**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tree of Life
time_limit_ms: 2000
memory_limit_mb: 512
Description: In the heart of an ancient kingdom grows the legendary Tree of Life — the only one of its kind and the source of magical power for the entire world. The tree consists of $n$ nodes. Each node of this tree is a magical source, connected to other such sources through magical channels (edges). In total, there are $n-1$ channels in the tree, with the $i$-th channel connecting nodes $v_i$ and $u_i$. Moreover, there exists a unique simple path through the channels between any two nodes in the tree.

However, the magical energy flowing through these channels must be balanced; otherwise, the power of the Tree of Life may disrupt the natural order and cause catastrophic consequences. The sages of the kingdom discovered that when two magical channels converge at a single node, a dangerous "magical resonance vibration" occurs between them. To protect the Tree of Life and maintain its balance, it is necessary to select several paths and perform special rituals along them. A path is a sequence of distinct nodes $v_1, v_2, \ldots, v_k$, where each pair of adjacent nodes $v_i$ and $v_{i+1}$ is connected by a channel. When the sages perform a ritual along such a path, the resonance vibration between the channels $(v_i, v_{i+1})$ and $(v_{i+1}, v_{i+2})$ is blocked for each $1 \leq i \leq k - 2$.

The sages' task is to select the minimum number of paths and perform rituals along them to block all resonance vibrations. This means that for every pair of channels emanating from a single node, there must exist at least one selected path that contains both of these channels.

Help the sages find the minimum number of such paths so that the magical balance of the Tree of Life is preserved, and its power continues to nourish the entire world!

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 4 \cdot 10^4$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \leq n \leq 5 \cdot 10^5$) — the number of nodes in the Tree of Life.

The $i$-th of the following $n - 1$ lines of each test case contains two integers $v_i$ and $u_i$ ($1 \leq v_i < u_i \leq n$) — the channel connecting nodes $v_i$ and $u_i$.

It is guaranteed that there exists a unique simple path through the channels between any two nodes.

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

Output: For each test case, output a single integer — the minimum number of paths that the sages need to select to prevent a catastrophe.

Examples:
input:
5
4
1 2
2 3
3 4
2
1 2
4
1 2
1 3
1 4
8
3 7
2 4
1 2
2 5
3 6
1 3
3 8
6
2 3
1 2
3 6
1 5
1 4
output:
1
0
3
7
3

Note: In the first test case, there are two pairs of channels emanating from a single node: $(1, 2)$ and $(2, 3)$, $(2, 3)$ and $(3, 4)$. It is sufficient to perform the ritual along the path $1-2-3-4$. Thus, the answer is $1$.

In the second test case, there are no pairs of channels emanating from a single node, so the answer is $0$.

In the third test case, rituals can be performed along the paths $2-1-3$, $2-1-4$, and $3-1-4$.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
vector<int>D[500009];
int dp[500009];
long long wyn,WYN;
void DFS(int v, int last)
{
    vector<int>A;
    for(int som : D[v])
    {
        if(som==last){continue;}
        DFS(som,v);
        A.push_back(dp[som]);
    }
    if(A.size()==0)
    {
        dp[v]=1;
        return;
    }
    wyn+=(long long)A.size()*(long long)(A.size()-1)/2;
    int s=0;
    for(int i=0;i<A.size();i++)
    {
        A[i]-=A.size()-1;
        A[i]=max(A[i],0);
        dp[v]+=max(A[i],1);
    }
    if(last==-1)
    {
        for(int i=1;i<A.size();i++)
        {
            if(A[i]>A[0])
            {
                swap(A[i],A[0]);
            }
        }
        int a=0,b=A[0];
        for(int i=1;i<A.size();i++)
        {
            a+=A[i];
        }
        if(a>=b)
        {
            wyn+=(a+b+1)/2;
        }
        else
        {
            wyn+=b;
        }
    }
}
int main()
{ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
    int N;
    cin>>N;
for(int I=1;I<=N;I++)
{
    int n;
    cin>>n;
    for(int i=1;i<=n;i++)
    {
        D[i].clear();
        dp[i]=0;
    }
    for(int i=1;i<n;i++)
    {
        int a,b;
        cin>>a>>b;
        D[a].push_back(b);
        D[b].push_back(a);
    }
    if(n<=2){cout<<0<<'\n';continue;}
    WYN=numeric_limits<long long>::max();
    for(int i=1;i<=n;i++)
    {
        if(D[i].size()>1)
        {
            wyn=0;
            DFS(i,-1);
            WYN=min(WYN,wyn);
        }
    }
    cout<<WYN<<'\n';
}
    return 0;
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios that maximize the failure probability. Here are some key points to consider:

1. **Small-scale Test Case**: The test case should be small enough to be easily manageable and fit within the constraints.
2. **Edge Cases**: Focus on trees with specific structures that can expose issues in the algorithm, such as:
   - Trees with a single node.
   - Trees with two nodes.
   - Trees with a linear structure (a chain).
   - Trees with a star structure (one central node with multiple children).
   - Trees with a more complex structure, such as a balanced binary tree.

Given the constraints and the need to focus on small-scale tests, let's create a test case that includes a star structure and a chain structure to challenge the algorithm.

### Test Case Explanation
- **Star Structure**: A tree where one node (the root) has multiple children. This structure can expose issues with how the algorithm handles high-degree nodes.
- **Chain Structure**: A tree where nodes form a single linear path. This structure can expose issues with how the algorithm handles paths.

### Test Input
```plaintext
3
4
1 2
1 3
1 4
3
1 2
2 3
5
1 2
2 3
3 4
4 5
```

### Explanation
1. **Test Case 1**: A star structure with 4 nodes.
   - Node 1 is the root with children 2, 3, and 4.
   - This structure should require 3 paths to block all resonance vibrations.

2. **Test Case 2**: A chain structure with 3 nodes.
   - Nodes 1, 2, and 3 form a linear path.
   - This structure should require 1 path to block all resonance vibrations.

3. **Test Case 3**: A chain structure with 5 nodes.
   - Nodes 1, 2, 3, 4, and 5 form a linear path.
   - This structure should require 1 path to block all resonance vibrations.

