**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import java.util.*;

public class TurtleAndThreeSequences {

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

        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            b[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            c[i] = sc.nextInt();
        }

        int maxSum = findMaxSum(n, m, a, b, c);
        System.out.println(maxSum);
    }

    private static int findMaxSum(int n, int m, int[] a, int[] b, int[] c) {
        // Edge case
        if (m > n) return -1;

        // dp[i][j] will store the maximum sum of c values for a subsequence of length j ending at index i
        int[][] dp = new int[n][m + 1];

        // Initialize dp array with -1 (indicating impossible states)
        for (int i = 0; i < n; i++) {
            Arrays.fill(dp[i], -1);
        }

        // Base case: Single element subsequences
        for (int i = 0; i < n; i++) {
            dp[i][1] = c[i];
        }

        // Fill DP table
        for (int length = 2; length <= m; length++) {
            for (int i = 0; i < n; i++) {
                if (dp[i][length - 1] != -1) {
                    for (int j = i + 1; j < n; j++) {
                        if (a[i] <= a[j] && b[i] != b[j]) {
                            dp[j][length] = Math.max(dp[j][length], dp[i][length - 1] + c[j]);
                        }
                    }
                }
            }
        }

        // Find the maximum sum for subsequences of length m
        int maxSum = -1;
        for (int i = 0; i < n; i++) {
            maxSum = Math.max(maxSum, dp[i][m]);
        }

        return maxSum;
    }
}

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

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

Think step by step.

