**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
/*
-   In the name of Allah, Most Gracious, Most Merciful.
-   Department of Computer Science and Engineering.
-   Mawlana Bhashani Science and Technology University.
-   Author : Mahmudunnabi_Fahim
-   Mail   : mahmudunnabifahim3579@gmail.com
-   Created: 20:37:35 28-12-2024
*/
#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define ld long double
#define pb push_back
#define all(x) x.begin(), x.end()
#define rev(x) x.begin(), x.end(), greater<int>()
#define ff first
#define ss second
#define mkp make_pair
#define mod 1000000007
void solve()
{
    int n;
    cin >> n;
    vector<int> a(n);
    for (int i = 0; i < n; i++)
        cin >> a[i];
    //sort(all(a));
    if(n==2)
    {
        sort(all(a));
        if(a[0]*2>a[1])
        cout<<"YES\n";
        else cout<<"NO\n";
        return;
    }
    for (int i = 0; i<n-1; i++)
    {
        if ((a[i]<a[i+1] and a[i]*2>a[i+1]) or (a[i]>a[i+1] and a[i+1]*2>a[i]))
        {
            cout << "YES\n";
            return;
        }
    }
    cout << "NO\n";
    return;
}
int32_t main()
{
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(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
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
int main(){
	int t;
	cin >> t;
	while(t--){
		int n;
		cin >> n;
		vector<int> a(n);
		for (int i = 0;i<n;++i){
			cin >> a[i];
		}
		bool check = false;
		for (int i = 0;i<n;++i){
			if(i<=n-3){
				if(a[i] + a[i+1] > a[i+2] and a[i] + a[i+2] > a[i+1] and a[i+1] + a[i+2] > a[i]){
					check = true;
					break;
				}
			}else if(i==n-2){
				int mx = max(a[i], a[i+1]);
				int mn = min(a[i], a[i+1]);
				if(mn + mn > mx){
					check = true;
					break;
				}
			}
		}
		if(check){
			cout << "YES\n";
		}else cout << "NO\n";
	}
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
t=int(input())
for i in range(t):
  n=int(input())
  e_a = input().split()
  a=[int(i) for i in e_a]
  l=False
  if len(a)==2 and a[0]==a[1]:
    print('YES')
    break
  for i in range(len(a)-2):
    if l:
      break
    if a[i]+a[i+1]>a[i+2] and a[i]+a[i+2]>a[i+1] and a[i+2]+a[i+1]>a[i]:
      l=True
      print('YES')
      break
  if not l:
    print('NO')

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

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

Think step by step.

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

### Explanation:
1. **First Test Case:**
   - Input: `1 2 2 3`
   - This array can be partitioned in multiple ways:
     - `[1, 2, 2]` and `[3]`: Both subarrays are stable.
     - `[1]` and `[2, 2, 3]`: Both subarrays are stable.
   - Expected Output: `YES`

2. **Second Test Case:**
   - Input: `1 1 1`
   - This array can only be partitioned into single-element subarrays:
     - `[1]`, `[1]`, and `[1]`: Each subarray is stable, but there is only one way to partition the array.
   - Expected Output: `NO`

3. **Third Test Case:**
   - Input: `1 2 3 4 5`
   - This array can be partitioned in multiple ways:
     - `[1, 2, 3]` and `[4, 5]`: Both subarrays are stable.
     - `[1]`, `[2, 3, 4]`, and `[5]`: Each subarray is stable.
   - Expected Output: `YES`

These test cases are designed to expose the bug in the provided code by focusing on edge cases and scenarios where the code might fail to correctly identify valid partitions.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
def _tri_(a, n):
    if n == 2:
        if a[0] < 2*a[1] or a[1] < 2*a[0]:
            return 'YES'
    for i in range(n-2):
        if a[i] + a[i+1] > a[i+2] and a[i] + a[i+2] > a[i+1] and a[i+2] + a[i+1] > a[i]:
            return 'YES'
    return 'NO'

t = int(input())
for _ in range(t):
    n = int(input())
    a = list(map(int, input().split(' ')))
    print(_tri_(a, n))

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

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

Think step by step.