This test input is designed to challenge the buggy code by presenting both a star structure and a chain structure, which are common edge cases in tree problems.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tree of Life
time_limit_ms: 2000
memory_limit_mb: 512
Description: In the heart of an ancient kingdom grows the legendary Tree of Life — the only one of its kind and the source of magical power for the entire world. The tree consists of $n$ nodes. Each node of this tree is a magical source, connected to other such sources through magical channels (edges). In total, there are $n-1$ channels in the tree, with the $i$-th channel connecting nodes $v_i$ and $u_i$. Moreover, there exists a unique simple path through the channels between any two nodes in the tree.

However, the magical energy flowing through these channels must be balanced; otherwise, the power of the Tree of Life may disrupt the natural order and cause catastrophic consequences. The sages of the kingdom discovered that when two magical channels converge at a single node, a dangerous "magical resonance vibration" occurs between them. To protect the Tree of Life and maintain its balance, it is necessary to select several paths and perform special rituals along them. A path is a sequence of distinct nodes $v_1, v_2, \ldots, v_k$, where each pair of adjacent nodes $v_i$ and $v_{i+1}$ is connected by a channel. When the sages perform a ritual along such a path, the resonance vibration between the channels $(v_i, v_{i+1})$ and $(v_{i+1}, v_{i+2})$ is blocked for each $1 \leq i \leq k - 2$.

The sages' task is to select the minimum number of paths and perform rituals along them to block all resonance vibrations. This means that for every pair of channels emanating from a single node, there must exist at least one selected path that contains both of these channels.

Help the sages find the minimum number of such paths so that the magical balance of the Tree of Life is preserved, and its power continues to nourish the entire world!

Input: Each test consists of multiple test cases. The first line contains a single integer $t$ ($1 \leq t \leq 4 \cdot 10^4$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \leq n \leq 5 \cdot 10^5$) — the number of nodes in the Tree of Life.

The $i$-th of the following $n - 1$ lines of each test case contains two integers $v_i$ and $u_i$ ($1 \leq v_i < u_i \leq n$) — the channel connecting nodes $v_i$ and $u_i$.

It is guaranteed that there exists a unique simple path through the channels between any two nodes.

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

Output: For each test case, output a single integer — the minimum number of paths that the sages need to select to prevent a catastrophe.

Examples:
input:
5
4
1 2
2 3
3 4
2
1 2
4
1 2
1 3
1 4
8
3 7
2 4
1 2
2 5
3 6
1 3
3 8
6
2 3
1 2
3 6
1 5
1 4
output:
1
0
3
7
3

Note: In the first test case, there are two pairs of channels emanating from a single node: $(1, 2)$ and $(2, 3)$, $(2, 3)$ and $(3, 4)$. It is sufficient to perform the ritual along the path $1-2-3-4$. Thus, the answer is $1$.

In the second test case, there are no pairs of channels emanating from a single node, so the answer is $0$.

In the third test case, rituals can be performed along the paths $2-1-3$, $2-1-4$, and $3-1-4$.

Buggy Code:
import sys
from typing import List, Tuple

class TreeOfLifeBalancer:
    def __init__(self, total_nodes: int):
        self.total_nodes = total_nodes
        self.parent_node = [0] * (total_nodes + 1)
        self.paths_required = [0] * (total_nodes + 1)
        self.extra_path_capacity = [0] * (total_nodes + 1)
        self.adjacency_list = [[] for _ in range(total_nodes + 1)]
        self.total_resonance_value = 0

    def add_edge(self, node_a: int, node_b: int) -> None:
        self.adjacency_list[node_a].append(node_b)
        self.adjacency_list[node_b].append(node_a)

    def calculate_initial_resonance_value(self) -> None:
        for i in range(1, self.total_nodes + 1):
            degree = len(self.adjacency_list[i])
            self.total_resonance_value += degree * (degree - 1) // 2

    def depth_first_search(self, current_node: int) -> None:
        self.paths_required[current_node] = 0
        self.extra_path_capacity[current_node] = 0

        node_degree_minus_parent = len(self.adjacency_list[current_node]) - 1
        total_subtree_paths = 0
        maximum_subtree = (0, 0)

        for connected_node in self.adjacency_list[current_node]:
            if self.parent_node[current_node] != connected_node:
                self.parent_node[connected_node] = current_node
                self.depth_first_search(connected_node)

                subtree_paths = self.paths_required[connected_node]

                if subtree_paths < node_degree_minus_parent:
                    adjustment = min(
                        self.extra_path_capacity[connected_node],
                        (node_degree_minus_parent + 1 - subtree_paths) // 2
                    )
                    self.extra_path_capacity[connected_node] -= adjustment
                    subtree_paths += adjustment * 2

                if subtree_paths >= node_degree_minus_parent:
                    self.total_resonance_value -= node_degree_minus_parent
                    subtree_paths -= node_degree_minus_parent
                    total_subtree_paths += subtree_paths
                    maximum_subtree = max(maximum_subtree, (subtree_paths, self.extra_path_capacity[connected_node]))
                else:
                    self.total_resonance_value -= subtree_paths

                self.extra_path_capacity[current_node] += self.extra_path_capacity[connected_node]

        if maximum_subtree[0] * 2 > total_subtree_paths:
            adjustment = self.extra_path_capacity[current_node] - maximum_subtree[1]
            self.extra_path_capacity[current_node] -= adjustment
            total_subtree_paths += adjustment * 2

        balanced_paths = min(total_subtree_paths // 2, total_subtree_paths - maximum_subtree[0])
        self.extra_path_capacity[current_node] += balanced_paths
        self.paths_required[current_node] = node_degree_minus_parent + total_subtree_paths - 2 * balanced_paths

    def compute_minimum_paths(self) -> int:
        self.depth_first_search(1)
        return self.total_resonance_value - self.extra_path_capacity[1]

    def reset_tree_data(self) -> None:
        for i in range(1, self.total_nodes + 1):
            self.adjacency_list[i].clear()
            self.parent_node[i] = 0

def main():
    input = sys.stdin.read
    data = input().split()
    idx = 0
    number_of_test_cases = int(data[idx])
    idx += 1

    for _ in range(number_of_test_cases):
        total_nodes = int(data[idx])
        idx += 1

        tree_balancer = TreeOfLifeBalancer(total_nodes)

        for _ in range(total_nodes - 1):
            node_a = int(data[idx])
            node_b = int(data[idx + 1])
            idx += 2
            tree_balancer.add_edge(node_a, node_b)

        tree_balancer.calculate_initial_resonance_value()
        print(tree_balancer.compute_minimum_paths())

        tree_balancer.reset_tree_data()

