**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

Buggy Code:
#include <bits/stdc++.h>
#define PB push_back
#define F first
#define S second
#define all(x) x.begin(), x.end()
#define bit(x, i) ((x >> i) & 1)
#define il (node * 2)
#define ir (il + 1)
#define mid (l + r)/2
#define maxn
#define ll long long
#define pii pair <int, int>
#define MOD 1000000007
#define Task "try_codeforce"
using namespace std;
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
const int N = 100005;
int n, q, a[N], b[N];
void solve(){
    cin >> n >> q;
    for (int i = 1; i <= n; i ++)
        cin >> b[i];
    for (int i = 1; i <= n; i ++){
        char c;
        cin >> c;
        a[i] =  c == 'L';
    }
    while (q --){
        int x, mx = 0, ok = 1;
        cin >> x;
        a[x] ^= 1;
        for (int i = 1; i <= n; i ++){
            mx = max(mx, b[i]);
            if (i < n && a[i] == 1 && a[i + 1] == 0 && mx > i) ok = 0;
//            cout << a[i];
        }
        if (ok) cout << "YES\n";
        else cout << "NO\n";
    }
}
signed main(){
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
    if(fopen(Task".inp", "r")) {
        freopen(Task".inp", "r", stdin);
        freopen(Task".out", "w", stdout);
    }
    int ntest;
    cin >> ntest;
    while (ntest --)
        solve();
    cerr << "\nTime elapsed: " << 1000.0 * clock() / CLOCKS_PER_SEC << " ms.\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:
```plaintext
1
5 2
4 1 3 5 2
RLRLL
2
4
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

Buggy Code:
t=int(input())
for i in range(t):
    a=input().split()
    n=int(a[0])
    q=int(a[1])
    b=input().split()
    for k in range(n):
        b[k]=int(b[k])

    s=list(input())
    for j in range(q):
        f=0
        query=int(input())
        if (s[query-1]=="R"):
            s[query-1]="L"
        else:
            s[query-1]="R"
        for l in range(len(s)-1):
            if (s[l]=="L" and s[l+1]=="R"):
                maxx=max(b[:l+1])
                minn=min(b[l+1:])
                if (maxx>minn):
                    f=1
                    break
        if (f==1):
            print("NO")
        else:
            print("YES")

**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 3
5 4 3 2 1
RLRLL
2
4
3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

Buggy Code:
def f(n, p0, s):
    if p0 == sorted(p0) or 'LR' not in s:
        return "YES"
    if s == 'LR' and p0 != sorted(p0):
        return "NO"
    while 'LR' in s:
        if p0 == sorted(p0):
            return "YES"
        if s == 'LR' and p0 != sorted(p0):
            return "NO"
        cut = s.find('LR')
        sr = s[cut+1:]
        pl, pr = p0[:cut+1], p0[cut+1:]
        if max(pl) > min(pr):
            return "NO"
        s = sr
        p0 = pr
    else:
        return "YES"

t = int(input())
for _ in range(t):
    n, q = list(map(int, input().split()))
    p = list(map(int, input().split()))
    s = str(input())
    for i in range(q):
        ch = int(input())
        s = list(s)
        s[ch-1] = 'L' if s[ch-1] == 'R' else 'R'
        s = ''.join(s)
        print(f(n, p, s))

# YES
# YES
# NO
# NO
# YES
# NO
# NO
# NO
# YES
# YES

**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 3
5 4 3 2 1
RLRLL
2
4
3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

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

public class main2 {
    public static PrintWriter out;
    public static class MyScanner {
        BufferedReader br;
        StringTokenizer scan;

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

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

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

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

        double nextDouble() {
            return Double.parseDouble(next());
        }

        String nextLine(){
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }

    }

    public static void main(String[] args) {
        MyScanner scan = new MyScanner();
        out = new PrintWriter(new BufferedOutputStream(System.out));
        int tests = scan.nextInt();
        for(int i = 0; i<tests; i++){

            int n = scan.nextInt();
            int q = scan.nextInt();

            Integer[] arr = new Integer[n];
            Integer[] maxArr = new Integer[n];
            int max = -1;
            for(int j = 0; j<n; j++){
                int val = scan.nextInt();
                arr[j] = val;
                if (val > max){
                    max = val;
                }
                maxArr[j] = max;
            }
            Arrays.sort(arr);
            boolean[] safeCuts = new boolean[n];
            for(int j = 0; j<n; j++){
                if(maxArr[j] == arr[j]){
                    safeCuts[j] = true;
                }else{
                    safeCuts[j] = false;
                }
            }

            char[] leftRights = scan.next().toCharArray();

            int badCuts = 0;
            for(int j = 0; j<n-1; j++){
                if(leftRights[j] == 'L' && leftRights[j+1] == 'R' && (!safeCuts[j])){
                    badCuts++;
                }
            }

            for(int j = 0; j<q; j++){
                int query = scan.nextInt()-1;
                if(leftRights[query] == 'L'){
                    leftRights[query] = 'R';
                    if(leftRights[query-1] == 'L' && (!safeCuts[query-1])){
                        badCuts++;
                    }
                    if(leftRights[query+1] == 'R' && (!safeCuts[query])){
                        badCuts--;
                    }
                } else if(leftRights[query] == 'R'){
                    leftRights[query] = 'L';
                    if(leftRights[query-1] == 'L' && (!safeCuts[query-1])){
                        badCuts--;
                    }
                    if(leftRights[query+1] == 'R' && (!safeCuts[query])){
                        badCuts++;
                    }
                }
                if(badCuts == 0){
                    out.println("YES");
                }else{
                    out.println("NO");
                }

            }

        }

        out.close();
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

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

public class main2 {
    public static PrintWriter out;
    public static class MyScanner {
        BufferedReader br;
        StringTokenizer scan;

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

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

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

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

        double nextDouble() {
            return Double.parseDouble(next());
        }

        String nextLine(){
            String str = "";
            try {
                str = br.readLine();
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }

    }

    public static void main(String[] args) {
        MyScanner scan = new MyScanner();
        out = new PrintWriter(new BufferedOutputStream(System.out));
        int tests = scan.nextInt();
        for(int i = 0; i<tests; i++){

            int n = scan.nextInt();
            int q = scan.nextInt();

            Integer[] arr = new Integer[n];
            Integer[] maxArr = new Integer[n];
            int max = -1;
            for(int j = 0; j<n; j++){
                int val = scan.nextInt();
                arr[j] = val;
                if (val > max){
                    max = val;
                }
                maxArr[j] = max;
            }
            boolean flag = false;
            if(n > 190000 && arr[0] == 1 && arr[1] == 2 && arr[3] == 4){
                flag = true;
            }
            Arrays.sort(arr);
            boolean[] safeCuts = new boolean[n];
            for(int j = 0; j<n; j++){
                if(maxArr[j] == arr[j]){
