are 10, 15, 20, and 25 burles respectively.
- **b = [1, 2, 3, 4]**: The work unit costs for the engineers are 1, 2, 3, and 4 burles respectively.

### Expected Output:
The optimal distribution of work should be:
- Engineer 1: 4 units (10 - 4*1 = 6)
- Engineer 2: 3 units (15 - 3*2 = 9)
- Engineer 3: 2 units (20 - 2*3 = 14)
- Engineer 4: 4 units (25 - 4*4 = 9)

However, the code might fail to distribute the work optimally, especially since the total required work is just slightly more than the sum of the limits.

### Final Test Input:
```plaintext
4 13
10 15 20 25
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
n, k = map(int, input().split())  # Correctly parse n and k
a = list(map(int, input().split()))  # Correctly parse the bonus list a
b = list(map(int, input().split()))  # Correctly parse the work cost list b

c = [0] * n  # Initialize an array to store the work each engineer will do

# Iterate over the engineers from the last one to the first one
for i in range(n - 1, -1, -1):
    # Calculate the maximum work this engineer can do based on their bonus and cost
    temp = min(k, a[i] // b[i])  # Engineer can do either the remaining work or what they are willing to do
    k -= temp  # Reduce the remaining work required
    c[i] = temp  # Assign work to engineer i

# Print the result
print(" ".join(map(str, c)))  # Output the work done by each engineer

**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 10
10 20 30 40 50
5 4 3 2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
import java.util.*;

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

        int noOfEngineers = sc.nextInt();
        int noOfWorks = sc.nextInt();

        int[] bonus = new int[noOfEngineers];
        int[] workValue = new int[noOfEngineers];
        for (int i = 0; i < noOfEngineers; i++) {
            bonus[i] = sc.nextInt();
        }
        for (int i = 0; i < noOfEngineers; i++) {
            workValue[i] = sc.nextInt();
        }

        List<int[]> engineers = new ArrayList<>();
        for (int i = 0; i < noOfEngineers; i++) {
            engineers.add(new int[]{bonus[i], workValue[i], i});
        }

        engineers.sort((a, b) -> {
            if (b[0] != a[0]) return b[0] - a[0];
            return a[2] - b[2];
        });

        int[] workDone = new int[noOfEngineers];
        for (int[] engineer : engineers) {
            int index = engineer[2];
            int maxWork = engineer[0] / engineer[1];
            int workAssigned = Math.min(maxWork, noOfWorks);
            workDone[index] = workAssigned;
            noOfWorks -= workAssigned;
            if (noOfWorks <= 0) break;
        }

        for (int work : workDone) {
            System.out.print(work + " ");
        }

        sc.close();
    }
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to consider scenarios where the sorting and work assignment logic might fail. The primary issue in the code is the sorting criterion, which only considers the bonus and the engineer's index. This can lead to suboptimal work assignments when the work value (cost) is not taken into account properly.

### Step-by-Step Analysis:

1. **Sorting Criteria**: The code sorts engineers primarily by bonus and secondarily by index. This means that engineers with higher bonuses are given priority, regardless of their work value (cost).

2. **Work Assignment**: The code then assigns the maximum possible work to each engineer based on their bonus and work value, without considering the overall efficiency (i.e., the benefit per unit of work).

3. **Edge Case**: To expose the bug, we need a scenario where the engineers with higher bonuses have higher work values, making it suboptimal to assign them the maximum possible work first.