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
5
4
1 2
2 3
3 4
2
1 2
4
1 2
1 3
1 4
8
3 7
2 4
1 2
2 5
3 6
1 3
3 8
6
2 3
1 2
3 6
1 5
1 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

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

#define ll long long
#define pb push_back
#define pii pair<int,int>
#define sz size()

const int N = 2e5;
const ll mod = 1e9+7;
const int INF = 1e9;

void solve() {
    int n,k;
    cin>>n>>k;
    int a[n+1];
    map<int,int> cnt;
    for(int i = 1; i <= n; i++) {
        cin>>a[i];
        cnt[a[i]]++;
    }
    sort(a+1,a+n+1);
    vector<int> v;
    v.pb(a[1]);
    for(int i = 2; i <= n; i++) {
        if(v.back()!=a[i]) {
            v.pb(a[i]);
        }
    }
    int m=v.sz;
    int pref[m];
    memset(pref,0,sizeof(pref));
    pref[0]=cnt[v[0]];
    for(int i = 1; i < m; i++) {
        pref[i]=pref[i-1]+cnt[v[i]];
    }
    // for(int i = 0 ; i < m; i++) {
    //     cout<<pref[i]<<" ";
    // }
    // cout<<"\n";
    int ans=0;
    for(int i = 0; i < m; i++) {
        int j=i;
        while(j<m and v[j]+1==v[j+1])
            j++;
        // cout<<i<<" "<<j<<"\n";
        if(j-k+1<0) {
            int x=pref[j];
            if(i-1>=0)
                x-=pref[i-1];
            ans=max(ans,x);
        }
        for(int _k = i; _k <= j-k+1; _k++) {
            // cout<<_k+k-1<<" "<<_k<<" "<<pref[_k+k-1]-pref[_k-1]<<"\n";
            int x=pref[_k+k-1];
            if(_k-1>=0)
                x-=pref[_k-1];
            ans=max(ans,x);
        }
        i=j;
    }
    cout<<ans<<"\n";
}

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

    int t=1;
    cin>>t;
    while(t--) {
        solve();
    }
    return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
5 3
1 2 3 4 5
4 2
1 2 2 3
6 1
1 1 1 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

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

#define int long long
#define fr first
#define sn second
#define pb push_back
#define all(x) (x).begin(), (x).end()
#define MODULO 1000000007
#define v(x) vector<x>
#define endl '\n'
#define CY cout << "YES" << endl
#define CN cout << "NO" << endl

template <typename T> void inp(v(T)& vec, int n = 1) {for (T value; n--;) cin >> value,vec.pb(value);}
template <typename T> void out(v(T)& vec, int n, int m) {while(n<m) {cout << vec[n] << " "; n++;} cout << endl;}

void __print(int x) {cerr << x;}
void __print(unsigned x) {cerr << x;}
void __print(unsigned int x) {cerr << x;}
void __print(float x) {cerr << x;}
void __print(double x) {cerr << x;}
void __print(long double x) {cerr << x;}
void __print(char x) {cerr << '\'' << x << '\'';}
void __print(const char *x) {cerr << '\"' << x << '\"';}
void __print(const string &x) {cerr << '\"' << x << '\"';}
void __print(bool x) {cerr << (x ? "true" : "false");}
template<typename T, typename V>
void __print(const pair<T, V> &x) {cerr << '{'; __print(x.first); cerr << ','; __print(x.second); cerr << '}';}
template<typename T>
void __print(const T &x) {int f = 0; cerr << '{'; for (auto &i: x) cerr << (f++ ? "," : ""), __print(i); cerr << "}";}
void _print() {cerr << "]\n";}
template <typename T, typename... V>
void _print(T t, V... v) {__print(t); if (sizeof...(v)) cerr << ", "; _print(v...);}
#ifndef ONLINE_JUDGE
#define debug(x...) cerr << "[" << #x << "] = ["; _print(x)
#else
#define debug(x...)
#endif

void solve()
{
    int n; cin >> n;
    int k; cin>>k;
    v(int)x(n);
    map<int,int>mp;
    for (int i = 0; i < n; i++)
    {
        cin>>x[i];
        mp[x[i]]++;
    }
    debug(mp);
    sort(all(x));
    int ans=0;

    for(const auto &pair:mp)
    {
        if (pair.first>x[n-1])
        {
            break;
        }

        int sum=0;
        for (int i = pair.first; i <= pair.first+k-1; i++)
        {
            sum=sum+mp[i];
            debug(sum);
            debug(mp);
        }
        ans=max(sum,ans);

    }

    cout<<ans<<endl;

}

int32_t main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    int t; cin >> t;
    while (t--)
        solve();
    return 0;
}

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

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

Think step by step.