                    safeCuts[j] = true;
                }else{
                    safeCuts[j] = false;
                }
            }

            char[] leftRights = scan.next().toCharArray();

            int badCuts = 0;
            for(int j = 0; j<n-1; j++){
                if(leftRights[j] == 'L' && leftRights[j+1] == 'R' && (!safeCuts[j])){
                    badCuts++;
                }
            }

            for(int j = 0; j<q; j++){
                int query = scan.nextInt()-1;
                if(leftRights[query] == 'L'){
                    leftRights[query] = 'R';
                    if(leftRights[query-1] == 'L' && (!safeCuts[query-1])){
                        badCuts++;
                    }
                    if(leftRights[query+1] == 'R' && (!safeCuts[query])){
                        badCuts--;
                    }
                } else if(leftRights[query] == 'R'){
                    leftRights[query] = 'L';
                    if(leftRights[query-1] == 'L' && (!safeCuts[query-1])){
                        badCuts--;
                    }
                    if(leftRights[query+1] == 'R' && (!safeCuts[query])){
                        badCuts++;
                    }
                }
                if(flag){
                    out.println(badCuts);
                }

                if(badCuts == 0){
                    out.println("YES");
                }else{
                    out.println("NO");
                }

            }

        }

        out.close();
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: QED's Favorite Permutation
time_limit_ms: 2000
memory_limit_mb: 256
Description: QED is given a permutation$^{\text{∗}}$ $p$ of length $n$. He also has a string $s$ of length $n$ containing only characters $\texttt{L}$ and $\texttt{R}$. QED only likes permutations that are sorted in non-decreasing order. To sort $p$, he can select any of the following operations and perform them any number of times:

  * Choose an index $i$ such that $s_i = \texttt{L}$. Then, swap $p_i$ and $p_{i-1}$. It is guaranteed that $s_1 \neq \texttt{L}$.    * Choose an index $i$ such that $s_i = \texttt{R}$. Then, swap $p_i$ and $p_{i+1}$. It is guaranteed that $s_n \neq \texttt{R}$.

He is also given $q$ queries. In each query, he selects an index $i$ and changes $s_i$ from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$). Note that the changes are persistent.

After each query, he asks you if it is possible to sort $p$ in non- decreasing order by performing the aforementioned operations any number of times. Note that before answering each query, the permutation $p$ is reset to its original form.

$^{\text{∗}}$A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).

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

The first line of each test case contains two integers $n$ and $q$ ($3 \leq n \leq 2 \cdot 10^5$, $1 \leq q \leq 2 \cdot 10^5$) – the length of the permutation and the number of queries.

The following line contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \leq p_i \leq n$, $p$ is a permutation).

The following line contains $n$ characters $s_1s_2 \ldots s_n$. It is guaranteed that $s_i$ is either $\texttt{L}$ or $\texttt{R}$, $s_1 = \texttt{R}$, and $s_n = \texttt{L}$.

The following $q$ lines contain an integer $i$ ($2 \leq i \leq n-1$), denoting that $s_i$ is changed from $\texttt{L}$ to $\texttt{R}$ (or from $\texttt{R}$ to $\texttt{L}$).

It is guaranteed that the sum of $n$ and $q$ over all test cases does not exceed $2 \cdot 10^5$.

Output: For each query, output "YES" (without quotes) if it is possible, and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yES", "yes" and "Yes" will be recognized as a positive response).

Examples:
input:
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
output:
YES
YES
NO
NO
YES
NO
NO
NO
YES
YES

Note: In the first testcase, $s = \texttt{RRRLL}$ after the first query. QED may sort $p$ using the following operations:

  * Initially, $p = [1,4,2,5,3]$.    * Select $i = 2$ and swap $p_2$ with $p_{3}$. Now, $p = [1,2,4,5,3]$.    * Select $i = 5$ and swap $p_5$ with $p_{4}$. Now, $p = [1,2,4,3,5]$.    * Select $i = 4$ and swap $p_4$ with $p_{3}$. Now, $p = [1,2,3,4,5]$, which is in non-decreasing order.

It can be shown that it is impossible to sort the array after all three updates of the first testcase.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define int long long
#define endl '\n'
#define IOS ios::sync_with_stdio(0);cin.tie(0);cout.tie(0);
#define i1 (int)1
const int N = 1e6+10;
const int mod = 1e9+7;

int a[N];
int f[N];

void solve(){
	int n,m;
	cin >> n >> m;
	for(int i=1;i<=n;i++){
		cin >> a[i];
		f[i]=0;
	}

	int util=-1;
	int cnt=0;
	for(int i=1;i<=n;i++){
		if(a[i]!=i && util<=a[i]){
			if(util<i)
				cnt++;
			util=max(util,a[i]);
		}
		if(util>=i)
			f[i]=cnt;
//		cout << f[i] << " ";
	}
//	cout << endl;

	string s;
	cin >> s;
	s=" "+s;

	set<int> wa;

	for(int i=2;i<=n-1;i++){
		if(f[i]!=0){

			if(f[i-1]==f[i] && s[i]!=s[i-1] && s[i-1]=='L'){
				wa.insert(i);
				wa.insert(i-1);
			}
			if(f[i+1]==f[i] && s[i]!=s[i+1] && s[i]=='L'){
				wa.insert(i);
				wa.insert(i+1);
			}

		}
	}

	for(int i=0;i<m;i++){
		int x;
		cin >> x;
		if(s[x]=='L')
			s[x]='R';
		else s[x]='L';

		if(f[x]!=0 && f[x-1]==f[x] && s[x]!=s[x-1] && s[x-1]=='L'){
			wa.insert(x);
			wa.insert(x-1);
		}
		else if(f[x]!=0 && f[x-1]==f[x]){
			wa.erase(x);
			wa.erase(x-1);
		}
		if(f[x]!=0 && f[x+1]==f[x] &&s[x]!=s[x+1] && s[x]=='L'){
			wa.insert(x);
			wa.insert(x+1);
		}
		else if(f[x]!=0 && f[x+1]==f[x]){
			wa.erase(x);
			wa.erase(x+1);
		}

		int hy=x;
		if((x-1)>1){
			x=x-1;
		}
		if(f[x]!=0 && f[x-1]==f[x] && s[x]!=s[x-1] && s[x-1]=='L'){
			wa.insert(x);
			wa.insert(x-1);
		}
		else if(f[x]!=0 && f[x-1]==f[x]){
			wa.erase(x);
			wa.erase(x-1);
		}
		if(f[x]!=0 && f[x+1]==f[x] &&s[x]!=s[x+1] && s[x]=='L'){
			wa.insert(x);
			wa.insert(x+1);
		}
		else if(f[x]!=0 && f[x+1]==f[x]){
			wa.erase(x);
			wa.erase(x+1);
		}
		x=hy;
		if((x+1)<n)
			x=x+1;
		if(f[x]!=0 && f[x-1]==f[x] && s[x]!=s[x-1] && s[x-1]=='L'){
			wa.insert(x);
			wa.insert(x-1);
		}
		else if(f[x]!=0 && f[x-1]==f[x]){
			wa.erase(x);
			wa.erase(x-1);
		}
		if(f[x]!=0 && f[x+1]==f[x] &&s[x]!=s[x+1] && s[x]=='L'){
			wa.insert(x);
			wa.insert(x+1);
		}
		else if(f[x]!=0 && f[x+1]==f[x]){
			wa.erase(x);
			wa.erase(x+1);
		}

		if(wa.size()==0)
			cout << "YES" << endl;
		else cout << "NO" << endl;

	}

	wa.clear();
}

signed main()
{
	IOS;

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

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

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

Think step by step.

Response:
```plaintext
3
5 3
1 4 2 5 3
RLRLL
2
4
3
8 5
1 5 2 4 8 3 6 7
RRLLRRRL
4
3
5
3
4
6 2
1 2 3 4 5 6
RLRLRL
4
5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
typedef long long i64;
constexpr int mod = 1e9 + 7;
struct Trie {
    int n, idx;
    vector<vector<int>> ch;
    vector<vector<int>> mxch;
    Trie(int n) {
        this->n = n;
        idx = 0;
        ch.resize(30 * (n + 1), vector<int>(2));
        mxch.resize(30 * (n + 1), vector<int>(2));
    }
    void insert(int x, int id) {
        int u = 0;
        for (int i = 30; ~i; i--) {
            int &v = ch[u][x >> i & 1];
            mxch[u][x >> i & 1] = max(mxch[u][x >> i & 1], id);
            if (!v)
                v = ++idx;
            u = v;
        }
    }
    int query(int x, int k) {
        int u = 0, res = 0;
        for (int i = 30; ~i; i--) {
            int v = x >> i & 1;
            int p = k >> i & 1;
            if (p == 1) {
                res = max(res, mxch[u][v]);
                u = ch[u][!v];
            } else {
                u = ch[u][v];
            }
            if (!u)
                return res;
        }
        return res;
    }
};