Response:
```plaintext
5 3
1 2 2 3 4
1 2 2 3 4
10 20 30 40 50
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import java.util.*;

public class TurtleAndThreeSequences {

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

        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            b[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            c[i] = sc.nextInt();
        }

        // DP table to store max sum with specific subsequence length ending at index i
        int[][] dp = new int[n][m + 1];

        // Initialize with -1 (impossible case)
        for (int[] row : dp) {
            Arrays.fill(row, -1);
        }

        // Start with subsequences of length 1
        for (int i = 0; i < n; i++) {
            dp[i][1] = c[i];
        }

        // Fill DP table
        for (int k = 2; k <= m; k++) {  // For each subsequence length
            for (int i = 0; i < n; i++) {  // Ending at index i
                for (int j = 0; j < i; j++) {  // Consider extending from index j
                    if (a[j] <= a[i] && b[j] != b[i] && dp[j][k - 1] != -1) {
                        dp[i][k] = Math.max(dp[i][k], dp[j][k - 1] + c[i]);
                    }
                }
            }
        }

        // Find the maximum sum for subsequences of length m
        int maxSum = -1;
        for (int i = 0; i < n; i++) {
            maxSum = Math.max(maxSum, dp[i][m]);
        }

        System.out.println(maxSum == -1 ? -1 : maxSum);
    }
}

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

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

Think step by step.

Response:
```plaintext
5 3
1 2 3 4 5
1 1 1 1 1
10 20 30 40 50
```

This test input is designed to expose the bug in the code. The sequences `a` and `c` are strictly increasing, which means that the only way to form a valid subsequence of length `m` (in this case, `3`) is to ensure that the `b` sequence also meets the distinct condition. However, in this test case, all elements in the `b` sequence are the same (`1`), which means it is impossible to form a valid subsequence of length `3` that satisfies the conditions. The expected output should be `-1`, but the buggy code might not handle this edge case correctly.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
int a[5000],b[5000],c[5000];
int main()
{
	int n,m;cin>>n>>m;
	for(int i=1;i<=n;i++)cin>>a[i];
	for(int i=1;i<=n;i++)cin>>b[i];
	for(int i=1;i<=n;i++)cin>>c[i];
	int maxn=0;
	if(pow(n,m)>1e8)
	{
		cout<<-1;
		return 0;
	}
	if(m==1)
	{
		for(int i=1;i<=n;i++)
		{
			maxn=max(c[i],maxn);
		}
		cout<<maxn;
		return 0;
	}
	if(m==2)
	{
		for(int i=1;i<=n;i++)
		{
			for(int j=i+1;j<=n;j++)
			{
				if(a[i]<=a[j]&&(b[i]!=b[j]))
				{
					maxn=max(maxn,(c[i]+c[j]));
				}
			}
		}
		if(maxn)cout<<maxn;
		else cout<<-1;
	}
	if(m==3)
	{
		for(int i=1;i<=n;i++)
		{
			for(int j=i+1;j<=n;j++)
			{
				for(int k=j+1;k<=n;k++)
				{
					if(a[i]<=a[j]&&a[j]<=a[k]&&(b[i]!=b[j])&&(b[j]!=b[k])&&(b[i]!=b[k]))
					{
						maxn=max(maxn,(c[i]+c[j]+c[k]));
					}
				}
			}
		}
		if(maxn)cout<<maxn;
		else cout<<-1;
	}
	if(m==4)
	{
		for(int i=1;i<=n;i++)
		{
			for(int j=i+1;j<=n;j++)
			{
				for(int k=j+1;k<=n;k++)
				{
					for(int l=k+1;l<=n;l++)
					{
						if(a[i]<=a[j]&&a[j]<=a[k]&&a[k]<=a[l]&&(b[i]!=b[j])&&(b[j]!=b[k])&&(b[i]!=b[k])&&(b[i]!=b[l])&&(b[j]!=b[l])&&(b[k]!=b[l]))
						{
							maxn=max(maxn,(c[i]+c[j]+c[k]+c[l]));
						}
					}
				}
			}
		}
		if(maxn)cout<<maxn;
		else cout<<-1;
	}
	if(m==5)
	{
		for(int i=1;i<=n;i++)
		{
			for(int j=i+1;j<=n;j++)
			{
				for(int k=j+1;k<=n;k++)
				{
					for(int l=k+1;l<=n;l++)
					{
						for(int r=l+1;r<=n;r++)
						{
							if(a[i]<=a[j]&&a[j]<=a[k]&&a[k]<=a[l]&&a[l]<=a[r]&&(b[i]!=b[j])&&(b[j]!=b[k])&&(b[i]!=b[k])&&(b[i]!=b[l])&&(b[j]!=b[l])&&(b[k]!=b[l])&&(b[i]!=b[r])&&(b[j]!=b[r])&&(b[l]!=b[r])&&(b[k]!=b[r]))
							{
								maxn=max(maxn,(c[i]+c[j]+c[k]+c[l]+c[r]));
							}
						}
					}
				}
			}
		}
		if(maxn)cout<<maxn;
		else cout<<-1;
	}
	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 3
1 2 3 4 5
1 2 3 2 1
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: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
// author: towrist

#include <bits/stdc++.h>
#define int long long

using namespace std;

/*
for any fixed A and B and any subset of {(i, j): 1<=i<=n, 1<=j<=n, i!=j} of length>=4n has atleast one element (a, b) such that (a!=A) and (b!=B)
*/

const int INF=1e12;

bool debg=false;

void solve()
{
    int n, m;
    cin >> n >> m;
    vector<int> a(n + 1, 0), b(n + 1, 0), c(n + 1, 0);
    for (int j = 1; j <= n; j++)
        cin >> a[j];
    for (int j = 1; j <= n; j++)
        cin >> b[j];
    for (int j = 1; j <= n; j++)
        cin >> c[j];

    // stores the top 4n {i, j} pairs from right and left for each index
    vector<vector<int>> left(n + 1, vector<int>(n+1, -INF));
    // LEFT[i][j] = maximum c-value of an index to the left of i whose b-value is j and a value>=i: O(n**2) compute ez
    vector<vector<pair<int, pair<int, int>>>> LEFT(n + 1);
    auto RIGHT=LEFT;
    auto right=left;

    auto in=[](const pair<int, int>& p, int q)
    {
        return (q==p.first || q==p.second);
    };

    // compute left and LEFT in this iteration
    {
        int best_val[n+1][n+1]={0};
        set<pair<int, pair<int, int>>> best[n+1];
        int curr[n+1][n+1]={0};
        for (int j=1; j<=n; j++)
        {
            for (int i=j-1; i>=1; i--) if (a[i]<=a[j] && b[i]!=b[j]) left[j][b[i]]=max(left[j][b[i]], c[i]);

            set<pair<int, pair<int, int>>> ptr;
            for (int i=1; i<=a[j]; i++) if (!best[i].empty()) ptr.insert(*best[i].rbegin());
            while (LEFT[j].size()<4*n && !ptr.empty())
            {
                auto pa=*ptr.rbegin();
                ptr.erase(pa);
                {
                    auto &s=best[curr[pa.second.first][pa.second.second]];
                    auto it=s.find(pa);
                    assert(it!=s.end());
                    if (it!=s.begin()) ptr.insert(*prev(it));
                }
                if (not(in(pa.second, b[j]))) LEFT[j].push_back(pa);
            }
            for (int i=1; i<=n; i++)
            {
                if (i!=b[j])
                {
                    if (best_val[i][b[j]]<left[j][i]+c[j])
                    {
                        best[curr[i][b[j]]].erase({best_val[i][b[j]], {i, b[j]}});
                        best_val[i][b[j]]=left[j][i]+c[j];
                        curr[i][b[j]]=a[j];
                        best[curr[i][b[j]]].insert({best_val[i][b[j]], {i, b[j]}});
                    }
                }
            }
        }
    }

    // compute right and RIGHT in this iteration
    {
        int best_val[n+1][n+1]={0};
        set<pair<int, pair<int, int>>> best[n+1];
        int curr[n+1][n+1]={0};
        for (int j=n; j>=1; j--)
        {
            for (int i=j+1; i<=n; i++) if (a[i]>=a[j] && b[i]!=b[j]) right[j][b[i]]=max(right[j][b[i]], c[i]);

            set<pair<int, pair<int, int>>> ptr;
            for (int i=a[j]; i<=n; i++) if (!best[i].empty()) ptr.insert(*best[i].rbegin());
            while (RIGHT[j].size()<4*n && !ptr.empty())
            {
                auto pa=*ptr.rbegin();
                ptr.erase(pa);
                {
                    auto &s=best[curr[pa.second.first][pa.second.second]];
                    auto it=s.find(pa);
                    assert(it!=s.end());
                    if (it!=s.begin()) ptr.insert(*prev(it));
                }
                if (not(in(pa.second, b[j]))) RIGHT[j].push_back(pa);
            }
            for (int i=1; i<=n; i++)
            {
                if (i!=b[j])
                {
                    if (best_val[b[j]][i]<right[j][i]+c[j])
                    {
                        best[curr[b[j]][i]].erase({best_val[b[j]][i], {b[j], i}});
                        best_val[b[j]][i]=right[j][i]+c[j];
                        curr[b[j]][i]=a[j];
                        best[curr[b[j]][i]].insert({best_val[b[j]][i], {b[j], i}});
                    }
                }
            }
        }
    }

    if (debg)
    {
       for (int j=1; j<=n; j++)
       {
        cout<<"j: "<<j<<'\n';
        cout<<"RIGHT[]\n";
        for (auto p: RIGHT[j]) cout<<p.first<<" "<<p.second.first<<" "<<p.second.second<<'\n';
       }
    }
    int res=0;
    if (m==1)
    {
        for (int j=1; j<=n; j++) res=max(res, c[j]);
    }
    else if (m==2)
    {
        for (int j=2; j<=n; j++) for (int i=1; i<=n; i++) if (i!=b[j]) res=max(res, c[j]+left[j][i]);
    }
    else if (m==3)
    {
        for (int j=1; j<=n; j++) for (auto p: RIGHT[j]) res=max(res, p.first+c[j]);
    }
    else if (m==4)
    {
        for (int j=2; j<=n; j++)
        {
            for (int i=1; i<=n; i++)
            {
                if (i!=b[j])
                {
                    for (auto p: RIGHT[j])
                    {
                        if (in(p.second, i)) continue;
                        res=max(res, c[j]+left[j][i]+p.first);
                    }
                }
            }
        }
    }
    else
    {
        for (int j=3; j<=n; j++)
        {
            for (auto pl: LEFT[j])
            {
                for (auto pr: RIGHT[j])
                {
                    if (in(pl.second, pr.second.first)||in(pl.second, pr.second.second)) continue;
                    res=max(res, c[j]+pl.first+pr.first);
                }
            }
        }
    }
    cout<<(res==0 ? -1 : res)<<'\n';
    return;
}

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

    int t;
    // cin>>t;
    t = 1;

    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 3
1 2 3 4 5
1 2 3 4 5
100 200 300 400 500
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import heapq
from typing import List, Tuple, Set
from functools import namedtuple

State = namedtuple('State', ['neg_sum_c', 'last_a', 'used_b', 'last_index', 'current_length'])

def main():
    import sys
    input = sys.stdin.read
    data = input().split()

    n = int(data[0])
    m = int(data[1])
    a = list(map(int, data[2:n+2]))
    b = list(map(int, data[n+2:2*n+2]))
    c = list(map(int, data[2*n+2:3*n+2]))

    # Create a list of indices sorted by c in decreasing order
    indices = sorted(range(n), key=lambda x: -c[x])

    # Initialize variables
    answer = -1
    heap = []
    visited = set()

    # Helper function to add states to heap
    def add_state(sum_c, last_a, used_b, last_index, current_length):
        state = State(-sum_c, last_a, frozenset(used_b), last_index, current_length)
        if state not in visited:
            heapq.heappush(heap, state)
            visited.add(state)

    # Initialize heap with single elements
    for idx in indices:
        add_state(c[idx], a[idx], {b[idx]}, idx, 1)

    while heap:
        state = heapq.heappop(heap)
        if state.current_length == m:
            answer = max(answer, -state.neg_sum_c)
            continue
        for next_idx in indices:
            if next_idx > state.last_index and a[next_idx] >= state.last_a and b[next_idx] not in state.used_b:
                add_state(-state.neg_sum_c + c[next_idx], a[next_idx], state.used_b | {b[next_idx]}, next_idx, state.current_length + 1)

    print(answer)

if __name__ == "__main__":
    main()

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases that can maximize the failure probability. Specifically, we need to consider scenarios where:

1. The subsequence length \( m \) is greater than the number of distinct values in \( b \).
2. The values in \( a \) are such that it is difficult to maintain the non-decreasing order.
3. The values in \( c \) are such that the maximum sum can be achieved in multiple ways, but the code might not handle all cases correctly.

Given the constraints \( 1 \le n \le 3000 \) and \( 1 \le m \le 5 \), we will create a small-scale test input. Here's a step-by-step approach to generate such a test input:

1. **Choose a small \( n \) and \( m \)**: Let's choose \( n = 6 \) and \( m = 3 \).
2. **Create sequence \( a \)**: Ensure that it has a mix of increasing and equal values to test the non-decreasing condition.
3. **Create sequence \( b \)**: Ensure that it has some repeated values to test the distinct condition.
4. **Create sequence \( c \)**: Ensure that the values are such that the maximum sum can be achieved in multiple ways, but not all ways are valid.

Here's a test input that meets these criteria:

```plaintext
6 3
1 2 2 3 3 4
1 2 2 3 3 4
10 5 10 1 1 2
```

### Explanation:
- **n = 6** and **m = 3**: We are testing a subsequence of length 3.
- **a = [1, 2, 2, 3, 3, 4]**: The sequence has a mix of increasing and equal values.
- **b = [1, 2, 2, 3, 3, 4]**: The sequence has repeated values, making it challenging to maintain the distinct condition.
- **c = [10, 5, 10, 1, 1, 2]**: The values are such that the maximum sum can be achieved in multiple ways, but not all ways are valid.

This test input should expose the bug in the code, particularly in handling the distinct condition for \( b \) and the non-decreasing condition for \( a \).


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
from itertools import combinations

# Read input values
n, m = map(int, input().split())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
c = list(map(int, input().split()))

# Initialize the maximum sum to a large negative number
ans = -1

# Generate all combinations of indices of length m
for p in combinations(range(n), m):
    # Check if the subsequence of a is non-decreasing
    if all(a[p[i]] <= a[p[i + 1]] for i in range(m - 1)):
        # Check if the subsequence of b has all distinct elements
        if len(set(b[i] for i in p)) == m:
            # Calculate the sum of the subsequence from c
            s = sum(c[i] for i in p)
            # Update the maximum sum if the current sum is greater
            if s > ans:
                ans = s

# Output the maximum sum found
print(ans)

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

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

Think step by step.

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

1. **Non-decreasing subsequence of `a`**: The subsequence of `a` should be non-decreasing.
2. **Distinct elements in `b`**: The subsequence of `b` should have all distinct elements.
3. **Maximizing the sum of `c`**: The subsequence should maximize the sum of `c`.

We need to create a scenario where the code might fail to find the correct subsequence or might incorrectly handle the conditions.

### Challenging Test Input

Let's consider a test case where the non-decreasing condition for `a` and the distinct condition for `b` are tightly constrained, and the values in `c` are such that the maximum sum is not immediately obvious.

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

### Explanation

- **`a` sequence**: `[1, 1, 2, 2, 3]`
  - The subsequence must be non-decreasing, so valid subsequences could be `[1, 1, 2]`, `[1, 2, 2]`, `[1, 2, 3]`, etc.
- **`b` sequence**: `[1, 2, 3, 4, 5]`
  - All elements in `b` are distinct, so any subsequence of length 3 will satisfy this condition.
- **`c` sequence**: `[10, 20, 30, 40, 50]`
  - The values in `c` are increasing, so the maximum sum will be achieved by the subsequence that includes the largest values of `c`.

### Correct Output

The correct output for this test case should be:

- The subsequence `[1, 2, 3]` with indices `[2, 3, 4]` (since `a[2] = 1`, `a[3] = 2`, `a[4] = 2` and `b[2] = 2`, `b[3] = 3`, `b[4] = 4` are distinct).
- The sum of `c` for this subsequence is `20 + 30 + 40 = 90`.

### Why This Test Case is Challenging

- The non-decreasing condition for `a` is tight, with multiple repeated values.
- The distinct condition for `b` is straightforward but must be checked carefully.
- The values in `c` are increasing, which means the code must correctly identify the subsequence that maximizes the sum.

This test case should expose any bugs in the code that might incorrectly handle the non-decreasing condition or the distinct condition for `b`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import java.util.*;

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

        int n = sc.nextInt();
        int m = sc.nextInt();

        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            b[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            c[i] = sc.nextInt();
        }

        int maxSum = -1;

        List<Integer> indices = new ArrayList<>();
        for (int i = 0; i < n; i++) indices.add(i);

        List<List<Integer>> combinations = generateCombinations(indices, m);

        for (List<Integer> subseq : combinations) {

            if (isValidSubsequence(subseq, a, b)) {

                int sum = 0;
                for (int index : subseq) {
                    sum += c[index];
                }

                maxSum = Math.max(maxSum, sum);
            }
        }

        System.out.println(maxSum);
        sc.close();
    }

    static boolean isValidSubsequence(List<Integer> subseq, int[] a, int[] b) {

        for (int i = 1; i < subseq.size(); i++) {
            if (a[subseq.get(i)] < a[subseq.get(i - 1)]) {
                return false;
            }
        }

        Set<Integer> bSet = new HashSet<>();
        for (int index : subseq) {
            if (!bSet.add(b[index])) {
                return false;
            }
        }
        return true;
    }

    static List<List<Integer>> generateCombinations(List<Integer> indices, int m) {
        List<List<Integer>> combinations = new ArrayList<>();
        generateCombinationsHelper(indices, m, 0, new ArrayList<>(), combinations);
        return combinations;
    }

    static void generateCombinationsHelper(List<Integer> indices, int m, int start, List<Integer> current, List<List<Integer>> result) {
        if (current.size() == m) {
            result.add(new ArrayList<>(current));
            return;
        }

        for (int i = start; i < indices.size(); i++) {
            current.add(indices.get(i));
            generateCombinationsHelper(indices, m, i + 1, current, result);
            current.remove(current.size() - 1);
        }
    }
}

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

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

Think step by step.

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

1. **Non-decreasing sequence in `a`**: The subsequence selected from `a` must be non-decreasing.
2. **Distinct elements in `b`**: The subsequence selected from `b` must have all distinct elements.
3. **Maximize the sum of `c`**: The subsequence should maximize the sum of `c` values.

The code provided generates all possible combinations of indices and checks each combination for validity. The bug might be in the combination generation or the validity check. To expose this, we can create a test case where the number of valid combinations is limited, making it easier to identify issues in the combination generation or validity check.

### Test Input
Let's create a small-scale test input with `n = 5` and `m = 3`. We will ensure that there are a few valid combinations to check the correctness of the code.

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

### Explanation
- **`n = 5` and `m = 3`**: We need to find a subsequence of length 3.
- **`a = [1, 2, 3, 4, 5]`**: This sequence is already non-decreasing, so any subsequence of length 3 will be valid in terms of the first condition.
- **`b = [1, 2, 3, 4, 5]`**: All elements are distinct, so any subsequence of length 3 will be valid in terms of the second condition.
- **`c = [10, 20, 30, 40, 50]`**: The values are increasing, so the maximum sum will be from the subsequence `[3, 4, 5]`.

### Expected Output
The expected output for this test input is:
```plaintext
120
```

This test input is designed to be simple enough to be checked manually but challenging enough to ensure the code handles the combination generation and validity check correctly. If the code is faulty, it will likely fail to generate or validate the correct subsequence, leading to an incorrect output.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import java.util.*;

public class TurtleAndThree {

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

        int n = sc.nextInt();
        int m = sc.nextInt();

        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            b[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            c[i] = sc.nextInt();
        }

        // Generate all combinations of indices of length m
        List<List<Integer>> allCombinations = generateCombinations(n, m);

        int maxSum = -1;

        // Evaluate each combination
        for (List<Integer> combination : allCombinations) {
            if (isValidSubsequence(a, b, combination)) {
                int currentSum = 0;
                for (int index : combination) {
                    currentSum += c[index];
                }
                maxSum = Math.max(maxSum, currentSum);
            }
        }

        // Print the result
        System.out.println(maxSum);

        sc.close();
    }

    private static List<List<Integer>> generateCombinations(int n, int m) {
        List<List<Integer>> result = new ArrayList<>();
        generateCombinations(new ArrayList<>(), 0, n, m, result);
        return result;
    }

    private static void generateCombinations(List<Integer> current, int start, int n, int m, List<List<Integer>> result) {
        if (current.size() == m) {
            result.add(new ArrayList<>(current));
            return;
        }

        for (int i = start; i < n; i++) {
            current.add(i);
            generateCombinations(current, i + 1, n, m, result);
            current.remove(current.size() - 1);
        }
    }

    private static boolean isValidSubsequence(int[] a, int[] b, List<Integer> indices) {
        Set<Integer> seen = new HashSet<>();
        int prevA = -1;

        for (int index : indices) {
            if (a[index] < prevA || !seen.add(b[index])) {
                return false;
            }
            prevA = a[index];
        }

        return true;
    }
}

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

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

Think step by step.

Response:
```plaintext
5 3
1 2 3 4 5
1 2 1 2 3
10 20 30 40 50
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