Response:
```plaintext
3
10 2
1 1 1 1 1 2 2 2 2 2
5 1
1 2 3 4 5
9 3
1 1 2 2 3 3 4 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

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

signed main() {
    int t;
    cin >> t;
    while (t--) {
        int n, k;
        cin >> n >> k;
        vector<int> cards(n);
        for (int i = 0; i < n; i++) {
            cin >> cards[i];
        }
        sort(cards.begin(), cards.end());
        unordered_map<int, int> freq;
        int maxCards = 0, distinct = 0, left = 0;
        for (int right = 0; right < n; right++) {
            if (freq[cards[right]] == 0) {
                distinct++;
            }
            freq[cards[right]]++;

            while (distinct > k) {
                freq[cards[left]]--;
                if (freq[cards[left]] == 0) {
                    distinct--;
                }
                left++;
            }
            if (right > 0 && (cards[right] == cards[right - 1] || cards[right] == cards[right - 1] + 1)) {
                maxCards = max(maxCards, right - left + 1);
            }
        }

        cout << maxCards << endl;
    }

    return 0;
}

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

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

Think step by step.

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

**Explanation:**
- This test case is designed to expose a potential issue in the buggy code where the algorithm incorrectly handles the condition of taking cards with the same or consecutive numbers.
- The input consists of a deck of 5 cards with numbers 1, 2, 3, 4, and 5, and Monocarp can take up to 2 distinct numbers.
- The expected output should be 4, as Monocarp can take cards with numbers 1, 2, 2, and 3 (or 2, 3, 3, and 4), but the buggy code might fail to correctly handle the transition between distinct numbers, leading to an incorrect result.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
for _ in range(int(input())):
    n,k = map(int,input().split())
    lst = list(map(int,input().split()))
    dict = {}
    value = []
    for i in lst:
        if i in dict:
            dict[i] += 1
        else:
            dict[i] = 1
            value.append(i)

    value.sort()

    i=0
    last =-1
    ans=0
    sum=0
    for j in range(len(value)):
        if last+1 == value[j]:
            sum += dict[value[j]]
            if j-i+1 > k:
                sum -= dict[value[i]]
        else:
            sum = dict[value[j]]
            i = j
        last = value[j]
        ans = max(ans,sum)
    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
10 3
1 2 3 4 5 1 2 3 4 5
5 1
1 2 3 4 5
7 2
1 1 2 2 3 3 3
4 2
1 1 2 2
6 3
1 2 3 2 1 3
```

### Explanation:
1. **Test Case 1:**
   - **Input:** `10 3 1 2 3 4 5 1 2 3 4 5`
   - **Expected Output:** `6`
   - **Explanation:** Monocarp can take cards with numbers 1, 2, and 3. He can take all cards with 1, 2, and 3, totaling 6 cards.

2. **Test Case 2:**
   - **Input:** `5 1 1 2 3 4 5`
   - **Expected Output:** `1`
   - **Explanation:** Monocarp can only take one card since he can only take cards with the same number, and there are no duplicates.

3. **Test Case 3:**
   - **Input:** `7 2 1 1 2 2 3 3 3`
   - **Expected Output:** `4`
   - **Explanation:** Monocarp can take cards with numbers 1 and 2. He can take all cards with 1 and 2, totaling 4 cards.

4. **Test Case 4:**
   - **Input:** `4 2 1 1 2 2`
   - **Expected Output:** `4`
   - **Explanation:** Monocarp can take all cards with numbers 1 and 2, totaling 4 cards.

5. **Test Case 5:**
   - **Input:** `6 3 1 2 3 2 1 3`
   - **Expected Output:** `6`
   - **Explanation:** Monocarp can take cards with numbers 1, 2, and 3. He can take all cards with 1, 2, and 3, totaling 6 cards.

These test cases are designed to cover various edge cases, including sequences with repeated numbers, sequences with no repeated numbers, and sequences where the number of distinct numbers is exactly equal to the limit `k`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import sys; input = lambda: sys.stdin.buffer.readline().decode().strip(); print = lambda *args, **kwargs: sys.stdout.write(" ".join(map(str, args)) + "\n", **kwargs)
def InputList(Dtype=int): return list(map(Dtype, input().split()))
from random import randint; RND = randint(1, 10**9);
from collections import deque, defaultdict
from math import lcm, inf, gcd, ceil
from bisect import bisect_left, bisect_right
from functools import lru_cache, cache
from types import GeneratorType
from heapq import heapify, heappop, heappush

def bootstrap(f, stack=[]):
	def wrappedfunc(*args, **kwargs):
		if stack: return f(*args, **kwargs)
		else:
			to = f(*args, **kwargs)
			while True:
				if type(to) is GeneratorType:
					stack.append(to)
					to = next(to)
				else:
					stack.pop()
					if not stack: break
					to = stack[-1].send(to)
			return to
	return wrappedfunc

def solve():
	# print(f"=============================")
	N, k = InputList()
	nums = InputList()
	cnt = defaultdict(int)
	for n in nums: cnt[RND^n] += 1
	distinct = [x^RND for x in cnt.keys()]
	distinct.sort()
	# print(distinct)
	# def ok(x):
		# pass

	# def binarySearch()

	r = 0 # right pointer
	window = 0
	start = distinct[0]
	end = bisect_left(distinct, start+k-1)
	if end < len(distinct) and distinct[end] > start+k-1: end -= 1
	for j in range(r, end+1):
		if j >= len(distinct): break
		window += cnt[distinct[j]^RND]
	r = end + 1
	best = window

	for i in range(1, len(distinct)):
		window -= cnt[distinct[i-1]^RND] # remove the previous left pointer
		start = distinct[i]
		end = bisect_left(distinct, start+k-1) # inclusive of end we need
		if end < len(distinct) and distinct[end] > start+k-1: end -= 1

		for j in range(r, end+1):
			if j >= len(distinct): break
			window += cnt[distinct[j]^RND]
		r = end + 1
		# if end < len(distinct): print(f"start: {start}, end: {distinct[end]}, window: {window}")
		# else: print(f"start: {start}, endInd: {end}, window: {window}")
		best = max(best, window)
	return best

# print(solve())
for i in range(t:=int(input())):
	print(solve())

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's consider the following steps:

1. **Understand the Bug**: The buggy code uses a random number `RND` to XOR the numbers in the input list. This is a non-standard approach and can lead to incorrect results if the same number appears multiple times or if the XOR operation results in the same value for different numbers.

2. **Edge Cases**:
   - **Duplicate Numbers**: If the same number appears multiple times, the XOR operation can result in the same value for different indices.
   - **Consecutive Numbers**: If the numbers are consecutive, the XOR operation can still result in the same value for different indices.
   - **Large Numbers**: Large numbers can also cause issues with the XOR operation.

3. **Test Case**:
   - We will create a test case with a small number of cards where the numbers are such that the XOR operation with `RND` can lead to incorrect results.