void solve() {
    i64 n, k;
    cin >> n >> k;
    vector<i64> a(n + 1);
    for (i64 i = 1; i <= n; i++) {
        cin >> a[i];
    }
    auto check = [&](i64 x) {
        Trie trie(n);
        i64 sum = 0;
        int pos = 0;
        for (i64 i = 1; i <= n; i++) {
            pos = max(pos, trie.query(a[i], x + 1));
            sum += pos;
            trie.insert(a[i], i);
        }
        return sum >= k;
    };
    i64 l = 0, r = 1e6;
    while (l < r) {
        i64 mid = (l + r) / 2;
        if (check(mid)) {
            r = mid;
        } else {
            l = mid + 1;
        }
    }
    cout << l << '\n';
    return;
}
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0), cout.tie(0);
    i64 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
4 1
1 1 1 1
4 6
1 2 4 8
5 10
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
// They say that life is always easier
// After you let yourself come undone
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
const int N = 1e5 + 10;
int n, a[N];
ll k;
multiset<int> st, val;
void Add(int k) {
	auto it = st.insert(k);
	bool l = it != st.begin();
	bool r = next(it) != st.end();
	if(l) val.insert(*prev(it) ^ k);
	if(r) val.insert(*next(it) ^ k);
	if(l && r) val.erase(*prev(it) ^ *next(it));
}
void Del(int k) {
	auto it = st.lower_bound(k);
	bool l = it != st.begin();
	bool r = next(it) != st.end();
	if(l) val.erase(val.lower_bound(*prev(it) ^ k));
	if(r) val.erase(val.lower_bound(*next(it) ^ k));
	if(l && r) val.insert(*prev(it) ^ *next(it));
	st.erase(it);
}
ll Check(ll mid) {
	st.clear(), val.clear();
	ll ans = 0; int l = n + 1;
	for(int i = n; i >= 1; --i) {
		while(l > 1 && (val.empty() || *val.begin() > mid))
			Add(a[--l]);
		if(!val.empty() && *val.begin() <= mid)
			ans += l;
		Del(a[i]);
	}
	return ans;
}
void Solve() {
	cin >> n >> k;
	for(int i = 1; i <= n; ++i) cin >> a[i];
	ll lef = 0, rig = 2e9, ans = 0;
	while(lef <= rig) {
		ll mid = lef + rig >> 1;
		if(Check(mid) >= k) ans = mid, rig = mid - 1;
		else lef = mid + 1;
	}
	printf("%lld\n", ans);
}
int main() {
	//freopen("data.in", "r", stdin);
	//freopen("code.out", "w", stdout);
	cin.tie(0)->sync_with_stdio(0);
	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
3 1
1 1 1
4 2
1 2 2 3
5 10
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
#include <bits/stdc++.h>
#define fi first
#define se second
using namespace std;
#define pb push_back
#define ls (u << 1)
#define rs (u << 1 | 1)

// #define debug(args...) {}
#define debug(args...) { \
     string _s = #args; replace(_s.begin(), _s.end(), ',', ' '); \
     stringstream _ss(_s); istream_iterator<string> _it(_ss); err(_it, args); cout << '\n';}
void err(istream_iterator<string> it){}
template<typename T, typename... Args>
void err(istream_iterator<string> it, T a, Args... args) {
    cout << *it << " = " << a << ' ';
    err(++it, args...);
}

typedef long long LL;
typedef pair<int, int> PII;
struct Trie {
    vector<array<int, 2>> tr;
    vector<int> mx;
    int tot;
    Trie (int n) {
        tr.assign(n * 31 + 5, {});
        mx.assign(n * 31 + 5, 0);
        tot = 0;
    }
    void insert(int x, int id) {
        int u = 0;
        for (int i = 29; i >= 0; i--) {
            int one = x >> i & 1;
            if (!tr[u][one]) tr[u][one] = ++tot;
            u = tr[u][one];
            mx[u] = max(mx[u], id);
        }
    }
    int query(int x, int y) {
        y--;
        int ok = 0, u = 0;
        for (int i = 29; i >= 0; i--) {
            int nxt = -1, mxid = 0;
            int kx = x >> i & 1, ky = y >> i & 1;
            for (int j = 0; j < 2; j++) {
                if (!ok && (kx ^ j) > ky) continue;
                if (mx[tr[u][j]] > mxid) {
                    nxt = j;
                    mxid = mx[tr[u][j]];
                }
            }
            if (nxt == -1) return 0;
            // debug(u, nxt, kx, ky, tr[u][nxt]);
            if ((kx ^ nxt) < ky) ok = true;
            u = tr[u][nxt];
        }
        return mx[u];
    }
};
void solve() {
    LL n, k; cin >> n >> k;
    vector<int> a(n + 5);
    for (int i = 1; i <= n; i++) cin >> a[i];
    auto check = [&](int x) {
        Trie trie(n);
        LL cnt = n * (n - 1) / 2;
        int lstl = 1;
        for (int r = 1; r <= n; r++) {
            if (r > 1) {
                int l = trie.query(a[r], x) + 1;
                l = max(l, lstl);
                cnt -= r - l;
                lstl = l;
            }
            trie.insert(a[r], r);
        }
        return cnt < k;
    };
    int l = 0, r = 1e9;
    while (l < r) {
        int mid = l + r + 1 >> 1;
        if (check(mid)) l = mid;
        else r = mid - 1;
    }
    cout << l << "\n";
}

int main() {
    ios::sync_with_stdio(0);
    cin.tie(0);
    int tt = 1; cin >> tt;
    while (tt--) 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
3 2
1 1 1
4 4
1 2 3 4
5 10
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
import heapq

def find_kth_smallest_xor(test_cases):
    results = []
    for case in test_cases:
        n, k = case['params']['n'], case['params']['k']
        arr = case['params']['arr']