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

**Explanation:**

1. **Test Case 1:**
   - Input: `3\n1 1 2`
   - Expected Output: `NO`
   - Reason: The only possible partition is `[1, 1, 2]`, but this set is not stable because `1 + 1` is not greater than `2`.

2. **Test Case 2:**
   - Input: `4\n1 2 2 3`
   - Expected Output: `YES`
   - Reason: There are multiple valid partitions:
     - `[1, 2, 2], [3]` (both subsegments are stable)
     - `[1], [2, 2, 3]` (both subsegments are stable)
     - `[1], [2], [2, 3]` (all subsegments are stable)

3. **Test Case 3:**
   - Input: `5\n1 1 1 2 2`
   - Expected Output: `YES`
   - Reason: There are multiple valid partitions:
     - `[1, 1, 1], [2, 2]` (both subsegments are stable)
     - `[1, 1], [1, 2, 2]` (both subsegments are stable)
     - `[1], [1], [1, 2, 2]` (all subsegments are stable)

These test cases are designed to expose the bug in the provided code by testing edge cases where the stability condition is not met and where multiple valid partitions exist.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
import java.util.*;

public class JavaCode {
    public static void main(String[] args){
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-- > 0){
            int n = sc.nextInt();
            int a[] = new int[n];
            boolean flag = false;
            for(int i =0 ; i < n ; i++){
                a[i] = sc.nextInt();
                if(i != 0 && Math.abs(a[i-1]-a[i]) == 3){
                    if(a[i] != 1 && a[i] != 2 && a[i] != 3){
                        if(a[i-1] != 1 && a[i-1] != 2 && a[i-1] != 3){
                            flag = true;
                        }
                    }

                }else if(i != 0 && Math.abs(a[i-1]-a[i]) < 3){
                    flag = true;
                }
            }
            if(flag){
                System.out.println("YES");

            }else{
                System.out.println("NO");
            }

        }
    }

}

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

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

Think step by step.

Response:
```plaintext
5
4
1 2 3 4
4
4 5 6 7
3
1 1 1
2
100000 100000
5
1 3 5 7 9
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
import java.util.*;
public class A_Tender_Carpenter {
    public static void main(String args[]) {
        Scanner sc=new Scanner(System.in);
        long t=sc.nextLong();
        while(t-->0) {
            int n=sc.nextInt();
            int arr[]=new int[n];
            for(int i=0;i<n;i++) {
                arr[i]=sc.nextInt();
            }
            HashMap<Integer,Integer> map=new HashMap<>();
            for(int i=0;i<n;i++) {
                map.put(arr[i],map.getOrDefault(arr[i],0)+1);
            }
            boolean found=false;
            for(int num:map.values()) {
                if(num>=2) {
                    System.out.println("YES");
                    found=true;
                    break;
                }
            }
            if(!found) {
                Arrays.sort(arr);
                if(arr[0]+arr[1]>=arr[2]) {
                    System.out.println("YES");
                }
                else {
                    System.out.println("NO");
                }
            }
        }
        sc.close();
    }
}

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

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

Think step by step.

Response:
```plaintext
3
3
1 1 2
4
1 2 3 6
2
100000 100000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define fast_io ios::sync_with_stdio(0); cin.tie(0); cout.tie(0)
typedef long long ll;
typedef vector<ll> vi;

void solve() {
    int n;
    cin >> n;
    vi v(n);
    for (int i = 0; i < n; ++i) {
        cin >> v[i];
    }
    if(n==2)
        {cout<<"yes"<<'\n';
    return ;}

    bool found = false;
    for (int x = 0; x < n - 1; ++x) {
        if ((v[x] < v[x + 1] && 2 * v[x] > v[x + 1]) ||
            (v[x] > v[x + 1] && v[x] < 2 * v[x + 1])) {
            cout << "yes" << endl;
            found = true;
            break;
        }
    }

    if (!found) {
        cout << "no" << endl;
    }
}

int main() {
    fast_io;
    int t;
    cin >> t;
    while (t--) {
        solve();
    }
    return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided buggy code, we need to craft a test input that highlights the issues in the logic. The primary issue in the code is that it only checks pairs of adjacent elements to determine if they can form a non-degenerate triangle, but it does not consider the stability of the entire subsegment.