Let's create a test case where the numbers are consecutive and appear multiple times. This will test the edge case where the XOR operation might not work as expected.

```plaintext
1
5 2
1 1 2 2 3
```

In this test case:
- There are 5 cards with numbers 1, 1, 2, 2, 3.
- The maximum number of distinct numbers that can be taken is 2.
- The correct answer should be 4 (taking two 1s and two 2s).

This test case should expose the bug in the buggy code because the XOR operation might not correctly handle the duplicate numbers and consecutive numbers.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
def game(cards, k):
    cards.sort()
    max_lenght = 0
    start = 0
    for i in range(len(cards)):
        while cards[min(i + max_lenght+1, len(cards))-1] - cards[i] <= k-1:
            if i + max_lenght > len(cards) - 1:
                break
            for x, y in enumerate(cards[i:min(i + max_lenght+1, len(cards))-1]):
                if cards[x+i+1] - cards[x + i] > 1:
                    break
            else:
                max_lenght += 1

    return max_lenght

A = int(input())
for j in range(A):
    n, k = map(int, input().split())
    deck = list(map(int, input().split()))
    print(game(deck, k))

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

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

Think step by step.

Response:
```plaintext
4
10 2
1 2 3 4 5 6 7 8 9 10
5 1
1 1 1 1 1
9 3
1 2 3 1 2 3 1 2 3
3 2
1 2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import java.util.*;

public class Main {
    static Scanner sc = new Scanner(System.in);
    static long MOD = (long) Math.pow(10, 9) + 7;

    public static long[] makearr(int n) {
        long arr[] = new long[n];
        for (int i = 0; i < n; i++) {
            arr[i] = sc.nextLong();
        }
        return arr;
    }

    public static void main(String[] args) {
        int t = sc.nextInt();
        while (t-- > 0) {
            int n = sc.nextInt();
            int k = sc.nextInt();
            long arr[] = makearr(n);
            long fre[] = new long[20000007];

            for (long i : arr) {
                fre[(int) i]++;
            }

            long maxSum = 0, currentSum = 0;
            for (int j = 0; j < k; j++) {
                currentSum += fre[j];
            }
            maxSum = currentSum;

            for (int j = k; j < fre.length; j++) {
                currentSum = currentSum - fre[j - k] + fre[j];
                maxSum = Math.max(maxSum, currentSum);
            }

            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
1
10 3
1 2 3 1 2 3 1 2 3 4
```

### Explanation:
This test case is designed to challenge the buggy code in the following ways:

1. **Distinct Numbers**: The distinct numbers in the array are 1, 2, 3, and 4. The maximum number of distinct numbers that can be taken is 3, which matches the value of `k`.

2. **Repeating Numbers**: The numbers 1, 2, and 3 are repeated multiple times, which means Monocarp can potentially take many cards without violating the rules.

3. **Edge Case for `k`**: The value of `k` is 3, which means Monocarp can take cards with numbers 1, 2, and 3. However, the array also contains a number 4, which cannot be taken without violating the rule of not exceeding `k` distinct numbers.

4. **Order of Numbers**: The numbers are arranged in a way that Monocarp can take a sequence of 1, 2, 3, 1, 2, 3, 1, 2, 3, but he cannot take the last 4 without violating the rule.

The expected output for this test case should be 9, as Monocarp can take all the cards with numbers 1, 2, and 3, but not the card with number 4. The buggy code, however, might incorrectly handle the frequency array or the sliding window, leading to an incorrect result.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import java.util.*;

public class mat {

    public static int rr(int n){
        if(n<3){
            return 1;
        }
        return 1+rr(n/3);
    }
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while (t-- > 0) {
           int n = sc.nextInt();
           int k = sc.nextInt();
           int arr[]= new int[n];
           for(int i=0;i<n;i++){
            arr[i]= sc.nextInt();
           }
           Arrays.sort(arr);
           int a=1;
           int c=1;
           int max=0;
           int z=-1;
        //    2 2 3 3 3 4 4 4 5 5
           for(int i=1;i<n;i++){
            if(arr[i]-arr[i-1]>1){
                max = Math.max(max,c);
                a=1;
                c=1;
            }else if(arr[i]-arr[i-1]==1 && a<k){
                a++;
                c++;
                z=i;
            }else if(arr[i]-arr[i-1]==1 && a>=k){
                max = Math.max(max,c);
                if(z==-1){
                    z=i;
                }
                c-=z;
                c++;
                z=i-z;
            }else{
                c++;
            }
           }
           max = Math.max(max,c);
           System.out.println(max);

        }
        sc.close();
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
int main(){
    long long tt;
    cin>>tt;
    while(tt>0){
        long long n,k;
        cin>>n>>k;
        vector<int>v;
        map<long long,long long>mp;
        for(long long i=0;i<n;i++){
            long long x;
            cin>>x;
            if(mp.find(x)==mp.end()) v.push_back(x);
            mp[x]++;
        }
        int ans=0;
        int mx=0;
        sort(v.begin(),v.end());
        int count=0;
        for(int i=0;i<v.size();i++){
            if(count==0){
                mx+=(mp[v[i]]);
                count++;
                ans=max(ans,mx);
            }
            else{
                if(v[i]-v[i-1]==1){
                    if(count<k){
                        mx+=(mp[v[i]]);
                        count++;
                    }
                    else{
                        ans=max(ans,mx);
                        mx-=(mp[v[i-k]]);
                        mx+=(mp[v[i]]);
                    }
                }
                else{
                    ans=max(ans,mx);
                    mx=0;
                    count=0;
                }
            }
        }

         cout<<(max(ans,mx))<<endl;

        tt--;
    }
}

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

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

Think step by step.