        # Create a list to store all XOR values
        xor_values = []

        # Generate all subarrays of length >= 2 and compute their XOR values
        for i in range(n):
            for j in range(i + 1, n):
                min_xor = float('inf')
                for m in range(i, j + 1):
                    for p in range(m + 1, j + 1):
                        min_xor = min(min_xor, arr[m] ^ arr[p])
                xor_values.append(min_xor)

        # Sort the XOR values
        xor_values.sort()

        # Find the k-th smallest XOR value
        result = xor_values[k - 1] if k <= len(xor_values) else None
        results.append(result)

    return results

# Example usage
test_cases = [
    {
        'params': {'n': 5, 'k': 2, 'arr': [1, 2, 3, 4, 5]}
    },
    {
        'params': {'n': 2, 'k': 1, 'arr': [4, 3]}
    },
    {
        'params': {'n': 4, 'k': 6, 'arr': [1, 2, 4, 8]}
    },
    {
        'params': {'n': 5, 'k': 9, 'arr': [1, 2, 3, 4, 5]}
    }
]

results = find_kth_smallest_xor(test_cases)
for res in results:
    print(res)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios that can maximize the failure probability. The primary issue in the code is the inefficient and incorrect way it calculates the minimum XOR value for each subarray. The nested loops make the code very slow and prone to errors, especially for large inputs.

Here are the steps to create a challenging test input:

1. **Small but Complex Array**: Use an array with a small number of elements but with values that can create a wide range of XOR results.
2. **Edge Cases**: Include arrays with repeated values, arrays with all zeros, and arrays with large values.
3. **Multiple Test Cases**: Provide multiple test cases to cover different scenarios.

Given these considerations, here is a challenging test input:

```plaintext
4
3 2
0 0 0
4 6
1 2 3 4
5 10
1 1 2 2 3
6 15
1 2 3 4 5 6
```

### Explanation:
1. **Test Case 1**:
   - Array: `[0, 0, 0]`
   - k: `2`
   - This test case checks the behavior when all elements are the same. The expected output is `0` because the XOR of any two identical elements is `0`.

2. **Test Case 2**:
   - Array: `[1, 2, 3, 4]`
   - k: `6`
   - This test case checks a small array with distinct values. The expected output is `3` because the possible XOR values are `[1, 2, 3, 3, 1, 2]` and the 6th smallest value is `3`.

3. **Test Case 3**:
   - Array: `[1, 1, 2, 2, 3]`
   - k: `10`
   - This test case checks an array with repeated values. The expected output is `1` because the possible XOR values are `[0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3]` and the 10th smallest value is `1`.

4. **Test Case 4**:
   - Array: `[1, 2, 3, 4, 5, 6]`
   - k: `15`
   - This test case checks a slightly larger array with distinct values. The expected output is `1` because the possible XOR values are `[1, 3, 1, 7, 1, 2, 3, 7, 3, 1, 2, 3, 1, 7, 7]` and the 15th smallest value is `1`.

This test input should expose the inefficiencies and potential bugs in the given code, particularly the nested loops and the incorrect calculation of the minimum XOR values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
from dataclasses import dataclass
from typing import Tuple, List
import bisect

@dataclass
class State:
    xor_values: List[int]
    n: int

@dataclass
class Parameters:
    k: int

def compute_result_exists() -> bool:
    return True

def count_smaller_or_equal(xor_values: List[int], x: int) -> int:
    count = 0
    for i, val in enumerate(xor_values):
        if val <= x:
            count += (len(xor_values) - i)
    return count

def binary_search(state: State, parameters: Parameters) -> int:
    left, right = 0, max(state.xor_values)
    while left < right:
        mid = (left + right) // 2
        if count_smaller_or_equal(state.xor_values, mid) >= parameters.k:
            right = mid
        else:
            left = mid + 1
    return left

def read_test_case() -> Tuple[State, Parameters]:
    n, k = map(int, input().split())
    a = list(map(int, input().split()))
    xor_values = [a[i] ^ a[i+1] for i in range(n-1)]
    xor_values.sort()  # Sort for efficient counting
    return State(xor_values=xor_values, n=n), Parameters(k=k)

def print_result(result: int):
    print(result)

def main():
    t = int(input())
    for _ in range(t):
        state, parameters = read_test_case()
        if compute_result_exists():
            result = binary_search(state, parameters)
            print_result(result)

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 code, we need to focus on edge cases and scenarios that can maximize the failure probability. The primary issue in the code is in the `count_smaller_or_equal` function, which incorrectly counts the number of elements that are less than or equal to a given value. Specifically, it counts all elements from the current index to the end of the list, which is incorrect for this problem.

Let's consider the following edge cases:
1. **Small Arrays with Repeated Values**: This can help test the function's ability to handle identical values correctly.
2. **Arrays with Large Values**: This can test the function's ability to handle large XOR results.
3. **Arrays with Specific Patterns**: This can test the function's ability to handle specific patterns that might cause incorrect counting.

Given the constraints and the need for a small-scale test input, we can create a test case with repeated values and a specific pattern to expose the bug.

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

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

Explanation:
1. **Test Case 1**: `4 1 1 1 1`
   - The only possible subarray is `[1, 1, 1, 1]` with XOR values `[0, 0, 0]`.
   - The 1st smallest value is `0`.

2. **Test Case 2**: `4 2 1 2 3 4`
   - Subarrays and their XOR values:
     - `[1, 2] -> 3`
     - `[2, 3] -> 1`
     - `[3, 4] -> 7`
     - `[1, 2, 3] -> 1`
     - `[2, 3, 4] -> 5`
     - `[1, 2, 3, 4] -> 4`
   - The 2nd smallest value is `1`.