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

# define ll long long
# define I inline int
# define V inline void
# define rg register int
using namespace std;
const int N=3000+10,S=32;
int n,m;
int a[N],b[N],c[N];
int co[N];
ll f[N][S][6],g[N][S][6];
inline int lowbit(int x){
	return x&(-x);
}
inline void put(int x,ll y,int j,int k){
	for(rg i=x;i<=3000;i+=lowbit(i)) g[i][j][k]=max(g[i][j][k],y);
}
inline ll get(int x,int j,int k)
{
	ll sum=-1e18;
	for(rg i=x;i>0;i-=lowbit(i)) sum=max(sum,g[i][j][k]);
	return sum;
}
int main()
{
	scanf("%d%d",&n,&m);
	for(rg i=1;i<=n;i++) scanf("%d",&a[i]);
	for(rg i=1;i<=n;i++) scanf("%d",&b[i]);
	for(rg i=1;i<=n;i++) scanf("%d",&c[i]);
	ll ans=-1;
	for(rg t=1;t<=500;t++)
	{
		for(rg i=0;i<=n;i++) co[i]=0;
		for(rg i=1;i<=n;i++)
		if(!co[b[i]]) co[b[i]]=rand()%5+1;
//		for(rg i=1;i<=n;i++) printf("%d ",co[b[i]]);
//		printf("\n");
		for(rg i=0;i<=n;i++)
		for(rg j=0;j<32;j++)
		for(rg k=0;k<=5;k++) f[i][j][k]=g[i][j][k]=-1e16;
		f[0][0][0]=0;
		put(1,0,0,0);
		for(rg i=1;i<=n;i++)
		{
			for(rg j=0;j<32;j++)
			{
				for(rg k=0;k<5;k++)
				{
					int u=1<<(co[b[i]]-1);
					if(!(j&u)) f[i][j|u][k+1]=max(f[i][j|u][k+1],get(a[i],j,k)+c[i]);
				}
			}
			for(rg j=0;j<32;j++)
			{
				for(rg k=0;k<5;k++)
				{
					put(a[i],f[i][j][k],j,k);
				}
			}
		}
		for(rg j=0;j<32;j++) ans=max(ans,get(n,j,m));
	}
	printf("%lld",ans);
	return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases that can challenge the algorithm's assumptions or logic. Specifically, we should look for cases where the subsequence selection criteria are stringent, and the random assignment of `co[b[i]]` values might not cover all necessary conditions.

Here are the steps to generate a challenging test input:

1. **Ensure the length constraints**: The input should have \( n \) and \( m \) within the specified ranges.
2. **Create a sequence `a` that is sorted or has a specific pattern**: This will ensure the first condition is met.
3. **Create a sequence `b` with values that are difficult to satisfy the distinct condition**: This will ensure the second condition is challenging to meet.
4. **Create a sequence `c` with values that maximize the sum if the subsequence is selected correctly**.

Let's construct a test input:

- \( n = 5 \)
- \( m = 3 \)
- Sequence `a` should be sorted or have a specific pattern: \( a = [1, 2, 3, 4, 5] \)
- Sequence `b` should have values that are difficult to satisfy the distinct condition: \( b = [1, 1, 2, 2, 3] \)
- Sequence `c` should have values that maximize the sum if the subsequence is selected correctly: \( c = [100, 100, 100, 100, 100] \)

This input will challenge the algorithm to find a subsequence of length 3 that satisfies both conditions. The random assignment of `co[b[i]]` values might not cover all necessary conditions, leading to a failure in finding a valid subsequence.

Here is the test input:

```plaintext
5 3
1 2 3 4 5
1 1 2 2 3
100 100 100 100 100
```

This input ensures that the subsequence selection criteria are stringent, and the random assignment of `co[b[i]]` values might not cover all necessary conditions, potentially exposing the bug in the code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
/**
 * https://codeforces.com/contest/2003/submission/278736438
 *
 * (c) 2024 Diego Sogari
 */
#include <bits/stdc++.h>

using namespace std;

#ifdef ONLINE_JUDGE
#define debug
#else
#include "debug.h"
init();
#endif

void println(auto &&...args) { ((cout << args << ' '), ...) << endl; }

template <typename T> struct Num {
  T x;
  Num() { cin >> x; }
  Num(T a) : x(a) {}
  operator T &() { return x; }
  operator T() const { return x; }
};
using Int = Num<int>;

struct Digraph : vector<vector<int>> {
  const int n, m;
  Digraph(int n, int m = 0) : vector<vector<int>>(n + 1), n(n), m(m) {
    for (auto &[u, v] : vector<array<Int, 2>>(m)) {
      add(u, v);
    }
  }
  void add(int u, int v) { (*this)[u].push_back(v); }
};

mt19937 engine(random_device{}());

void solve(int t) { // O(n^2)
  Int n, m;
  vector<Int> a(n), b(n), c(n);
  Digraph g(n);
  for (int i = 0; i < n; i++) { // O(n^2)
    for (int j = 0; j < i; j++) {
      if (a[j] <= a[i]) {
        g.add(j, i);
      }
    }
  }
  map<int, unsigned> ids;
  for (auto &bi : b) {
    ids[bi] = engine();
  }
  unordered_set<int> vis;
  vector<map<unsigned, int>> memo(n + 1);
  auto f = [&](auto &self, int u, int rem, int taken) -> int {
    taken ^= ids[b[u]];
    auto it = memo[u].find(taken);
    if (it != memo[u].end()) {
      return it->second;
    }
    int ans = 0;
    if (rem > 1) {
      ans = -1;
      for (auto &v : g[u]) {
        if (vis.insert(b[v]).second) {
          ans = max(ans, self(self, v, rem - 1, taken));
          vis.erase(b[v]);
        }
      }
    }
    if (ans >= 0) {
      ans += c[u];
    }
    memo[u].emplace(taken, ans);
    return ans;
  };
  int ans = -1;
  for (int i = 0; i <= n - m; i++) { // O(n^2)
    vis.insert(b[i]);
    ans = max(ans, f(f, i, m, 0));
    vis.erase(b[i]);
  }
  println(ans);
}

int main() {
  cin.tie(nullptr)->tie(nullptr)->sync_with_stdio(false);
#ifdef ONLINE_JUDGE
  int t = 1;
#else
  Int t;
#endif
  for (int i = 1; i <= t; i++) {
    solve(i);
  }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios that are likely to cause issues. The code involves creating a directed graph and using dynamic programming to find the maximum sum of a subsequence that meets the given conditions. Here are the key points to consider:

1. **Subsequence Length and Constraints**:
   - The subsequence length \( m \) should be small (since \( 1 \le m \le 5 \)).
   - The sequences \( a \), \( b \), and \( c \) should be designed to test the constraints effectively.

2. **Conditions to Test**:
   - The subsequence should be non-decreasing in \( a \).
   - The elements in \( b \) should be distinct in the subsequence.
   - The sum of \( c \) should be maximized.

3. **Edge Cases**:
   - Test with sequences where it is impossible to form a valid subsequence.
   - Test with sequences where multiple valid subsequences exist, but the code might not choose the one with the maximum sum.
   - Test with sequences where the constraints are just barely met.

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

```plaintext
7 3
1 1 2 2 3 3 3
1 1 2 2 3 3 3
10 20 30 40 50 60 70
```

### Explanation:
- **Input**:
  - \( n = 7 \)
  - \( m = 3 \)
  - \( a \) sequence: \( [1, 1, 2, 2, 3, 3, 3] \)
  - \( b \) sequence: \( [1, 1, 2, 2, 3, 3, 3] \)
  - \( c \) sequence: \( [10, 20, 30, 40, 50, 60, 70] \)

- **Expected Output**:
  - The maximum sum of a valid subsequence of length 3 should be 120, which can be achieved by selecting indices \( [2, 4, 6] \) (values \( [20, 40, 60] \)).

- **Why This Test Case is Challenging**:
  - The \( a \) sequence has repeated values, which can lead to issues if the code does not handle non-strict inequalities correctly.
  - The \( b \) sequence has repeated values, which must be handled to ensure the subsequence condition is met.
  - The \( c \) sequence is designed to have a clear maximum sum, but the code must correctly identify the valid subsequence.

This test case should help expose any bugs in the code related to handling repeated values and ensuring the subsequence conditions are met.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
def find_neighbors(idx, path_lb):
    global la, lb, n
    neighbors = []
    val = la[idx]
    for i in range(idx + 1, n):
        if la[i] >= val and lb[i] not in path_lb:
            neighbors.append(i)
    return neighbors

def dfs_maxsum(node, current_sum, path_lb, current_length):
    global la, lb, lc, max_sum, m

    if current_length == m:
        max_sum = max(max_sum, current_sum)
        return

    for neighbor in find_neighbors(node, path_lb):
        dfs_maxsum(neighbor, current_sum + lc[neighbor], path_lb + [lb[neighbor]], current_length + 1)

# Ввод данных
n, m = map(int, input().split())
la = list(map(int, input().split()))
lb = list(map(int, input().split()))
lc = list(map(int, input().split()))

result = -1

for start_node in range(n):
    max_sum = -1
    dfs_maxsum(start_node, lc[start_node], [lb[start_node]], 1)
    result = max(result, max_sum)

print(result)

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the given code, we need to focus on edge cases that can highlight issues in the logic. Here are the steps to think through this:

1. **Sequence Length and Subsequence Length**: Ensure that the length of the sequences is small enough to be manageable but large enough to test the logic thoroughly. The subsequence length \( m \) should be set to a value that is challenging to satisfy the conditions.