Response:
```plaintext
3
5 2
1 2 3 4 5
10 3
1 2 2 3 3 3 4 4 4 4
20 4
1 2 2 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 6 6
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
#include <bits/stdc++.h>
#define ll long long
#define con cout << "NO\n";
#define coy cout << "YES\n";
#define co1 cout << "-1\n";
#define all(v) v.begin(), v.end()
#define int long long
#define IOS                  \
ios::sync_with_stdio(0); \
cin.tie(0);              \
cout.tie(0);
const int N = 2e5 + 10;
using namespace std;
ll t, n, k,a[N], b[N];
void solve()
{
        int n, k;
        cin >> n >> k;

        vector<int> v(n);
        for (int i = 0; i < n; i++) {
            cin >> v[i];
        }
        sort(v.begin(), v.end());

        int mx = 0;
        int left = 0;
        map<int, int> mp;

        for (int i = 0; i < n; i++) {
            mp[v[i]]++;
            while (mp.size() > k) {
                mp[v[left]]--;
                if (mp[v[left]] == 0) {
                    mp.erase(v[left]);
                }
                left++;
            }
            if (i > 0 && v[i] > v[i - 1] + 1) {
                left = i;
                mp.clear();
            }
            mx= max(mx, i - left + 1);
        }

        cout << mx << endl;

}
 main() {
    IOS

    int t;
    cin >> t;

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

    return 0;
}

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the provided code, we need to focus on edge cases where the code's logic might fail. Specifically, the code has a condition that resets the `left` pointer and clears the map when the current card's value is more than one greater than the previous card's value. This condition might not handle certain edge cases correctly.

Let's consider the following scenarios:

1. **Consecutive Numbers with Gaps**: The code might not handle gaps correctly, especially when the gap is exactly 2.
2. **Multiple Gaps**: The code might fail when there are multiple gaps in the sequence.
3. **Edge Cases with Small `k`**: The code might not handle small values of `k` correctly, especially when `k` is 1 or 2.

Based on these scenarios, let's create a test input that includes a sequence with gaps and small `k` values to expose the bug.

### Test Input
```plaintext
3
5 2
1 3 3 3 3
6 2
1 2 4 4 4 4
7 3
1 3 4 4 4 4 4
```

### Explanation
1. **Test Case 1**:
   - `n = 5`, `k = 2`
   - Sequence: `1 3 3 3 3`
   - The sequence has a gap of 2 between 1 and 3. The expected output should be 5 because Monocarp can take all 3s after taking one 1.

2. **Test Case 2**:
   - `n = 6`, `k = 2`
   - Sequence: `1 2 4 4 4 4`
   - The sequence has a gap of 2 between 2 and 4. The expected output should be 5 because Monocarp can take one 1, one 2, and then all 4s.

3. **Test Case 3**:
   - `n = 7`, `k = 3`
   - Sequence: `1 3 4 4 4 4 4`
   - The sequence has a gap of 2 between 1 and 3. The expected output should be 7 because Monocarp can take one 1, one 3, and then all 4s.

This test input should expose the bug in the code by challenging its handling of gaps and small `k` values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import sys
from collections import Counter

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

    idx = 0
    t = int(data[idx])
    idx += 1
    results = []

    for _ in range(t):
        n = int(data[idx])
        k = int(data[idx + 1])
        idx += 2
        a = list(map(int, data[idx:idx + n]))
        idx += n

        freq = Counter(a)
        unique_sorted = sorted(freq.keys())

        max_sum = 0
        left = 0
        current_sum = 0
        distinct_count = 0
        num_counts = {}

        for right in range(len(unique_sorted)):
            num = unique_sorted[right]
            if freq[num] == 0:
                continue
            if num not in num_counts:
                distinct_count += 1
                num_counts[num] = 0
            num_counts[num] += freq[num]
            current_sum += freq[num]

            while distinct_count > k or (unique_sorted[right] - unique_sorted[left] > k - 1):
                left_num = unique_sorted[left]
                num_counts[left_num] -= freq[left_num]
                current_sum -= freq[left_num]
                if num_counts[left_num] == 0:
                    del num_counts[left_num]
                    distinct_count -= 1
                left += 1

            if distinct_count <= k and unique_sorted[right] - unique_sorted[left] <= k - 1:
                max_sum = max(max_sum, current_sum)

        results.append(str(max_sum))

    print('\n'.join(results))

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 expose the bug in the provided buggy code, we need to focus on edge cases that challenge the logic of the sliding window and the way the distinct counts are managed. The bug seems to be in how the window is adjusted and the distinct counts are maintained. Here are the key points to consider:

1. **Sliding Window Adjustment**: The window should correctly adjust to ensure that the number of distinct numbers does not exceed \( k \) and the difference between the largest and smallest numbers in the window does not exceed \( k - 1 \).

2. **Handling Duplicates**: The code should correctly handle cases where the same number appears multiple times.

3. **Edge Cases**: Test with small \( n \) and \( k \) values to ensure the logic is sound.

Let's create a small-scale test input that can expose the bug:

- **Test Case 1**: A simple case where \( k = 1 \) and all numbers are different.
- **Test Case 2**: A case where \( k = 2 \) and the numbers are in a pattern that should be challenging to handle.
- **Test Case 3**: A case with repeated numbers and \( k = 2 \).

Here is the test input:

```plaintext
3
5 1
1 2 3 4 5
10 2
1 2 1 2 1 2 3 4 3 4
9 2
1 1 2 2 3 3 4 4 5
```

Explanation:
1. **Test Case 1**: \( n = 5 \), \( k = 1 \)
   - The numbers are 1, 2, 3, 4, 5.
   - Monocarp can only take one number, so the maximum number of cards is 1.

2. **Test Case 2**: \( n = 10 \), \( k = 2 \)
   - The numbers are 1, 2, 1, 2, 1, 2, 3, 4, 3, 4.
   - Monocarp can take a maximum of 6 cards (1, 2, 1, 2, 1, 2).

3. **Test Case 3**: \( n = 9 \), \( k = 2 \)
   - The numbers are 1, 1, 2, 2, 3, 3, 4, 4, 5.
   - Monocarp can take a maximum of 8 cards (1, 1, 2, 2, 3, 3, 4, 4).

This test input should help in identifying the bug in the sliding window logic and the management of distinct counts.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import sys
from collections import Counter

input = sys.stdin.readline
output = []

for _ in range(int(input())):
    n, k = map(int, input().split())
    a = list(map(int, input().split()))
    cnt = Counter(a)

    uq = sorted(cnt.keys())
    ans = 0
    cur = 0
    d = 0
    l = 0
    for r in range(len(uq)):
        if uq[r] not in cnt.keys():
            continue
        cur += cnt[uq[r]]
        d += 1
        while uq[r] - uq[l] >= k:
            if uq[l] not in cnt.keys():
                break
            cur -= cnt[uq[l]]
            d -= 1
            l += 1
        ans = max(ans, cur)
    output.append(ans)

sys.stdout.write('\n'.join(map(str, output)) + '\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
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import java.util.*;
import java.util.stream.Collectors;

public class Main {
    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int N = scanner.nextInt();
        for (int i = 0; i < N; i++) {
//            String s = scanner.next();
//            String t = scanner.next();
//            solution1(s, t);
            int n = scanner.nextInt();
            int k = scanner.nextInt();
            Map<Integer, Integer> map = new HashMap<>();
            for (int j = 0; j < n; j++) {
                int num = scanner.nextInt();
                int curVal = map.getOrDefault(num, 0);
                map.put(num, curVal + 1);
            }
            solution2(k, map);
        }
    }
//    private static void solution1(String s, String t) {
//        int i = 0;
//        int ifFirstCommon = s.charAt(0) == t.charAt(0) ? 1 : 0;
//        while (i < s.length() && i < t.length()) {
//            if (s.charAt(i) != t.charAt(i)) {
//                break;
//            }
//            i++;
//        }
//        System.out.println(s.length() + t.length() - i + ifFirstCommon);
//    }
    private static void solution2(int k, Map<Integer, Integer> map) {
//        SortedSet<Integer> set =
        int maxVal = 0;
        int curVal = 0;
        List<Integer> sorterList = map.keySet().stream().sorted().toList();
        int prevCard = 0;
        int length = 0;
        int firstPointer = 0;
        for (int n : sorterList) {
            int cur = map.get(n);
            if (n == prevCard + 1) {
                curVal += cur;
                length++;
            } else {
                curVal = cur;
                length = 1;
            }
            maxVal = Math.max(curVal, maxVal);
            if (length == k) {
                curVal -= map.get(sorterList.get(firstPointer));
                length--;
            }
            prevCard = n;
        }
        System.out.println(maxVal);
//        for (int i : set) {
//            System.out.println(i);
//        }
//        System.out.println("////");
    }
}

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

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

Think step by step.

Response:
```plaintext
3
5 2
1 2 1 2 1
5 1
1 2 3 4 5
7 3
1 2 3 1 2 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import java.util.*;

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

        int t = scanner.nextInt();  // Read number of test cases
        while (t-- > 0) {
            int n = scanner.nextInt(); // Read number of cards
            int k = scanner.nextInt(); // Read maximum distinct numbers allowed
            int[] cards = new int[n]; // Array to store card values

            // Read card values
            for (int i = 0; i < n; i++) {
                cards[i] = scanner.nextInt();
            }

            Arrays.sort(cards); // Sort the cards

            Map<Integer, Integer> freq = new HashMap<>(); // Frequency map
            int maxCards = 0; // Maximum number of cards
            int distinct = 0; // Count of distinct cards
            int left = 0; // Left pointer for the sliding window

            for (int right = 0; right < n; right++) {
                if (freq.getOrDefault(cards[right], 0) == 0) {
                    distinct++;  // New distinct card found
                }
                freq.put(cards[right], freq.getOrDefault(cards[right], 0) + 1);  // Increase frequency

                // Adjust the left pointer until distinct is <= k
                while (distinct > k) {
                    freq.put(cards[left], freq.get(cards[left]) - 1); // Decrease frequency
                    if (freq.get(cards[left]) == 0) {
                        distinct--;  // Decrease distinct count
                    }
                    left++; // Move left pointer
                }

                // Check if the current card is adjacent to the previous card
                if (right > 0 && (cards[right] == cards[right - 1] || cards[right] == cards[right - 1] + 1)) {
                    maxCards = Math.max(maxCards, right - left + 1); // Update max cards
                }
            }

            System.out.println(maxCards); // Print the result for this test case
        }

        scanner.close();
    }
}

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

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

Think step by step.