### Test Input:

Let's create a test input where the engineers with higher bonuses have higher work values, and the optimal solution requires a different distribution of work.

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

### Explanation:

- **n = 4**: There are 4 engineers.
- **k = 10**: The project requires 10 units of work.
- **a = [10, 9, 8, 7]**: The bonuses for the engineers.
- **b = [1, 2, 3, 4]**: The work values (costs) for the engineers.

### Expected Output:

The optimal distribution of work should be:
- Engineer 1 (bonus 10, work value 1): 4 units of work
- Engineer 2 (bonus 9, work value 2): 3 units of work
- Engineer 3 (bonus 8, work value 3): 2 units of work
- Engineer 4 (bonus 7, work value 4): 1 unit of work

This distribution ensures that the project is completed (total work = 4 + 3 + 2 + 1 = 10) and maximizes the benefit for each engineer.

### Actual Output from Buggy Code:

The buggy code will likely assign more work to engineers with higher bonuses, which may result in a suboptimal distribution. For example, it might assign:

- Engineer 1: 10 units of work
- Engineer 2: 0 units of work
- Engineer 3: 0 units of work
- Engineer 4: 0 units of work

This distribution does not complete the project (total work = 10 + 0 + 0 + 0 = 10, but the work is not distributed optimally).

### Final Test Input:

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
import java.util.*;

public class Main{
    public static void main(String[] args){
        HashSet<ArrayList<Integer>> hs1 = new HashSet<>();
        Scanner sc = new Scanner(System.in);
        int tc = 1;
        while(tc-->0){
            int n = sc.nextInt(), k = sc.nextInt();
            long a[]=new long[n];
            long b[]=new long[n];
            for(int i=0;i<n;i++){
                a[i]=sc.nextInt();
            }
            for(int i=0;i<n;i++){
                b[i]=sc.nextInt();
            }
            long ans[]=new long[n];
            for(int i=n-1;i>=0;i--){
                ans[i]=Math.min(k,a[i]/b[i]);
                k-=(a[i]/b[i]);
                k=Math.max(k,0);
            }
            for(int i=0;i<n;i++){
                System.out.print(ans[i]+" ");
            }
            System.out.println();
        }
        sc.close();
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on scenarios where the code's logic fails to correctly distribute the work units. The buggy code incorrectly calculates the amount of work each engineer will do by subtracting the maximum possible work from the total required work in a reverse order. This can lead to incorrect results when the work distribution needs to be optimized based on the benefits.

Here are the steps to identify a suitable test case:

1. **Identify the Bug**: The bug is in the logic where the code subtracts the maximum possible work from the total required work in a reverse order without considering the actual benefit or the feasibility of completing the project.

2. ** edge cases**:
   - **Case 1**: When the total required work is exactly equal to the sum of the maximum possible work each engineer can do.
   - **Case 2**: When the total required work is slightly more than the sum of the maximum possible work each engineer can do.
   - **Case 3**: When the total required work is much less than the sum of the maximum possible work each engineer can do, but the distribution needs to be optimized to maximize benefits.

3. **Construct the Test Case**:
   - **Number of Engineers (n)**: 3
   - **Total Work Units (k)**: 10
   - **Bonuses (a)**: 6, 7, 8
   - **Work Unit Costs (b)**: 1, 2, 3

This test case is chosen because:
- The total required work (10) is less than the sum of the maximum possible work (6/1 + 7/2 + 8/3 = 6 + 3 + 2 = 11).
- The distribution needs to be optimized to maximize benefits, which the buggy code fails to do correctly.

Here is the test input:

```plaintext
3 10
6 7 8
1 2 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

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

#define ll long long
#define vll vector<ll>
#define vvll vector<vector<ll>>
#define fi(a, n) for (ll i = a; i < n; i++)
#define fj(a, n) for (ll j = a; j < n; j++)
#define cy cout<<"YES"<<endl
#define cn cout<<"NO"<<endl

void solve() {
    ll n;
    cin >> n;
    ll k; cin >> k;
    vll a(n),b(n);
    fi(0,n) cin >> a[i];
    fi(0,n) cin >> b[i];

    ll cnt=0, res=0;
    fi(0,n) cnt += a[i]/b[i];

    vll ans(n,0);

    if(cnt < k){
        fi(0,n) cout <<0 <<" ";
        cout<<endl;
        return;
    }
    else
    {
        for(ll i=n-1; i>=0; i--)
        {

            if((res + a[i]/b[i]) > k){
                //res = k;
                ans[i] = k-res;
            }
            else ans[i] = a[i]/b[i];

            res += a[i]/b[i];
        }

        fi(0,n) cout<< ans[i]<< " ";
        cout << endl;
    }

}

int main() {
    ll t=1;
    //cin >> t;
    while (t--) {
        solve();
    }

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
5 10
10 10 10 10 10
1 2 3 4 5
```

### Explanation:
1. **Number of Engineers and Work Units**:
   - `n = 5` (5 engineers)
   - `k = 10` (10 units of work required)

2. **Bonuses**:
   - `a = [10, 10, 10, 10, 10]` (Each engineer gets a bonus of 10 burles)

3. **Work Unit Costs**:
   - `b = [1, 2, 3, 4, 5]` (Cost of one unit of work for each engineer)

### Why This Test Case is Challenging:
- **Equal Bonuses**: All engineers have the same bonus, which means the decision is based solely on the work unit cost.
- **Increasing Work Unit Costs**: The work unit cost increases linearly, making it more expensive for later engineers to contribute.
- **Total Work Units**: The total required work units (10) can be distributed in a way that maximizes the benefit for all engineers, but the buggy code might not handle this correctly.

### Expected Output:
The optimal distribution of work units should be:
- Engineer 1: 10 units (since the cost is 1, and 10/1 = 10)
- Engineer 2: 0 units (since 10/2 = 5, but 10 units are already covered by Engineer 1)
- Engineer 3: 0 units (same reason)
- Engineer 4: 0 units (same reason)
- Engineer 5: 0 units (same reason)

Thus, the expected output is:
```plaintext
10 0 0 0 0
```

However, the buggy code might not handle this correctly and might distribute the work units in a suboptimal way, 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: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

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

void solve() {
    int n, k; cin >> n >> k;

    vector<ll> bonus(n); for (int i = 0; i<n; i++) cin >> bonus[i];

    vector<ll> cost(n); for (int i = 0; i<n; i++) cin >> cost[i];

    vector<ll> maxHours(n); for (int i = 0; i<n; i++) maxHours[i] = bonus[i]/cost[i];

    vector<ll> prefix(n);

    prefix[0] = maxHours[0];
    for (int i = 1; i<n; i++) prefix[i] = prefix[i-1] + maxHours[i];

    // ok cool

    // now sim it happening

    vector<ll> willDo(n);

    for (int i = 0; i<n; i++) {
        ll othersDo = prefix[n-1] - prefix[i];

        ll hours = k-othersDo;

        if (hours <= maxHours[i]) {
            k-=hours;
            willDo[i] = hours;
        }
        else {
            for (int i = 0; i<n; i++) cout << "0 ";
            return;
        }
    }

    for (int i = 0; i<n; i++) cout << willDo[i] << " ";
}

int main() {
	solve();
}

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

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

Think step by step.

Response:
```plaintext
3 10
10 10 10
1 2 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
n, k = map(int, input().split())
a = list(map(int, input().split()))
b = list(map(int, input().split()))

# Sorting engineers by potential value to cost ratio
engineers = sorted(range(n), key=lambda i: -b[i])
work = [0] * n

for i in engineers:
    max_work = a[i] // b[i]
    if k > 0:
        assign = min(max_work, k)
        work[i] = assign
        k -= assign

if k > 0:  # If the work is not fulfilled
    work = [0] * n

print(*work)

**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 10
10 20 30 40
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

Buggy Code:
def countCommitment(a, b, n, k):
    max_tasks = []
    curr = 0
    for i in range(n):
        curr += a[i] // b[i]
        max_tasks.append(curr)

    if max_tasks[-1] < k:
        return [0] * n

    tasks = []
    t = k
    for i in range(n-1):
        remaining_tasks = max_tasks[-1] - max_tasks[i]
        if remaining_tasks >= (t-1):
            tasks.append(1 if a[i] // b[i] > 0 else 0)
        else:
            tasks.append(t - remaining_tasks)
        t -= tasks[-1]
    tasks.append(t)
    return tasks

n, k = map(int, input().split())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
tasks = countCommitment(a, b, n, k)
print(' '.join(map(str, tasks)))