2. **Non-Increasing \( a \) Values**: Create a sequence \( a \) where it is difficult to find a non-decreasing subsequence of length \( m \).

3. **Duplicate \( b \) Values**: Ensure that the sequence \( b \) has duplicate values, which can make it hard to find a subsequence with distinct \( b \) values.

4. **High \( c \) Values**: Assign high values to the \( c \) sequence to make the maximum sum more challenging to achieve.

5. **Ensure Validity**: The test case should be valid according to the problem constraints.

Let's construct such a test input:

```plaintext
6 3
3 2 2 3 4 4
1 2 2 1 3 3
10 20 30 40 50 60
```

### Explanation:
- **n = 6**: The length of the sequences is 6.
- **m = 3**: The required length of the subsequence is 3.
- **a = [3, 2, 2, 3, 4, 4]**: This sequence is non-increasing, making it difficult to find a non-decreasing subsequence of length 3.
- **b = [1, 2, 2, 1, 3, 3]**: This sequence has duplicate values, making it challenging to find a subsequence with distinct \( b \) values.
- **c = [10, 20, 30, 40, 50, 60]**: High values to make the maximum sum more significant.

This test input should expose the bug in the code by making it difficult to find a valid subsequence that satisfies both conditions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import random
import sys, os, io
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline

def f(u, v):
    return u << m ^ v

def update(i, j, x):
    i ^= l1
    tree[f(i, j)] = x
    i >>= 1
    while i:
        tree[f(i, j)] = max(tree[f(i << 1, j)], tree[f(i << 1 ^ 1, j)])
        i >>= 1
    return

def get_segment(s, t):
    s, t = s ^ l1, t ^ l1
    ans = []
    while s <= t:
        if s & 1:
            ans.append(s)
            s += 1
        s >>= 1
        if not t & 1:
            ans.append(t)
            t -= 1
        t >>= 1
    return ans

n, m = map(int, input().split())
a = list(map(int, input().split()))
b = list(map(int, input().split()))
c = list(map(int, input().split()))
l1 = pow(2, (n + 1).bit_length())
l2 = 2 * l1
pow2 = [1]
for _ in range(m):
    pow2.append(2 * pow2[-1])
l = pow2[m]
x = [0] * (n + 1)
inf = pow(10, 9) + 1
ans, z = -inf, 20
while z:
    z -= 1
    for i in range(1, n + 1):
        x[i] = pow2[random.randint(0, m - 1)]
    tree = [-inf] * (l2 << m)
    update(0, 0, 0)
    for i, j, k in zip(a, b, c):
        seg, u = get_segment(0, i), x[j]
        for v in range(l):
            if u & v:
                continue
            ma = -inf
            for w in seg:
                ma = max(ma, tree[f(w, v)])
            if 0 <= ma + k > tree[f(i ^ l1, u ^ v)]:
                update(i, u ^ v, ma + k)
    if ans < tree[f(1, l - 1)]:
        ans, z = tree[f(1, l - 1)], 20
ans = max(ans, -1)
print(ans)

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

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

Think step by step.

Response:
```plaintext
5 3
1 1 1 1 1
1 2 3 4 5
100 200 300 400 500
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
n, m = [int(i) for i in input().split(" ")]
a = [int(i) for i in input().split(" ")]
b = [int(i) for i in input().split(" ")]
c = [int(i) for i in input().split(" ")]

graph = [[0 for _ in range(n)] for _ in range(n)]

for i in range(n):
    for j in range(i + 1, n):
        if a[i] <= a[j]:
            graph[i][j] = 1

out = []

def make_paths(node, target=m, acc=0, b_path=()):
    if target == 1:
        out.append(acc)
    else:
        for next_node in (i for i in range(n) if graph[node][i] == 1 and b[i] not in b_path):
            make_paths(next_node, target=target - 1, acc=acc + c[next_node], b_path=b_path + (b[next_node],))

for i in range(n):
    make_paths(i, acc=c[i], b_path=(b[i],))

if out:
    print(max(out))
else:
    print(-1)

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

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

Think step by step.

Response:
```plaintext
5 2
1 2 3 4 5
1 1 2 2 3
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: Turtle and Three Sequences
time_limit_ms: 3000
memory_limit_mb: 256
Description: Piggy gives Turtle three sequences $a_1, a_2, \ldots, a_n$, $b_1, b_2, \ldots, b_n$, and $c_1, c_2, \ldots, c_n$.

Turtle will choose a subsequence of $1, 2, \ldots, n$ of length $m$, let it be $p_1, p_2, \ldots, p_m$. The subsequence should satisfy the following conditions:

  * $a_{p_1} \le a_{p_2} \le \cdots \le a_{p_m}$;    * All $b_{p_i}$ for all indices $i$ are pairwise distinct, i.e., there don't exist two different indices $i$, $j$ such that $b_{p_i} = b_{p_j}$.

Help him find the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$, or tell him that it is impossible to choose a subsequence of length $m$ that satisfies the conditions above.

Recall that a sequence $a$ is a subsequence of a sequence $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements.

Input: The first line contains two integers $n$ and $m$ ($1 \le n \le 3000$, $1 \le m \le 5$) — the lengths of the three sequences and the required length of the subsequence.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le n$) — the elements of the sequence $a$.

The third line contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le n$) — the elements of the sequence $b$.

The fourth line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($1 \le c_i \le 10^4$) — the elements of the sequence $c$.

Output: Output a single integer — the maximum value of $\sum\limits_{i = 1}^m c_{p_i}$. If it is impossible to choose a subsequence of length $m$ that satisfies the conditions above, output $-1$.

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

input:
7 3
1 4 5 2 3 6 7
1 2 2 1 1 3 2
1 5 6 7 3 2 4
output:
13

input:
5 3
1 2 3 4 5
1 1 2 1 2
5 4 3 2 1
output:
-1

Note: In the first example, we can choose $p = [1, 2]$, then $c_{p_1} + c_{p_2} = 1 + 4 = 5$. We can't choose $p = [2, 4]$ since $a_2 > a_4$, violating the first condition. We can't choose $p = [2, 3]$ either since $b_2 = b_3$, violating the second condition. We can choose $p = [1, 4]$, but $c_1 + c_4 = 4$, which isn't maximum.

In the second example, we can choose $p = [4, 6, 7]$.

In the third example, it is impossible to choose a subsequence of length $3$ that satisfies both of the conditions.

Buggy Code:
import java.util.*;

public class TurtleSubsequence {
    private static int maxSum = -1;
    public static void maxSumOfSubsequence(int index, int chosenCount, int n, int m, int[] a, int[] b, int[] c, int currentSum, int lastA, Set<Integer> bSet) {
        if (chosenCount == m) {
            maxSum = Math.max(maxSum, currentSum);
            return;
        }
        if (index >= n) {
            return;
        }
        maxSumOfSubsequence(index + 1, chosenCount, n, m, a, b, c, currentSum, lastA, bSet);
        if (a[index] >= lastA && !bSet.contains(b[index])) {
            bSet.add(b[index]);
            maxSumOfSubsequence(index + 1, chosenCount + 1, n, m, a, b, c, currentSum + c[index], a[index], bSet);
            bSet.remove(b[index]);
        }
    }

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

        int n = sc.nextInt();
        int m = sc.nextInt();

        int[] a = new int[n];
        int[] b = new int[n];
        int[] c = new int[n];

        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            b[i] = sc.nextInt();
        }
        for (int i = 0; i < n; i++) {
            c[i] = sc.nextInt();
        }

        maxSum = -1;
        Set<Integer> bSet = new HashSet<>();
        maxSumOfSubsequence(0, 0, n, m, a, b, c, 0, -1, bSet);
        System.out.println(maxSum);
    }
}

**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 2
1 2 3 4 5
1 1 2 2 3
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: Set
time_limit_ms: 2000
memory_limit_mb: 512
Description: Define the binary encoding of a finite set of natural numbers $T \subseteq \\{0,1,2,\ldots\\}$ as $f(T) = \sum\limits_{i \in T} 2^i$. For example, $f(\\{0,2\\}) = 2^0 + 2^2 = 5$ and $f(\\{\\}) = 0$. Notice that $f$ is a bijection from all such sets to all non-negative integers. As such, $f^{-1}$ is also defined.