### Key Points to Consider:
1. **Single Element Subsegments**: The code incorrectly assumes that a single element is always a valid subsegment, which is true. However, it does not consider the stability of multi-element subsegments.
2. **Pairwise Comparison**: The code only checks pairs of adjacent elements, which is insufficient to determine the stability of a subsegment.
3. **Edge Cases**: We need to create a test case where the pairwise comparison fails to identify a valid partition, but a valid partition exists.

### Test Input:
Let's create a test input where the pairwise comparison fails to identify a valid partition, but a valid partition exists. Consider the following array:
- `[4, 5, 6, 7]`

In this array:
- The pairwise comparison will fail because:
  - `4 + 5 <= 9` (not a valid triangle)
  - `5 + 6 <= 11` (not a valid triangle)
  - `6 + 7 <= 13` (not a valid triangle)

However, the subsegments `[4, 5]` and `[6, 7]` are both stable:
- `[4, 5]` is stable because `4 + 4 > 5`, `4 + 5 > 4`, and `5 + 5 > 4`.
- `[6, 7]` is stable because `6 + 6 > 7`, `6 + 7 > 6`, and `7 + 7 > 6`.

### Test Input:
```plaintext
1
4
4 5 6 7
```

### Expected Output:
```plaintext
YES
```

This test input should expose the bug in the code because the pairwise comparison fails to identify the valid partition, but a valid partition exists.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

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

using ll= long long;
using lld= long double;
using ull= unsigned long long;

const lld pi= 3.141592653589793238;
const lld mod=1e9+7;

int binary_search(vector<int>& v,int target){
    int low = 0;
    int high = v.size() - 1;
    sort(v.begin(),v.end());
    while(low<=high){
        int mid =(low + high)/2;
        if(v[mid] < target){
            low = mid + 1;
        }
        else{
            high = mid - 1;
        }
    }
    if(v[low] == target) return low;
    if(v[high] == target) return high;
    return -1;
}

int lower_bound(vector<int>& v,int target){
    int low = 0,high = v.size()-1;
    int ans = -1;
    while(low<=high){
        int mid = (low + high)/2;
        if(v[mid] >= target){
            ans = mid;
            high = mid-1;
        }
        else low = mid +1;
    }
    return ans;
}

int upper_bound(vector<int>& v,int target){
    int low = 0,high = v.size()-1;
    int ans = -1;
    while (low <= high) {
        int mid = (low + high) / 2;
        if (v[mid] > target) {
            ans = mid;
            high = mid - 1;
        }
        else {
            low = mid + 1;
        }
    }

    return ans;
}

int gcd(int a,int b){
    if (b == 0) return a;
    return gcd(b, a % b);
}
int lcm(int a, int b){
    return (a/gcd(a,b)*b);
}

const int MAX=1e7+10;
vector<bool>prime(MAX,1);
void SieveOfEratosthenes(){
    prime[0]=prime[1]=false;
    for(int i=2;i< MAX;i++){
        if(prime[i]){
            for(int j=2*i;j<MAX;j+=i){
                prime[j]=false;
            }
        }
    }
}

bool isPrime(int n){
    if(n<=1) return false;
    if(n<=3) return true;
    if(n%2==0||n%3==0)return false;
    for(int i=5;i*i<=n;i=i+6){
        if(n%i==0||n%(i+2)==0)return false;
        return true;
    }
}

bool sort_inc(int a,int b){
    return a < b;
}

bool sort_dec(int a,int b){
    return a > b;
}

void sliding_window(vector<int>& v){
    int l = 0, r=0;
    int n = v.size() - 1;
    while(r < n){
        /*
        condition check