3. **Test Case 3**: `5 3 1 2 3 4 5`
   - Subarrays and their XOR values:
     - `[1, 2] -> 3`
     - `[2, 3] -> 1`
     - `[3, 4] -> 7`
     - `[4, 5] -> 1`
     - `[1, 2, 3] -> 1`
     - `[2, 3, 4] -> 5`
     - `[3, 4, 5] -> 6`
     - `[1, 2, 3, 4] -> 4`
     - `[2, 3, 4, 5] -> 4`
     - `[1, 2, 3, 4, 5] -> 3`
   - The 3rd smallest value is `3`.

This test input should help expose the bug in the `count_smaller_or_equal` function, as it will incorrectly count the number of elements in the list.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
from itertools import pairwise
import sys
input = sys.stdin.readline
MOD = 10**9 + 7

def read_array(factory):
    return [factory(num) for num in input().strip().split()]
def print_array(arr):
    print(" ".join(map(str, arr)))

def solve(k, arr):
    xors = []
    for i, j in pairwise(arr):
        xors.append(i ^ j)
    x = xors
    # print(xors)
    # while x:
    #     newx = []
    #     for i, j in pairwise(x):
    #         newx.append(min(i, j))
    #     x = newx
    #     print(x)

    stack = []
    l = len(xors)
    ans = [-1] * l
    for idx, i in enumerate(xors):
        while len(stack) > 0 and i < stack[-1][1]:
            idx2, _ = stack.pop()
            ans[idx2] = idx
        stack.append((idx, i))

    cnts = [1] * len(xors)
    for idx in range(len(xors)):
        if ans[idx] == -1:
            cnts[idx] *= len(xors) - idx
        else:
            i = ans[idx]
            cnts[i] += cnts[idx]
            cnts[idx] *= i - idx
    # print(cnts)

    z = list(zip(xors, cnts))
    z.sort(reverse=True)
    while k > 0:
        a, b = z.pop()
        k -= b

    return a

if __name__ == "__main__":
    # Read in input:
    t = int(input())

    # Call solution:
    for _ in range(t):
        _, k = read_array(int)
        arr = read_array(int)
        ans = solve(k, arr)
        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
3
3 2
1 2 3
4 6
1 2 4 8
5 10
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
import java.util.*;

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

        int t=s.nextInt();
        for(int j=0; j<t; j++) {
            int n=s.nextInt();
            int k=s.nextInt();
            int[] arr=new int[n];

            for(int i=0; i<n; i++){
                arr[i]=s.nextInt();
            }

            ArrayList<ArrayList<Integer>> subarrays = make(n, arr);

            int[] x = new int[subarrays.size()];

            for (int i = 0; i < subarrays.size(); i++) {
                x[i] = xor(subarrays.get(i));
            }

            Arrays.sort(x);
            System.out.println(x[k-1]);
        }

    }

    static int xor(ArrayList<Integer> a){
        int c=Integer.MAX_VALUE;

        for(int i=0; i<a.size()-1; i++){
            int z=a.get(i)^a.get(i+1);
            if(z<c) c=z;
        }

        return c;
    }

    static ArrayList<ArrayList<Integer>> make(int n, int[] a){
        ArrayList<ArrayList<Integer>> subarrays = new ArrayList<>();

        for (int len = 2; len <= n; len++) {
            for (int start = 0; start <= n - len; start++) {
                ArrayList<Integer> subarray = new ArrayList<>();
                for (int k = start; k < start + len; k++) {
                    subarray.add(a[k]);
                }
                subarrays.add(subarray);
            }
        }

        return subarrays;
    }
}

**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 1
1 1 1 1 1
4 6
1 2 4 8
2 1
0 0
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

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

public class CF_1983_F{
    //SOLUTION BEGIN
    int B = 24;
    void pre() throws Exception{
        to = new int[MAX][];
        maxInd = new int[MAX];
    }
    void solve(int TC) throws Exception{
        int N = ni();
        long K = nl();
        int[] V = new int[N];
        for(int i = 0; i< N; i++)V[i] = ni();

        int lo = 0, hi = (1<<B)-1;
        while(lo < hi){
            int mid = lo + (hi-lo)/2;
            if(count(N, V, mid) >= K)hi = mid;
            else lo = mid+1;
        }
        pn(hi);
    }
    long count(int N, int[] V, int v){
        long subCount = 0;
        NODES = 0;
        newNode();
        int root = newNode();
        int upto = -1;
        for(int i = 0; i< N; i++){
            int ind = maxIndexWithXorUptoX(root, V[i], v);
            upto = Math.max(upto, ind);
            subCount += upto+1;
            insert(root, i, V[i]);
        }
        return subCount;
    }
    int MAX = (int)2e7;
    int NODES;
    int[][] to;
    int[] maxInd;
    int newNode() {
        to[NODES] = new int[]{0, 0};
        maxInd[NODES] = -1;
        return NODES++;
    }
    void insert(int node, int ind, int X){
        maxInd[node] = ind;
        for(int b = B-1; b>= 0; b--){
            int bit = (X>>b)&1;
            if(to[node][bit] == 0)to[node][bit] = newNode();
            node = to[node][bit];
            maxInd[node] = ind;
        }
    }
    int maxIndexWithXorUptoX(int node, int val, int X){
        int max = -1;
        for(int b = B-1; b >= 0; b--){
            int bit = (val>>b)&1;
            if(((X>>b)&1)==1){
                max = Math.max(max, maxInd[to[node][bit]]);
                bit ^= 1;
            }
            node = to[node][bit];
        }
        max = Math.max(max, maxInd[node]);
        return max;
    }
    //SOLUTION END
    void hold(boolean b)throws Exception{if(!b)throw new Exception("Hold right there, Sparky!");}
    void exit(boolean b){if(!b)System.exit(0);}
    static void dbg(Object... o){System.err.println(Arrays.deepToString(o));}
    DecimalFormat df = new DecimalFormat("0.00000000000");
    double PI = 3.141592653589793238462643383279502884197169399;
    static boolean multipleTC = true, memory = true, fileIO = false;
    FastReader in;PrintWriter out;
    void run() throws Exception{
        long ct = System.currentTimeMillis();
        if (fileIO) {
            in = new FastReader("");
            out = new PrintWriter("");
        } else {
            in = new FastReader();
            out = new PrintWriter(System.out);
        }
        //Solution Credits: Taranpreet Singh
        int T = multipleTC? ni():1;
        pre();
        for (int t = 1; t <= T; t++) solve(t);
        out.flush();
        out.close();
        System.err.println("Runtime: "+(System.currentTimeMillis() - ct));
    }
    public static void main(String[] args) throws Exception{
        if(memory)new Thread(null, new Runnable() {public void run(){try{new CF_1983_F().run();}catch(Exception e){e.printStackTrace();System.exit(1);}}}, "1", 1 << 28).start();
        else new CF_1983_F().run();
    }
    int[][] make(int n, int e, int[] from, int[] to, boolean f){
        int[][] g = new int[n][];int[]cnt = new int[n];
        for(int i = 0; i< e; i++){
            cnt[from[i]]++;
            if(f)cnt[to[i]]++;
        }
        for(int i = 0; i< n; i++)g[i] = new int[cnt[i]];
        for(int i = 0; i< e; i++){
            g[from[i]][--cnt[from[i]]] = to[i];
            if(f)g[to[i]][--cnt[to[i]]] = from[i];
        }
        return g;
    }
    int[][][] makeS(int n, int e, int[] from, int[] to, boolean f){
        int[][][] g = new int[n][][];int[]cnt = new int[n];
        for(int i = 0; i< e; i++){
            cnt[from[i]]++;
            if(f)cnt[to[i]]++;
        }
        for(int i = 0; i< n; i++)g[i] = new int[cnt[i]][];
        for(int i = 0; i< e; i++){
            g[from[i]][--cnt[from[i]]] = new int[]{to[i], i, 0};
            if(f)g[to[i]][--cnt[to[i]]] = new int[]{from[i], i, 1};
        }
        return g;
    }
    int find(int[] set, int u){return set[u] = (set[u] == u?u:find(set, set[u]));}
    int digit(long s){int ans = 0;while(s>0){s/=10;ans++;}return ans;}
    long gcd(long a, long b){return (b==0)?a:gcd(b,a%b);}
    int gcd(int a, int b){return (b==0)?a:gcd(b,a%b);}
    int bit(long n){return (n==0)?0:(1+bit(n&(n-1)));}
    void p(Object... o){for(Object oo:o)out.print(oo+" ");}
    void pn(Object... o){for(int i = 0; i< o.length; i++)out.print(o[i]+(i+1 < o.length?" ":"\n"));}
    void pni(Object... o){for(Object oo:o)out.print(oo+" ");out.println();out.flush();}
    String n()throws Exception{return in.next();}
    String nln()throws Exception{return in.nextLine();}
    int ni()throws Exception{return Integer.parseInt(in.next());}
    long nl()throws Exception{return Long.parseLong(in.next());}
    double nd()throws Exception{return Double.parseDouble(in.next());}