You are given an integer $n$ along with $2^n-1$ sets $V_1,V_2,\ldots,V_{2^n-1}$.

Find all sets $S$ that satisfy the following constraint:

  * $S \subseteq \\{0,1,\ldots,n-1\\}$. Note that $S$ can be empty.    * For all non-empty subsets $T \subseteq \\{0,1,\ldots,n-1\\}$, $|S \cap T| \in V_{f(T)}$.

Due to the large input and output, both input and output will be given in terms of binary encodings of the sets.

Input: The first line of input contains a single integer $n$ ($1 \leq n \leq 20$).

The second line of input contains $2^n-1$ integers $v_1,v_2,\ldots,v_{2^n-1}$ ($0 \leq v_i < 2^{n+1}$) — the sets $V_i$ given in their binary encoding where $V_i = f^{-1}(v_i)$.

Output: The first line of output should contain an integer $k$ indicating the number of possible $S$.

In the following $k$ lines, you should output $f(S)$ for all possible $S$ in increasing order.

Examples:
input:
3
15 15 15 15 15 15 12
output:
4
3
5
6
7

input:
5
63 63 63 63 6 63 63 63 63 63 63 5 63 63 63 63 63 63 8 63 63 63 63 2 63 63 63 63 63 63 63
output:
1
19

Note: In the first test case, one possible $S$ is $f^{-1}(3) = \\{0,1\\}$. All the non-empty subsets $T \subseteq \\{0,1,2\\}$ and the corresponding $|S \cap T|$, $f(T)$ and $V_f(T)$ are as follows:

$T$| $|S\cap T|$| $f(T)$| $V_{f(T)}$   ---|---|---|---   $\\{0\\}$| $1$| $1$| $\\{0,1,2,3\\}$   $\\{1\\}$| $1$| $2$| $\\{0,1,2,3\\}$   $\\{2\\}$| $0$| $4$| $\\{0,1,2,3\\}$   $\\{0,1\\}$| $2$| $3$| $\\{0,1,2,3\\}$   $\\{0,2\\}$| $1$| $5$| $\\{0,1,2,3\\}$   $\\{1,2\\}$| $1$| $6$| $\\{0,1,2,3\\}$   $\\{0,1,2\\}$| $2$| $7$| $\\{2,3\\}$

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
const int MAX_N = 2e5 + 5;
int n, values[MAX_N], dp[MAX_N];
vector<int> results;
int main() {
 ios_base::sync_with_stdio(false);
 cin.tie(NULL);
 cout.tie(NULL);
 cin >> n;
 for (int i = 1; i < (1 << n); i++) {
 cin >> values[i];
 }
 dp[(1 << n) - 1] = 1;
 for (int i = (1 << n) - 2; i >= 0; i--) {
 for (int j = 0; j < n; j++) {
 if (i & (1 << j)) {
 if (__builtin_popcount(values[i]) == __builtin_popcount(values[i ^ (1 << j)] + dp[i ^ (1 << j)])) {
 dp[i] |= (1 << j);
 }
 }
 }
 }
 for (int i = 0; i < (1 << n); i++) {
 if (dp[i]) {
 results.push_back(i);
 }
 }
 cout << results.size() << '\n';
 for (auto result : results) {
 cout << result << '\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 code, we need to focus on edge cases and scenarios that can maximize the failure probability. Let's break down the problem and the code to understand where the bug might be.

### Understanding the Code
The code uses a dynamic programming (DP) approach to determine which sets \( S \) satisfy the given constraints. The DP array `dp` is used to keep track of valid sets. The key part of the code is the nested loop where it checks the intersection sizes and updates the DP array.

### Potential Issues
1. **Incorrect Use of `__builtin_popcount`**: The function `__builtin_popcount` returns the number of set bits in an integer. The code uses this function to compare the number of elements in sets, which might be incorrect if the sets are not properly encoded.
2. **Logical Errors in DP Updates**: The code might have logical errors in how it updates the DP array, especially in the condition `if (__builtin_popcount(values[i]) == __builtin_popcount(values[i ^ (1 << j)] + dp[i ^ (1 << j)]))`.

### Challenging Test Input
To expose these potential issues, we can create a test input that:
- Has a small \( n \) to keep the input and output manageable.
- Includes sets \( V_i \) that are tricky to handle, such as sets with varying sizes and specific patterns.

Let's create a test input with \( n = 4 \) and specific values for \( V_i \) that can expose the bug.

```plaintext
4
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15
```

### Explanation
- **\( n = 4 \)**: This is a small enough \( n \) to keep the input and output manageable.
- **\( V_i = 15 \) for all \( i \)**: The value 15 in binary is `1111`, which represents the set \(\{0, 1, 2, 3\}\). This means that every set \( T \) should intersect with \( S \) in a way that the size of the intersection is in \(\{0, 1, 2, 3\}\).

### Expected Output
The expected output for this input should be all possible sets \( S \) that satisfy the constraints. Since every set \( T \) can intersect with \( S \) in any size from 0 to 3, all subsets of \(\{0, 1, 2, 3\}\) should be valid.

```plaintext
16
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
```

This test input is designed to ensure that the code correctly handles the constraints and updates the DP array correctly. If the code fails to produce the correct output, it will expose the bug in the logic or the use of `__builtin_popcount`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 2000
memory_limit_mb: 512
Description: Define the binary encoding of a finite set of natural numbers $T \subseteq \\{0,1,2,\ldots\\}$ as $f(T) = \sum\limits_{i \in T} 2^i$. For example, $f(\\{0,2\\}) = 2^0 + 2^2 = 5$ and $f(\\{\\}) = 0$. Notice that $f$ is a bijection from all such sets to all non-negative integers. As such, $f^{-1}$ is also defined.

You are given an integer $n$ along with $2^n-1$ sets $V_1,V_2,\ldots,V_{2^n-1}$.

Find all sets $S$ that satisfy the following constraint:

  * $S \subseteq \\{0,1,\ldots,n-1\\}$. Note that $S$ can be empty.    * For all non-empty subsets $T \subseteq \\{0,1,\ldots,n-1\\}$, $|S \cap T| \in V_{f(T)}$.

Due to the large input and output, both input and output will be given in terms of binary encodings of the sets.

Input: The first line of input contains a single integer $n$ ($1 \leq n \leq 20$).

The second line of input contains $2^n-1$ integers $v_1,v_2,\ldots,v_{2^n-1}$ ($0 \leq v_i < 2^{n+1}$) — the sets $V_i$ given in their binary encoding where $V_i = f^{-1}(v_i)$.

Output: The first line of output should contain an integer $k$ indicating the number of possible $S$.

In the following $k$ lines, you should output $f(S)$ for all possible $S$ in increasing order.