        while( desird){

            l++;
        }
        ans = max(ans, r - l +1);
        */
        r++;
    }
    //cout<<ans<<endl;
}

signed main(){
    int t;cin>>t;
    while(t--){
        int n;cin>>n;
        vector<int> v;
        for(int i = 0;i<n;i++){
            int x;cin>>x;
            v.push_back(x);
        }
        int ct = 0;
        int l =0;
        int r = 0;
        while(r < n){
            r++;
            if(v[l] + v[l] > v[r] && v[r] + v[r] > v[l]){
                ct++;
                l = r;
            }
            l++;
        }
        if(ct>=2 || (n == 2)) cout<<"YES"<<endl;
        else cout<<"NO"<<endl;
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

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

    mx = [[0] * n for _ in range(n)]
    mn = [[0] * n for _ in range(n)]

    for i in range(n):
        mx[i][i] = a[i]
        mn[i][i] = a[i]
        for j in range(i + 1, n):
            mx[i][j] = max(mx[i][j - 1], a[j])
            mn[i][j] = min(mn[i][j - 1], a[j])

    r = 0
    def solve(i, j):
        global r

        if j >= n:
            r += 1
            return
        if i >= n:
            return

        while j < n and mn[i][j] * 2 > mx[i][j]:
            solve(i + 1, j + 1)
            j += 1

    solve(0, 0)
    print("YES" if r > 1 else "NO")

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
def solve():
    n = int(input())
    a=list(map(int,input().split()))
    even_count = sum(1 for x in a if x % 2 == 0)
    if n % 2 == 0 and even_count % 2 == 1:
        return "YES"
    else:
        a.sort()
        found_duplicate = False
        for i in range(1, n):
            if a[i] == a[i - 1]:
                found_duplicate = True
                break
        return "YES" if found_duplicate else "NO"
for _ in range(int(input())):
    print(solve())

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

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

Think step by step.

Response:
```plaintext
3
4
2 2 2 2
4
1 2 3 4
2
100000 1
```

### Explanation:
1. **Case 1: All Elements Are the Same**
   - Input: `4 2 2 2 2`
   - Expected Output: `YES`
   - Reason: This case tests the scenario where all elements in the array are the same. Since the set `{2}` is stable, there are multiple ways to partition the array (e.g., `[2, 2, 2, 2]`, `[2, 2], [2, 2]`, `[2], [2], [2], [2]`).

2. **Case 2: No Duplicates, All Elements Are Distinct**
   - Input: `4 1 2 3 4`
   - Expected Output: `NO`
   - Reason: This case tests the scenario where all elements are distinct. Since the set `{1, 2, 3, 4}` is not stable (e.g., `1, 2, 4` cannot form a non-degenerate triangle), the only way to partition the array is to split each element into a single subsegment, resulting in only one possible partition.

3. **Case 3: Two Elements, Both the Same**
   - Input: `2 100000 1`
   - Expected Output: `NO`
   - Reason: This case tests the scenario where there are only two elements, and they are distinct. Since the set `{100000, 1}` is not stable (e.g., `1, 1, 100000` cannot form a non-degenerate triangle), the only way to partition the array is to split each element into a single subsegment, resulting in only one possible partition.

These test cases are designed to expose the bugs in the provided buggy code by focusing on edge cases and scenarios that maximize the failure probability.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
import java.util.*;

public class Main {

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

        while (t-- > 0) {
            int n = scanner.nextInt();
            int[] arr = new int[n];
            for (int i = 0; i < n; i++) {
                arr[i] = scanner.nextInt();
            }