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

        public FastReader(String s) throws Exception{
            br = new BufferedReader(new FileReader(s));
        }

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

        String nextLine() throws Exception{
            String str;
            try{
                str = br.readLine();
            }catch (IOException e){
                throw new Exception(e.toString());
            }
            return str;
        }
    }
}

**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
2
3 1
1 2 3
3 3
1 2 3
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

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

#define pt "--> "
#define Fast() ios_base::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL)
typedef long long int ll ;
const ll N = 2e5 + 7 , M = 2e2 + 7 , lg = 30 , len = 174 , Sq = 350 , oo = 1e9 , mod = 998244353 ;
int dx[5] = {0 , -1 , 0 , 1 , 0} , dy[5] = {0 , 0 , -1 , 0 , 1} ;

struct node {
    node * child[2] = {nullptr, nullptr} ;
    int cnt = 0 ;
};
node * head = new node ;

ll n , k ;
ll a[N] ;

void add(ll x, ll val = 1) {
    node * cur = head ;
    cur->cnt += val ;
    for(int b = lg-1 ; ~b ; b--) {
        if(!cur->child[x >> b & 1])
            cur->child[x >> b & 1] = new node ;
        cur = cur->child[x >> b & 1] ;
        cur->cnt += val ;
    }
}

ll get(ll x, ll res = 0) {
    node * cur = head ;
    for(int b = lg-1 ; ~b ; b--) {
        if(!cur->child[x >> b & 1] || !(cur->child[x >> b & 1]->cnt)) {
            cur = cur->child[!(x >> b & 1)] ;
            res |= (1ll << b) ;
            continue ;
        }
        cur = cur->child[x >> b & 1] ;
    }
    return res ;
}

bool check(ll x, ll res = 0) {
    int l = 0 ;
    for(int r = 1 ; r <= n ; r++) {
        while(head->cnt && get(a[r]) <= x)
            add(a[++l], -1);
        add(a[r]) ;
        res += l ;
    }
    return res >= k ;
}

void dfs(node * cur) {
    cur->cnt = 0 ;
    for(int i = 0 ; i < 2 ; i++)
        if(cur->child[i])
            dfs(cur->child[i]) ;
}

void solve() {
    cin >> n >> k ;
    for(int i = 1 ; i <= n ; i++)
        cin >> a[i] ;

    ll l = 0, r = 1ll << 30, mid ;
    while(r-l > 1) {
        mid = l+r >> 1 ;
        (check(mid) ? r : l) = mid ;
        dfs(head) ;
    }
    cout << (check(r) ? r : l) << endl ;
}

int main() {
	Fast() ;

    int T ; cin >> T ;
    while(T--)
        solve() ;
}

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

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

Think step by step.

Response:
```plaintext
2
3 1
1 1 1
4 6
1 2 4 8
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

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

using namespace std;

#define int long long

const int LG = 30;
const int INF = 1e9;

struct Node {
  Node() {
    c[0] = c[1] = nullptr;
    mn = INF;
  }

  void insert(int val, int n, int pos = LG-1) {
    mn = min(mn, n);
    if (pos == -1) {
      return;
    }
    int bit = (val >> pos) & 1;
    if (c[bit] == nullptr) {
      c[bit] = new Node();
    }
    c[bit]->insert(val, n, pos - 1);
  }

  int getMin(int val, int X, int pos = LG-1) {
    if (pos == -1) {
      return mn;
    }
    int bitVal = (val >> pos) & 1;
    int bitX = (X >> pos) & 1;

    if (bitX == 0) {
      if (c[bitVal] != nullptr) {
        return c[bitVal]->getMin(val, X, pos - 1);
      } else {
        return INF;
      }
    } else {
      int ret = INF;
      if (c[1 - bitVal] != nullptr) {
        ret = min(ret, c[1 - bitVal]->getMin(val, X, pos - 1));
      }
      if (c[bitVal] != nullptr) {
        ret = min(ret, c[bitVal]->mn);
      }
      return ret;
    }
  }

  ~Node() {
    delete c[0];
    delete c[1];
  }

  Node* c[2];
  int mn;
};

int getCntLess(const vector<int>& a, const int n, const int X) {
  Node root;
  vector<int> nxt(n);
  for (int i = n-1; i >= 0; i--) {
    nxt[i] = root.getMin(a[i], X);
    root.insert(a[i], i);
  }
  nxt[n-1] = n;
  for (int i = n-2; i >= 0; i--) {
    nxt[i] = min(nxt[i], nxt[i+1]);
  }
  int cnt = 0;
  for (int i = 0; i < n; i++) {
    cnt += n - nxt[i];
  }
  return cnt;
}

void solve() {
  int n, kConst;
  cin >> n >> kConst;
  vector<int> a(n);
  for (int& x : a) {
    cin >> x;
  }
  int res = 0;
  for (int i = LG-1; i >= 0; i--) {
    int tryRes = res + (1 << i);
    int cntLess = getCntLess(a, n, tryRes - 1);
    if (cntLess < kConst) {
      res = tryRes;
    }
  }
  cout << res << '\n';
}

signed main() {
  int t;
  cin >> t;
  while (t--) {
    solve();
  }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases that can highlight issues in the algorithm. Here are the steps to consider:

1. **Small Test Case**: Since we need a small-scale test input, we will use a small array size.
2. **Edge Cases**: We should consider arrays with:
   - All elements being the same.
   - Elements that have a large difference in values.
   - Elements that have a specific pattern to maximize the XOR values.
3. **Multiple Test Cases**: We can include a few test cases to cover different scenarios.

Let's construct a test input that includes these considerations:

- **Test Case 1**: All elements are the same.
- **Test Case 2**: Elements have a large difference.
- **Test Case 3**: Elements have a specific pattern to maximize XOR values.

Here is the test input:

```plaintext
3
5 2
1 1 1 1 1
5 2
0 1 2 3 4
5 2
1 3 5 7 9
```

Explanation:
1. **Test Case 1**: All elements are the same. The sub-arrays will have XOR values of 0, which is the only possible value.
2. **Test Case 2**: Elements have a large difference. This will test how the algorithm handles different values and their XOR results.
3. **Test Case 3**: Elements have a specific pattern to maximize XOR values. This will test the algorithm's ability to handle different XOR values.

This input should help expose any issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
def find_min_xor(sub):
    min_xor = float('inf')
    sub.sort()
    for i in range(len(sub) - 1):
        min_xor = min(min_xor, sub[i] ^ sub[i + 1])
    return min_xor

for _ in range(int(input())):
    n = list(map(int, input().split()))
    main_list = list(map(int, input().split()))

    sub_xor_values = []

    for r in range(2, len(main_list) + 1):
        for i in range(len(main_list) - r + 1):
            sub = main_list[i:i + r]
            sub_xor_values.append(find_min_xor(sub))

    sub_xor_values.sort()
    print(sub_xor_values[n[1] - 1])

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
def find_min_xor(sub):
    min_xor = float('inf')
    for i in range(len(sub)):
        for j in range(i + 1, len(sub)):
            min_xor = min(min_xor, sub[i] ^ sub[j])
    return min_xor

for _ in range(int(input())):
    n = list(map(int, input().split()))
    main_list = list(map(int, input().split()))

    sub_xor_values = []

    for r in range(2, len(main_list) + 1):
        for i in range(len(main_list) - r + 1):
            sub = main_list[i:i + r]
            sub_xor_values.append(find_min_xor(sub))

    sub_xor_values.sort()
    print(sub_xor_values[n[1] - 1])

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

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

Think step by step.

Response:
```plaintext
3
5 1
1 2 3 4 5
4 6
1 2 4 8
6 10
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: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
import java.io.*;
import java.util.*;
public class Main {
    static long MAX_LEN = (long)1e18+50;
    static final String ENDL = "\n";
    public static void main(String[] args) throws Exception {
        InputOutput io = new InputOutput();
	    for(int t = io.nextInt(); t > 0; t--) {
            int n = io.nextInt();
            long k = io.nextLong();
            int[] val = io.nextIntArr(n);
            int lo = 0, hi = 1000000005;
            while(lo+1<hi) {
                int mid = lo+(hi-lo)/2;
                if(count(val,mid)<k) {
                    lo=mid;
                } else {
                    hi=mid;
                }
            }
            if(count(val,lo)<k) {
                io.append(hi);
            } else {
                io.append(lo);
            }
        }
        io.printAns();
    }
    static long count(int[] val, int tolerance) {
        BinaryTrie b = new BinaryTrie();
        int best = -1;
        long ans = 0;
        for(int i = 0; i < val.length; i++) {
            best = Math.max(best, b.find(val[i], tolerance));
            ans+=best+1;
            b.insert(i, val[i]);
        }
        // System.out.println("LINE 38: "+tolerance+" " + ans);
        return ans;
    }
    static class BinaryTrie {
        Node root = new Node(null);
        void insert(int index, int number) {
            Node cur = root;
            for(int i = 30; i >= 0; i--) {
                if((number&(1<<i))==0) {
		            if(cur.left==null) {
                        cur.left = new Node(cur);
                    }
                    cur=cur.left;
		        } else {
                    if(cur.right==null) {
                        cur.right = new Node(cur);
                    }
                    cur = cur.right;
		        }
	        }
            while(cur!=root) {
                cur.index=index;
                cur=cur.parent;
            }
	    }
        int find(int xor, int tolerance) {
            int ans = -1;
            Node cur = root;
            for(int i = 30; i >= 0; i--) {
                if((tolerance&(1<<i))==0) {
                    if((xor&(1<<i))==0) {
                        if(cur.left==null) return ans;
                        cur=cur.left;
                    } else {
                        if(cur.right==null) return ans;
                        cur=cur.right;
                    }
                } else {
                    if((xor&(1<<i))==0) {
                        if(cur.left!=null) ans=Math.max(ans, cur.left.index);
                        if(cur.right==null) return ans;
                        cur=cur.right;
                    } else {
                        if(cur.right!=null) ans = Math.max(ans, cur.right.index);
                        if(cur.left==null) return ans;
                        cur=cur.left;
                    }
                }
            }
            ans=Math.max(ans, cur.index);
            return ans;
        }
    }
    static class Node {
        Node parent;
		Node left, right;
		int index;
        Node(Node parent) {
            this.parent=parent;
        }
    }
    static class InputOutput {
        InputStream stream;
        byte[] buf = new byte[1<<16];
        int curChar;
        int numChars;
        PrintWriter pw;
        StringBuilder ans = new StringBuilder();
        public InputOutput() throws Exception {
            pw = new PrintWriter(System.out);
            stream = System.in;
        }
        public InputOutput(String name) throws Exception {
            pw = new PrintWriter(new FileWriter(name+".out"));
            stream = new FileInputStream(name+".in");
        }
        public int[] nextIntArr(int size) {
            int[] ret = new int[size];
            for(int i = 0; i < size; i++) ret[i]=nextInt();
            return ret;
        }
        public long[] nextLongArr(int size) {
            long[] ret = new long[size];
            for(int i = 0; i < size; i++) ret[i]=nextInt();
            return ret;
        }
        private int nextByte() {
            if (numChars == -1) { throw new InputMismatchException(); }
            if (curChar >= numChars) {
                curChar = 0;
                try {
                    numChars = stream.read(buf);
                } catch (IOException e) { throw new InputMismatchException(); }
                if (numChars == -1) {
                    return -1;  // end of file
                }
            }
            return buf[curChar++];
        }
        public String next() {
            int c;
            do { c = nextByte(); } while (c <= ' ');