Examples:
input:
3
15 15 15 15 15 15 12
output:
4
3
5
6
7

input:
5
63 63 63 63 6 63 63 63 63 63 63 5 63 63 63 63 63 63 8 63 63 63 63 2 63 63 63 63 63 63 63
output:
1
19

Note: In the first test case, one possible $S$ is $f^{-1}(3) = \\{0,1\\}$. All the non-empty subsets $T \subseteq \\{0,1,2\\}$ and the corresponding $|S \cap T|$, $f(T)$ and $V_f(T)$ are as follows:

$T$| $|S\cap T|$| $f(T)$| $V_{f(T)}$   ---|---|---|---   $\\{0\\}$| $1$| $1$| $\\{0,1,2,3\\}$   $\\{1\\}$| $1$| $2$| $\\{0,1,2,3\\}$   $\\{2\\}$| $0$| $4$| $\\{0,1,2,3\\}$   $\\{0,1\\}$| $2$| $3$| $\\{0,1,2,3\\}$   $\\{0,2\\}$| $1$| $5$| $\\{0,1,2,3\\}$   $\\{1,2\\}$| $1$| $6$| $\\{0,1,2,3\\}$   $\\{0,1,2\\}$| $2$| $7$| $\\{2,3\\}$

Buggy Code:
#include<bits/stdc++.h>
#define ep emplace_back
#define pii pair<int,int>
#define ll long long
#define F(i,l,r) for(int i=(l),i##end=(r);i<=i##end;++i)
#define G(i,l,r) for(int i=(l),i##end=(r);i>=i##end;--i)
#define x first
#define y second
#define ep emplace_back
using namespace std;
int n,a[1<<20];
vector<int> st[1<<20];
vector<int> out;
int tmp=0;
void dfs(int x)
{
 	if(x==0)
	{
		F(i,0,1) if((1<<i)&st[1].back()) out.ep(tmp|i);
		return ;
	}
	int mn=1e9;
	F(i,1,(1<<x)-1)
	{
	 	int sk=st[i].back()&st[i+(1<<x)].back();
		mn=min(mn,sk);
		st[i].ep(sk);
	}
	if(mn) dfs(x-1);
	F(i,1,(1<<x)-1) st[i].pop_back();
	tmp|=1<<x;mn=1e9;
	F(i,1,(1<<x)-1)
	{
	 	int sk=st[i].back()&(st[i+(1<<x)].back()>>1);
		mn=min(mn,sk);
		st[i].ep(sk);
	}
	if(mn) dfs(x-1);
	F(i,1,(1<<x)-1) st[i].pop_back();
	tmp-=1<<x;
}
int main()
{
	ios::sync_with_stdio(false);cin.tie(0);
 	cin>>n;
	F(i,1,(1<<n)-1)
	{
	 	cin>>a[i];st[i].ep(a[i]);
	}
	dfs(n-1);
	cout<<out.size()<<endl;sort(out.begin(),out.end());
	for(auto u:out) cout<<u<<endl;
}

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

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

Think step by step.

Response:
To create 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. **Understand the Bug**: The provided code seems to have issues with handling the intersection sizes and the bit manipulation. Specifically, the `dfs` function may not correctly handle the intersections and the bit masking.

2. **Edge Cases**:
   - **Empty Sets**: Test with an empty set.
   - **Single Element Sets**: Test with sets that have only one element.
   - **Full Sets**: Test with sets that include all elements.
   - **Mixed Sets**: Test with sets that have a mix of elements.

3. **Input Constraints**:
   - \( n \) should be small enough to manage but large enough to test the code's ability to handle multiple sets.
   - The sets \( V_i \) should be chosen to test the code's ability to handle different intersection sizes.

4. **Generate the Test Input**:
   - Choose \( n = 3 \) to keep the input small but still challenging.
   - Set \( V_i \) such that it includes a mix of valid and invalid sets to test the code's ability to correctly identify valid sets \( S \).

Given these considerations, here is a small-scale test input that should expose the bug:

```plaintext
3
7 7 7 7 7 7 14
```

### Explanation:
- **n = 3**: This means we are dealing with sets of elements from \(\{0, 1, 2\}\).
- **V_i = [7, 7, 7, 7, 7, 7, 14]**:
  - \( f(\{0, 1, 2\}) = 7 \) (binary: 111)
  - \( f(\{0, 1\}) = 3 \) (binary: 011)
  - \( f(\{0, 2\}) = 5 \) (binary: 101)
  - \( f(\{1, 2\}) = 6 \) (binary: 110)
  - \( f(\{0\}) = 1 \) (binary: 001)
  - \( f(\{1\}) = 2 \) (binary: 010)
  - \( f(\{2\}) = 4 \) (binary: 100)

The sets \( V_i \) are chosen to include all elements (7) and a mix of other valid and invalid sets (14). This should challenge the code to correctly handle the intersections and bit masking.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 2000
memory_limit_mb: 512
Description: Define the binary encoding of a finite set of natural numbers $T \subseteq \\{0,1,2,\ldots\\}$ as $f(T) = \sum\limits_{i \in T} 2^i$. For example, $f(\\{0,2\\}) = 2^0 + 2^2 = 5$ and $f(\\{\\}) = 0$. Notice that $f$ is a bijection from all such sets to all non-negative integers. As such, $f^{-1}$ is also defined.

You are given an integer $n$ along with $2^n-1$ sets $V_1,V_2,\ldots,V_{2^n-1}$.

Find all sets $S$ that satisfy the following constraint:

  * $S \subseteq \\{0,1,\ldots,n-1\\}$. Note that $S$ can be empty.    * For all non-empty subsets $T \subseteq \\{0,1,\ldots,n-1\\}$, $|S \cap T| \in V_{f(T)}$.

Due to the large input and output, both input and output will be given in terms of binary encodings of the sets.

Input: The first line of input contains a single integer $n$ ($1 \leq n \leq 20$).

The second line of input contains $2^n-1$ integers $v_1,v_2,\ldots,v_{2^n-1}$ ($0 \leq v_i < 2^{n+1}$) — the sets $V_i$ given in their binary encoding where $V_i = f^{-1}(v_i)$.

Output: The first line of output should contain an integer $k$ indicating the number of possible $S$.

In the following $k$ lines, you should output $f(S)$ for all possible $S$ in increasing order.

Examples:
input:
3
15 15 15 15 15 15 12
output:
4
3
5
6
7

input:
5
63 63 63 63 6 63 63 63 63 63 63 5 63 63 63 63 63 63 8 63 63 63 63 2 63 63 63 63 63 63 63
output:
1
19

Note: In the first test case, one possible $S$ is $f^{-1}(3) = \\{0,1\\}$. All the non-empty subsets $T \subseteq \\{0,1,2\\}$ and the corresponding $|S \cap T|$, $f(T)$ and $V_f(T)$ are as follows:

$T$| $|S\cap T|$| $f(T)$| $V_{f(T)}$   ---|---|---|---   $\\{0\\}$| $1$| $1$| $\\{0,1,2,3\\}$   $\\{1\\}$| $1$| $2$| $\\{0,1,2,3\\}$   $\\{2\\}$| $0$| $4$| $\\{0,1,2,3\\}$   $\\{0,1\\}$| $2$| $3$| $\\{0,1,2,3\\}$   $\\{0,2\\}$| $1$| $5$| $\\{0,1,2,3\\}$   $\\{1,2\\}$| $1$| $6$| $\\{0,1,2,3\\}$   $\\{0,1,2\\}$| $2$| $7$| $\\{2,3\\}$