            System.out.println(canPartitionInTwoWays(arr) ? "YES" : "NO");
        }
    }

    private static boolean canPartitionInTwoWays(int[] arr) {
        for (int num : arr) {
            if (!isStable(num, num, num)) {
                return false;
            }
        }
        List<List<Integer>> stableSubsegments = findStableSubsegments(arr);
        return hasTwoDifferentPartitions(arr, stableSubsegments);
    }

    private static List<List<Integer>> findStableSubsegments(int[] arr) {
        List<List<Integer>> stableSubsegments = new ArrayList<>();
        for (int i = 0; i < arr.length; i++) {
            List<Integer> subsegment = new ArrayList<>();
            for (int j = i; j < arr.length; j++) {
                subsegment.add(arr[j]);
                if (isStable(subsegment)) {
                    stableSubsegments.add(new ArrayList<>(subsegment));
                }
            }
        }
        return stableSubsegments;
    }

    private static boolean isStable(List<Integer> subsegment) {
        for (int i = 0; i < subsegment.size(); i++) {
            for (int j = i; j < subsegment.size(); j++) {
                for (int k = j; k < subsegment.size(); k++) {
                    if (!isStable(subsegment.get(i), subsegment.get(j), subsegment.get(k))) {
                        return false;
                    }
                }
            }
        }
        return true;
    }

    private static boolean isStable(int a, int b, int c) {
        return a + b > c && b + c > a && c + a > b;
    }

    private static boolean hasTwoDifferentPartitions(int[] arr, List<List<Integer>> stableSubsegments) {
        List<List<List<Integer>>> partitions = findPartitions(arr, stableSubsegments);
        return partitions.size() >= 2;
    }

    private static List<List<List<Integer>>> findPartitions(int[] arr, List<List<Integer>> stableSubsegments) {
        List<List<List<Integer>>> partitions = new ArrayList<>();
        backtrack(arr, 0, new ArrayList<>(), stableSubsegments, partitions);
        return partitions;
    }

    private static void backtrack(int[] arr, int start, List<List<Integer>> currentPartition,
                                  List<List<Integer>> stableSubsegments, List<List<List<Integer>>> partitions) {
        if (start == arr.length) {
            if (currentPartition.size() > 1) {
                partitions.add(new ArrayList<>(currentPartition));
            }
            return;
        }

        for (List<Integer> subsegment : stableSubsegments) {
            if (isValidSubsegment(arr, start, subsegment)) {
                currentPartition.add(subsegment);
                backtrack(arr, start + subsegment.size(), currentPartition, stableSubsegments, partitions);
                currentPartition.remove(currentPartition.size() - 1);
            }
        }
    }

    private static boolean isValidSubsegment(int[] arr, int start, List<Integer> subsegment) {
        if (start + subsegment.size() > arr.length) {
            return false;
        }
        for (int i = 0; i < subsegment.size(); i++) {
            if (arr[start + i] != subsegment.get(i)) {
                return false;
            }
        }
        return true;
    }
}

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

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

Think step by step.