            StringBuilder res = new StringBuilder();
            do {
                res.appendCodePoint(c);
                c = nextByte();
            } while (c > ' ');
            return res.toString();
        }
        public int nextInt() {  // nextLong() would be implemented similarly
            int c;
            do { c = nextByte(); } while (c <= ' ');

            int sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = nextByte();
            }

            int res = 0;
            do {
                if (c < '0' || c > '9') { throw new InputMismatchException(); }
                res = 10 * res + c - '0';
                c = nextByte();
            } while (c > ' ');
            return res * sgn;
        }
        public long nextLong() {  // nextLong() would be implemented similarly
            int c;
            do { c = nextByte(); } while (c <= ' ');

            long sgn = 1;
            if (c == '-') {
                sgn = -1;
                c = nextByte();
            }

            long res = 0;
            do {
                if (c < '0' || c > '9') { throw new InputMismatchException(); }
                res = 10 * res + c - '0';
                c = nextByte();
            } while (c > ' ');
            return res * sgn;
        }
        public double nextDouble() { return Double.parseDouble(next()); }
        public void append(int i) {
            ans.append(i + "\n");
        }
        public void append(long i) {
            ans.append(i + "\n");
        }
        public void append(String s) {
            ans.append(s + "\n");
        }
        public void printAns() {
            pw.print(ans);
            pw.close();
        }
    }
}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the given buggy code, we need to focus on edge cases that might cause the code to fail. Here are some considerations:

1. **Edge Cases for Small Arrays:**
   - Arrays with very few elements (e.g., 2 or 3 elements) can sometimes reveal bugs in the logic for handling small sub-arrays.

2. **Edge Cases for Large Values:**
   - Using large values in the array can test the limits of the bitwise operations and the data structures used.

3. **Edge Cases for Specific XOR Patterns:**
   - Arrays with specific XOR patterns can test the correctness of the XOR operations and the logic for finding the minimum XOR values.

4. **Edge Cases for Specific K Values:**
   - Test cases where `k` is the smallest or largest possible value for the given `n` can reveal issues in the binary search and counting logic.

Given these considerations, let's create a small-scale test input that is complete and valid:

```plaintext
4
3 1
1 2 3
3 2
1 2 3
4 1
1 2 4 8
4 6
1 2 4 8
```

### Explanation:
1. **Test Case 1:**
   - `n = 3`, `k = 1`
   - Array: `1, 2, 3`
   - This tests the smallest possible `k` for a small array. The sub-arrays are `[1, 2]`, `[2, 3]`, and `[1, 2, 3]`, with minimum XOR values `3`, `1`, and `1` respectively. The 1st smallest value is `1`.

2. **Test Case 2:**
   - `n = 3`, `k = 2`
   - Array: `1, 2, 3`
   - This tests a larger `k` for a small array. The sub-arrays are the same as above, with minimum XOR values `3`, `1`, and `1` respectively. The 2nd smallest value is `1`.

3. **Test Case 3:**
   - `n = 4`, `k = 1`
   - Array: `1, 2, 4, 8`
   - This tests a larger array with specific XOR patterns. The sub-arrays are `[1, 2]`, `[2, 4]`, `[4, 8]`, `[1, 2, 4]`, `[2, 4, 8]`, and `[1, 2, 4, 8]`, with minimum XOR values `3`, `6`, `12`, `1`, `4`, and `1` respectively. The 1st smallest value is `1`.

4. **Test Case 4:**
   - `n = 4`, `k = 6`
   - Array: `1, 2, 4, 8`
   - This tests the largest possible `k` for a given `n`. The sub-arrays are the same as above, with minimum XOR values `3`, `6`, `12`, `1`, `4`, and `1` respectively. The 6th smallest value is `12`.

These test cases should help expose any bugs in the given buggy code by covering a range of edge cases and specific scenarios.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: array-value
time_limit_ms: 4000
memory_limit_mb: 256
Description: You have an array of non-negative integers $a_1, a_2, \ldots, a_n$.

The value of a sub-array of length $\ge 2$, $a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$ is the minimum value of $a_i \oplus a_j$ such that $l \le i < j \le r$, where $\oplus$ is the xor (exclusive-or) operator.

You have to find the $k$-th smallest value over all sub-arrays of length $\ge 2$.

Input: The first line of the input contains multiple test cases $t$ ($1 \le t \le 2 \cdot 10^4$).

The first line of each test case contains integer numbers $n$ and $k$ ($2 \le n \le 10^5$, $1 \le k \le \frac{n\cdot(n-1)}{2}$).

The second line of the input contains $n$ non-negative integer numbers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — the array itself.

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

Output: Print the $k$-th smallest value obtained over all subarrays of length at least $2$.

Examples:
input:
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
output:
1
7
12
3

Note: In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$[1,2]: 3$

$[2,3]: 1$

$[3,4]: 7$

$[4,5]: 1$

$[1,2,3]: 1$

$[2,3,4]: 1$

$[3,4,5]: 1$

$[1,2,3,4]: 1$

$[2,3,4,5]: 1$

$[1,2,3,4,5]: 1$

The sorted order would be: $1, 1, 1, 1, 1, 1, 1, 1, 3, 7$. Therefore, the second smallest element would be $1$.

Buggy Code:
import java.util.*;

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

        int t=s.nextInt();
        for(int j=0; j<t; j++) {
            int n=s.nextInt();
            int k=s.nextInt();
            long[] arr=new long[n];

            for(int i=0; i<n; i++){
                arr[i]=s.nextInt();
            }

            ArrayList<ArrayList<Long>> subarrays = make(n, arr);

            long[] x = new long[subarrays.size()];

            for (int i = 0; i < subarrays.size(); i++) {
                x[i] = xor(subarrays.get(i));
            }

            Arrays.sort(x);
            System.out.println(x[k-1]);
        }

    }

//    static int xor(ArrayList<Integer> a){
//        int c=Integer.MAX_VALUE;
//
//        for(int i=0; i<a.size()-1; i++){
//            int z=a.get(i)^a.get(i+1);
//            if(z<c) c=z;
//        }
//
//        return c;
//    }

    static long xor(ArrayList<Long> a){
        long c=Integer.MAX_VALUE;

        for(int i=0; i<a.size(); i++){
            for(int j=0; j<a.size() && i!=j; j++) {
//                System.out.println(a.get(i));
//                System.out.println(a.get(j));
                long z = a.get(i) ^ a.get(j);
                if (z < c) c = z;
            }
        }

        return c;
    }

    static ArrayList<ArrayList<Long>> make(int n, long[] a){
        ArrayList<ArrayList<Long>> subarrays = new ArrayList<>();

        for (int len = 2; len <= n; len++) {
            for (int start = 0; start <= n - len; start++) {
                ArrayList<Long> subarray = new ArrayList<>();
                for (int k = start; k < start + len; k++) {
                    subarray.add(a[k]);
                }
                subarrays.add(subarray);
            }
        }

        return subarrays;
    }
}

**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
3 1
1 1 1
4 6
1 2 4 8
5 10
1 2 3 4 5
```