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

int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n;
    cin >> n;
    int total = (1 << n) -1;
    vector<unsigned int> V(total,0);
    for(int i=0;i<total;i++) cin >> V[i];
    // Handle singleton constraints
    // For each i from 0 to n-1, T = {i}, f(T) = 1<<i
    // So V_i corresponds to f(T) = 1<<i
    // We need to map f(T)=1<<i to index i in V
    // Wait, input is V_1 to V_{2^n -1}, indexed by mask from 1 to 2^n -1
    // So V[mask] where mask from 1 to 2^n -1
    // Now, for singleton T={i}, mask = 1<<i
    // So for each i, mask = 1<<i corresponds to V[mask]
    // Determine fixed bits and flexible bits
    vector<int> fixed_bits(n, -1); // -1: flexible, 0: not included, 1: included
    bool possible = true;
    for(int i=0;i<n;i++){
        int mask = 1<<i;
        unsigned int v = V[mask];
        bool has0 = (v & (1u << 0)) !=0;
        bool has1 = (v & (1u << 1)) !=0;
        if(has0 && !has1){
            fixed_bits[i] = 0;
        }
        else if(!has0 && has1){
            fixed_bits[i] = 1;
        }
        else if(!has0 && !has1){
            possible = false;
            break;
        }
        // else both are allowed, leave as -1
    }
    if(!possible){
        cout << 0;
        return 0;
    }
    // Now, collect flexible bits
    vector<int> flex_bits;
    unsigned int fixed_S =0;
    for(int i=0;i<n;i++){
        if(fixed_bits[i]==1){
            fixed_S |= (1<<i);
        }
        else if(fixed_bits[i]==-1){
            flex_bits.push_back(i);
        }
    }
    int m = flex_bits.size();
    // Now, iterate over all 2^m assignments
    // To store valid S's
    vector<int> valid_S;
    // Precompute all V_T
    // Convert V to vectors of allowed counts
    // We'll use bitmask to represent allowed counts
    // precompute popcount maximum
    // The maximum |S ∩ T| can be up to min(|S|, |T|)
    // Since n is small, up to 20
    // To speed up, we'll precompute all V_T
    // But it's already in V
    // Now, iterate over all possible flexible assignments
    for(int assign=0; assign<(1<<m); assign++){
        int S = fixed_S;
        for(int j=0; j<m; j++){
            if(assign & (1<<j)) S |= (1<<flex_bits[j]);
        }
        // Now, need to check all T's, |S ∩ T| ∈ V[T]
        // Since n=20, and 2^20 is ~1e6, we need to check efficiently
        // Idea: iterate over all T's in order of increasing mask
        bool ok = true;
        // To speed up, precompute popcount for S
        // Perhaps iterate over mask and check popcount
        // But n=20, and mask from 1 to (1<<n)-1
        // To speed up, iterate over all possible T's and check
        // Using fast popcount
        for(int mask=1; mask<(1<<n); mask++){
            // Compute |S ∩ T|
            // popcount(S & mask)
            int cnt = __builtin_popcount(S & mask);
            // Check if cnt is allowed in V[mask-1] (since V index is from 0 to (1<<n)-1, mask from 1 to (1<<n)-1)
            if( (V[mask-1] & (1u << cnt)) ==0 ){
                ok = false;
                break;
            }
        }
        if(ok){
            valid_S.push_back(S);
        }
    }
    // Now, sort valid_S in increasing order
    sort(valid_S.begin(), valid_S.end());
    // Output
    cout << valid_S.size() << "\n";
    for(auto s : valid_S){
        cout << s << "\n";
    }
}

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

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

Think step by step.

Response:
```plaintext
4
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Set
time_limit_ms: 2000
memory_limit_mb: 512
Description: Define the binary encoding of a finite set of natural numbers $T \subseteq \\{0,1,2,\ldots\\}$ as $f(T) = \sum\limits_{i \in T} 2^i$. For example, $f(\\{0,2\\}) = 2^0 + 2^2 = 5$ and $f(\\{\\}) = 0$. Notice that $f$ is a bijection from all such sets to all non-negative integers. As such, $f^{-1}$ is also defined.

You are given an integer $n$ along with $2^n-1$ sets $V_1,V_2,\ldots,V_{2^n-1}$.

Find all sets $S$ that satisfy the following constraint:

  * $S \subseteq \\{0,1,\ldots,n-1\\}$. Note that $S$ can be empty.    * For all non-empty subsets $T \subseteq \\{0,1,\ldots,n-1\\}$, $|S \cap T| \in V_{f(T)}$.

Due to the large input and output, both input and output will be given in terms of binary encodings of the sets.

Input: The first line of input contains a single integer $n$ ($1 \leq n \leq 20$).

The second line of input contains $2^n-1$ integers $v_1,v_2,\ldots,v_{2^n-1}$ ($0 \leq v_i < 2^{n+1}$) — the sets $V_i$ given in their binary encoding where $V_i = f^{-1}(v_i)$.

Output: The first line of output should contain an integer $k$ indicating the number of possible $S$.

In the following $k$ lines, you should output $f(S)$ for all possible $S$ in increasing order.

Examples:
input:
3
15 15 15 15 15 15 12
output:
4
3
5
6
7

input:
5
63 63 63 63 6 63 63 63 63 63 63 5 63 63 63 63 63 63 8 63 63 63 63 2 63 63 63 63 63 63 63
output:
1
19

Note: In the first test case, one possible $S$ is $f^{-1}(3) = \\{0,1\\}$. All the non-empty subsets $T \subseteq \\{0,1,2\\}$ and the corresponding $|S \cap T|$, $f(T)$ and $V_f(T)$ are as follows:

$T$| $|S\cap T|$| $f(T)$| $V_{f(T)}$   ---|---|---|---   $\\{0\\}$| $1$| $1$| $\\{0,1,2,3\\}$   $\\{1\\}$| $1$| $2$| $\\{0,1,2,3\\}$   $\\{2\\}$| $0$| $4$| $\\{0,1,2,3\\}$   $\\{0,1\\}$| $2$| $3$| $\\{0,1,2,3\\}$   $\\{0,2\\}$| $1$| $5$| $\\{0,1,2,3\\}$   $\\{1,2\\}$| $1$| $6$| $\\{0,1,2,3\\}$   $\\{0,1,2\\}$| $2$| $7$| $\\{2,3\\}$

Buggy Code:
def main():
    import sys
    input = sys.stdin.read
    data = input().split()

    n = int(data[0])
    v = list(map(int, data[1:1 + (1 << n) -1]))

    # Step 1: Determine parity constraints
    parity = {}
    for i in range(1, 1 << n):
        vi = v[i-1]
        V_i = set()
        j = 0
        while vi:
            if vi & 1:
                V_i.add(j)
            vi >>= 1
            j += 1
        # Determine parity
        all_even = all(x % 2 == 0 for x in V_i)
        all_odd = all(x % 2 == 1 for x in V_i)
        if all_even:
            parity[i] = 0
        elif all_odd:
            parity[i] = 1
        else:
            parity[i] = -1  # No constraint

    # Step 2: Collect constraints for Gaussian elimination
    equations = []
    for T, p in parity.items():
        if p == -1:
            continue
        eq = [0] * n
        for j in range(n):
            if T & (1 << j):
                eq[j] = 1
        eq.append(p)
        equations.append(eq)

    # Step 3: Gaussian elimination
    m = len(equations)
    mat = equations.copy()
    rank = 0
    for col in range(n):
        pivot = -1
        for row in range(rank, m):
            if mat[row][col] == 1:
                pivot = row
                break
        if pivot == -1:
            continue
        mat[pivot], mat[rank] = mat[rank], mat[pivot]
        for row in range(m):
            if row != rank and mat[row][col] == 1:
                for c in range(n + 1):
                    mat[row][c] ^= mat[rank][c]
        rank += 1

    # Find the basis
    basis = []
    free_vars = []
    for var in range(n):
        is_pivot = False
        for row in range(rank):
            if mat[row][var] == 1:
                is_pivot = True
                break
        if not is_pivot:
            free_vars.append(var)
    for b in range(1 << len(free_vars)):
        sol = [0] * n
        for i, var in enumerate(free_vars):
            sol[var] = (b >> i) & 1
        for row in range(rank):
            res = mat[row][-1]
            for col in range(n):
                if mat[row][col] == 1:
                    res ^= sol[col]
            if res != 0:
                break
        else:
            basis.append(sum(1 << i for i in range(n) if sol[i]))

    # Step 4: Validate solutions against original constraints
    valid_S = []
    for s in basis:
        valid = True
        for T in range(1, 1 << n):
            intersect_size = bin(s & T).count('1')
            Vi = set()
            j = 0
            vi = v[T-1]
            while vi:
                if vi & 1:
                    Vi.add(j)
                vi >>= 1
                j += 1
            if intersect_size not in Vi:
                valid = False
                break
        if valid:
            valid_S.append(s)

    # Output
    print(len(valid_S))
    for s in sorted(valid_S):
        print(s)

if __name__ == '__main__':
    main()

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

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

Think step by step.

Response:
```plaintext
4
15 15 15 15 15 15 15 15 15 15 15 15 15 15 15
```