**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 10
5 10 15 20
1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Bonus Project
time_limit_ms: 2000
memory_limit_mb: 512
Description: There is a team of $$$n$$$ software engineers numbered from $$$1$$$ to $$$n$$$. Their boss promises to give them a bonus if they complete an additional project. The project requires $$$k$$$ units of work in total. The bonus promised to the $$$i$$$-th engineer is $$$a_i$$$ burles. The boss doesn't assign specific tasks to engineers; it is expected that every engineer will voluntarily complete some integer amount of work units. The bonus will be paid to the entire team only if the project is completed; in other words, if the total amount of voluntary work units on the project is greater than or equal to $$$k$$$.

The amount of work that can be performed by each engineer is not limited. However, all engineers value their labour. The $$$i$$$-th engineer estimates one unit of their work as $$$b_i$$$ burles. If the bonus is paid, the benefit $$$s_i$$$ of the $$$i$$$-th engineer for completing $$$c$$$ units of work is defined as $$$s_i = a_i - c \cdot b_i$$$. If the bonus is not paid, the engineer will not volunteer to do any work.

Engineers work together for many years, so they know how the bonus is going to be distributed and how much their colleagues value the labour. That is, all $$$a_i$$$ and all $$$b_i$$$ are known to every engineer in the team.

Engineers are eager to get the bonus, so they agreed on the following process of work distribution between them:

- the first engineer says: "I will complete $$$c_1$$$ units of work", where $$$c_1$$$ is a non-negative integer;
- then, the second engineer says: "I will complete $$$c_2$$$ units of work", where $$$c_2$$$ is a non-negative integer;
- ... and so on;
- finally, the $$$n$$$-th engineer says: "I will complete $$$c_n$$$ units of work", where $$$c_n$$$ is a non-negative integer.

Every engineer voices $$$c_i$$$ in a way to maximize their own benefit $$$s_i$$$. If the expected benefit is going to be zero, an engineer will still agree to work to get the experience and to help their colleagues obtain the bonus. However, if the benefit is expected to be negative for some reason (an engineer needs to perform an excessive amount of work or the project is not going to be completed), that engineer will not work at all (completes zero amount of work units).

Given that every engineer acts perfectly, your task is to find out the numbers $$$c_i$$$ voiced by every engineer.

Input: The first line contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 1000$$$; $$$1 \le k \le 10^6$$$) — the number of engineers in the company and the number of work units the project requires, respectively.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$), where $$$a_i$$$ is the bonus which will be paid to the $$$i$$$-th engineer if the project is completed.

The third line contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$1 \le b_i \le 1000$$$), where $$$b_i$$$ is the work unit cost for the $$$i$$$-th engineer.

Output: Print $$$n$$$ integers $$$c_1, c_2, \dots, c_n$$$ ($$$0 \le c_i \le k$$$) — the amount of work completed by each engineer given that every engineer behaves optimally. Note that the answer is unique.

Examples:
input:
3 6
4 7 6
1 2 3
output:
1 3 2

input:
3 12
4 7 6
1 2 3
output:
0 0 0

input:
3 11
6 7 8
1 2 3
output:
6 3 2

Note: In the first example, engineers distributed the work across them and got the bonus, even though the benefit for the third engineer is zero.

In the second example, the bonus project requires too many work units to complete, so it's more beneficial for engineers not to work at all.

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

public class Main {
    private static final FastIO sc = new FastIO();

    public static void main(String[] args) {
        //int t = sc.nextInt();
        int t=1;

        while (t-- > 0) {
            solve();
        }
    }

    static void solve() {
        int n = sc.nextInt();
        long k=sc.nextLong();
        long [][] a=new long [n][3];
        long [][] ans=new long [n][2];
        int i=0;
        for(i=0;i<n;i++)
        {
        a[i][0]=sc.nextLong();
        a[i][2]=i;
        }
        for(i=0;i<n;i++)
        a[i][1]=sc.nextLong();
        Arrays.sort(a,(x,y)->{
            if(x[0]%x[1]!=y[0]%y[1])
            {
                        if(x[0]/x[1]!=y[0]/y[1])
                      return Long.compare(x[0]/x[1],y[0]/y[1]);
                      return Long.compare(y[1],x[1]);
            }
            return Long.compare(y[0]%y[1],x[0]%x[1]);
        });

        for(i=0;i<n && k>0;i++)
        {
               ans[i][0]=Math.min(a[i][0]/a[i][1],k);
               ans[i][1]=a[i][2];
               k-=ans[i][0];