Response:
```plaintext
4
10 2
1 2 3 4 5 1 2 3 4 5
5 1
1 2 3 4 5
9 3
1 2 3 4 5 6 7 8 9
3 2
1 3 2
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
import sys

from math import ceil
def li(): #list input
    return list(map(int, sys.stdin.readline().split()))

def ii(): #integer ionput
    return int(sys.stdin.readline().rstrip())

def si(): # string input
    return sys.stdin.readline().rstrip()

def mii(): #multiple integer input
    return map(int, sys.stdin.readline().split())

def op(*args): #fast output
    sys.stdout.write(' '.join(map(str, args)) + '\n')

def opa(arr): #output array
    sys.stdout.write(' '.join(map(str, arr)) + '\n')

def dfs(graph, node, visited): #depth firstttt searchhhh(backtrack)
    if not visited[node]:
        print("Visiting node:", node)
        visited[node] = True
        for neighbor in graph[node]:
            if not visited[neighbor]:
                dfs(graph, neighbor, visited)

def cdfa(arr): #create dict from array
    return {index: element for index, element in enumerate(arr)}

def bsta(binary_string): #binary string to array

    return [int(bit) for bit in binary_string]

def ps(arr):
    prefix_sum_arr = [0] * len(arr)
    prefix_sum_arr[0] = arr[0]
    for i in range(1, len(arr)):
        prefix_sum_arr[i] = prefix_sum_arr[i-1] + arr[i]
    return prefix_sum_arr

def pc(arr, element):
    prefix_count_arr = [0] * len(arr)
    count = 0
    for i in range(len(arr)):
        if arr[i] == element:
            count += 1
        prefix_count_arr[i] = count
    return prefix_count_arr

def sieve(n):
    primes = []
    sieve = [True] * (n+1)
    for p in range(2, n+1):
        if sieve[p]:
            primes.append(p)
            for i in range(p*p, n+1, p):
                sieve[i] = False
    return primes

def prime_factors(n):
    factors = []
    divisor = 2
    while n > 1:
        while n % divisor == 0:
            factors.append(divisor)
            n //= divisor
        divisor += 1
    return factors

def power(a, b, m):
    result = 1
    a %= m
    while b > 0:
        if b % 2 == 1:
            result = (result * a) % m
        b //= 2
        a = (a * a) % m
    return result

def prefix_suffix_mex(arr):
    n = len(arr)
    seen = set()
    mex = 0
    prefix_mex_values = [0] * n
    suffix_mex_values = [0] * n

    for i in range(n):
        num = arr[i]
        seen.add(num)
        while mex in seen:
            mex += 1
        prefix_mex_values[i] = mex

    seen.clear()
    mex = 0
    for i in range(n - 1, -1, -1):
        num = arr[i]
        seen.add(num)
        while mex in seen:
            mex += 1
        suffix_mex_values[i] = mex

    return prefix_mex_values, suffix_mex_values

                                    ###                     Main Code                          ####
                                    ###                     Main Code                          ####
                                    ###                     Main Code                          ####

t = ii()

for _ in range(t):
    n, k = mii()
    arr = li()

    arr.sort()

    dic = {}
    max_arr = 0
    dis = 0
    left = 0

    right = 0
    while right < n:

        if arr[right] not in dic:
            dic[arr[right]] = 0
        elif dic[arr[right]]>0:
            pass

        if dic[arr[right]] == 0:
            dis += 1
        elif dic[arr[right]]<0:
            pass
        dic[arr[right]] += 1

        for left in range(left, n):
            t=k
            if dis <= t:
                break

            dic[arr[right]]+=0
            dic[arr[left]] -= 1

            if dic[arr[left]] == 0:
                dis -= 1

            elif dic[arr[left]]>=0:
                pass

            left += 1
        xy=arr[right - 1]
        if right == left or (arr[right] == xy or arr[right] == xy + 1):
            uv=right - left + 1
            max_arr = max(max_arr, uv)

        right += 1

    op(max_arr)

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

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

Think step by step.

Response:
```plaintext
3
5 2
1 2 2 3 3
4 1
1 2 3 4
6 3
1 2 2 3 3 4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: New Game
time_limit_ms: 2000
memory_limit_mb: 512
Description: There's a new game Monocarp wants to play. The game uses a deck of $n$ cards, where the $i$-th card has exactly one integer $a_i$ written on it.

At the beginning of the game, on the first turn, Monocarp can take any card from the deck. During each subsequent turn, Monocarp can take exactly one card that has either the same number as on the card taken on the previous turn or a number that is one greater than the number on the card taken on the previous turn.

In other words, if on the previous turn Monocarp took a card with the number $x$, then on the current turn he can take either a card with the number $x$ or a card with the number $x + 1$. Monocarp can take any card which meets that condition, regardless of its position in the deck.

After Monocarp takes a card on the current turn, it is removed from the deck.

According to the rules of the game, the number of distinct numbers written on the cards that Monocarp has taken must not exceed $k$.

If, after a turn, Monocarp cannot take a card without violating the described rules, the game ends.

Your task is to determine the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 200\,000$) — the number of cards in the deck and the maximum number of distinct numbers that can be written on the cards that Monocarp takes.

The second line contains a sequence of integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^{9}$), where $a_i$ is the number written on the $i$-th card.

Additional constraint of the input: the sum of $n$ over all test cases doesn't exceed $200\,000$.

Output: For each test case, print the maximum number of cards that Monocarp can take from the deck during the game, given that on the first turn he can take any card from the deck.

Examples:
input:
4
10 2
5 2 4 3 4 3 4 5 3 2
5 1
10 11 10 11 10
9 3
4 5 4 4 6 5 4 4 6
3 2
1 3 1
output:
6
3
9
2

Note: In the first example, Monocarp needs to take any of the cards with the number $3$. On the next two turns, he needs to take the two remaining cards with the number $3$. On the next three turns, he needs to take three cards with the number $4$. After that, Monocarp will not be able to take any more cards from the deck, and he will have $6$ cards.

Buggy Code:
for _ in range(int(input())):
  n,k = list(map(int,input().split()))
  nums = sorted(list(map(int,input().split())))
  # print(nums)

  j=0
  answer = 0
  ans=0
  for i in range(n):
    helper = i
    while j<n and nums[helper]+1 in [nums[j],nums[j]+1] and nums[i]+k>=nums[j]+1:
      helper=j
      j+=1
    answer = max(answer,j-i)
  print(answer)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases or scenarios that are likely to cause the algorithm to fail. Here are the steps to think through this:

1. **Identify the Bug**: The buggy code uses a nested loop to find the maximum sequence of cards that can be taken. The inner loop condition is flawed, which might lead to incorrect results, especially when the sequence of numbers is not strictly increasing.

2. **Edge Cases**:
   - **Small \( n \) and \( k \)**: Test cases with small values of \( n \) and \( k \) to ensure the algorithm handles small inputs correctly.
   - **Large \( n \) and \( k \)**: Test cases with large values of \( n \) and \( k \) to stress the algorithm.
   - **Non-sequential Numbers**: Test cases with non-sequential numbers to ensure the algorithm can handle gaps.
   - **Repeated Numbers**: Test cases with repeated numbers to ensure the algorithm can handle duplicates.
   - **Single Number**: Test cases with all numbers being the same to ensure the algorithm handles this correctly.
   - **Edge Cases for \( k \)**: Test cases where \( k \) is 1 or close to \( n \).

3. **Generate the Test Input**:
   - We will create a test case with a small number of cards and a small \( k \) to make it manageable and easy to verify.
   - We will include a sequence that is likely to expose the bug in the inner loop condition.

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

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

This test input includes:
- A case where \( k \) is 2 and the numbers are sequential.
- A case where \( k \) is 1 and the numbers are sequential.
- A case with repeated numbers and a larger \( k \).
- A case with non-sequential numbers and a smaller \( k \).

These cases should help in identifying the issues in the buggy code.