        }
        if(k>0)
        {
            for(i=0;i<n;i++)
            System.out.print("0 ");
            System.out.println();
            return;
        }
        Arrays.sort(ans,(x,y)->Long.compare(x[1],y[1]));
        for(i=0;i<n;i++)
        System.out.print(ans[i][0]+" ");
        System.out.println();
    }

    static class Pair
    {
        int x;
        int y;
        Pair(int x,int y)
        {
            this.x=x;
            this.y=y;
        }
        @Override
        public boolean equals(Object o) {
            // Check if the object is of the same type
            if (this == o) return true;
            if (o == null || getClass() != o.getClass()) return false;
            Pair pair = (Pair) o;
            return (x == pair.x && y == pair.y) || (x == pair.y && y == pair.x);  // Ensure pairs are equal regardless of order
        }

        @Override
        public int hashCode() {
            return Objects.hash(Math.min(x, y), Math.max(x, y));  // Generate hash code based on ordered pair values
        }
    }

    static long modInverse(long x, long m) {
        return power(x, m - 2, m);
    }
    static long power(long x, long y, long m) {
        long res = 1;
        x = x % m;
        while (y > 0) {
            if ((y & 1) == 1) {
                res = (res * x) % m;
            }
            y = y >> 1;
            x = (x * x) % m;
        }
        return res;
    }

    static class FastIO {
        BufferedReader br;
        StringTokenizer st;

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

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

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

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

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases where the sorting logic or the distribution of work units might fail. The key issues in the code are:

1. **Incorrect Sorting Logic**: The sorting logic is based on a combination of the ratio `a[i][0]/a[i][1]` and the remainder `a[i][0]%a[i][1]`. This can lead to incorrect ordering of engineers, especially when the ratios are close but the remainders differ.

2. **Handling of Zero Benefits**: The code does not handle cases where the benefit for some engineers might be zero or negative, which can lead to incorrect work distribution.

3. **Work Distribution Logic**: The logic for distributing work units might not always ensure that the project is completed optimally.

To create a challenging test input, let's consider a scenario where the sorting logic might fail due to close ratios and remainders, and where the work distribution might not be optimal.

### Challenging Test Input
```plaintext
5 15
10 15 20 25 30
2 3 4 5 6
```

### Explanation
- **Number of Engineers (n)**: 5
- **Required Work Units (k)**: 15
- **Bonuses (a)**: [10, 15, 20, 25, 30]
- **Work Unit Costs (b)**: [2, 3, 4, 5, 6]

### Expected Output
The expected output should be a distribution of work units that maximizes the benefit for each engineer and ensures the project is completed.

### Reasoning
1. **Ratios and Remainders**:
   - Engineer 1: \( \frac{10}{2} = 5 \) with remainder 0
   - Engineer 2: \( \frac{15}{3} = 5 \) with remainder 0
   - Engineer 3: \( \frac{20}{4} = 5 \) with remainder 0
   - Engineer 4: \( \frac{25}{5} = 5 \) with remainder 0
   - Engineer 5: \( \frac{30}{6} = 5 \) with remainder 0

   Since all ratios are the same, the sorting logic might not correctly order the engineers, especially if it relies on the remainder.

2. **Optimal Work Distribution**:
   - The optimal distribution would be to assign work units to the engineers in a way that maximizes the benefit and ensures the project is completed.
   - The correct distribution should be: [5, 3, 3, 2, 2] (sum = 15).

This test case is designed to challenge the sorting and work distribution logic, especially when the ratios are close and the remainders are zero. If the code fails to handle this correctly, it will likely produce an incorrect distribution of work units.