Response:
```plaintext
5
4
2 2 2 2
4
1 2 3 4
5
1 1 2 3 5
6
1 1 2 2 3 3
2
100000 100000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Tender Carpenter
time_limit_ms: 1000
memory_limit_mb: 256
Description: In his dream, Cocoly would go on a long holiday with no worries around him. So he would try out for many new things, such as... being a carpenter. To learn it well, Cocoly decides to become an apprentice of Master, but in front of him lies a hard task waiting for him to solve.

Cocoly is given an array $$$a_1, a_2,\ldots, a_n$$$. Master calls a set of integers $$$S$$$ stable if and only if, for any possible $$$u$$$, $$$v$$$, and $$$w$$$ from the set $$$S$$$ (note that $$$u$$$, $$$v$$$, and $$$w$$$ do not necessarily have to be pairwise distinct), sticks of length $$$u$$$, $$$v$$$, and $$$w$$$ can form a non-degenerate triangle$$$^{\text{∗}}$$$.

Cocoly is asked to partition the array $$$a$$$ into several (possibly, $$$1$$$ or $$$n$$$) non-empty continuous subsegments$$$^{\text{†}}$$$, such that: for each of the subsegments, the set containing all the elements in it is stable.

Master wants Cocoly to partition $$$a$$$ in at least two different$$$^{\text{‡}}$$$ ways. You have to help him determine whether it is possible.

Input: Each test contains multiple test cases. The first line of the input contains a single integer $$$t$$$ ($$$1 \leq t \leq 200$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$2 \leq n \leq 200$$$) — the length of the array $$$a$$$.

The second line contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the elements in the array $$$a$$$.

Output: For each test case, print $$$\texttt{YES}$$$ if there are at least two ways to partition $$$a$$$, and $$$\texttt{NO}$$$ otherwise.

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

Examples:
input:
5
4
2 3 5 7
4
115 9 2 28
5
8 4 1 6 2
6
1 5 4 1 4 7
2
100000 100000
output:
YES
NO
NO
YES
YES

Note: In the first test case, here are two possible partitions:

- $$$[2, 3], [5, 7]$$$, since   $$$[2, 3]$$$ is stable because sticks of lengths $$$(2, 2, 2), (2, 2, 3), (2, 3, 3), (3, 3, 3)$$$ respectively can all form non-degenerate triangles.  $$$[5, 7]$$$ is stable because sticks of lengths $$$(5, 5, 5), (5, 5, 7), (5, 7, 7), (7, 7, 7)$$$ respectively can all form non-degenerate triangles.
- and $$$[2], [3, 5], [7]$$$, since   $$$[2]$$$ is stable because sticks of lengths $$$(2, 2, 2)$$$ respectively can form a non-degenerate triangle.  $$$[3, 5]$$$ is stable because sticks of lengths $$$(3, 3, 3), (3, 3, 5), (3, 5, 5), (5, 5, 5)$$$ respectively can all form non-degenerate triangles.  $$$[7]$$$ is stable because sticks of lengths $$$(7, 7, 7)$$$ respectively can form a non-degenerate triangle.

Note that some other partitions also satisfy the constraints, such as $$$[2], [3], [5], [7]$$$ and $$$[2], [3], [5, 7]$$$.

In the second test case, Cocoly can only partition each element as a single subsegment, resulting in $$$[115], [9], [2], [28]$$$. Since we only have one possible partition, the answer is $$$\texttt{NO}$$$.

In the third test case, please note that the partition $$$[8, 4], [1], [6], [2]$$$ does not satisfy the constraints, because $$$\{8, 4\}$$$ is not a stable set: sticks of lengths $$$4$$$, $$$4$$$, and $$$8$$$ cannot form a non-degenerate triangle.

Buggy Code:
import java.util.*;

public class Main {

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int t = scanner.nextInt(); // Number of test cases

        while (t-- > 0) {
            int n = scanner.nextInt(); // Length of the array
            int[] a = new int[n];

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

            System.out.println(canPartitionInTwoWays(a) ? "YES" : "NO");
        }

        scanner.close();
    }

    private static boolean canPartitionInTwoWays(int[] arr) {
        // Check for single-element instability
        for (int num : arr) {
            if (!isStable(num, num, num)) {
                return false;
            }
        }

        // Optimized stability check for consecutive elements
        for (int i = 2; i < arr.length; i++) {
            if (isStable(arr[i - 2], arr[i - 1], arr[i])) {
                return true; // Two consecutive elements form a stable set, allowing partitions
            }
        }

        // No stable sets found - check for duplicates (alternative approach)
        Set<Integer> uniqueElements = new HashSet<>();
        for (int num : arr) {
            if (!uniqueElements.add(num)) {
                return true; // Duplicates found, allowing partitions
            }
        }

        return false; // No stable sets or duplicates found, single partition only
    }

    private static boolean isStable(int a, int b, int c) {
        return (a + b > c) && (a + c > b) && (b + c > a);
    }
}

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

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

Think step by step.

Response:
```plaintext
5
4
1 2 3 4
3
1 1 1
5
10 10 10 10 10
4
1 2 2 3
2
100 200
```

### Explanation:
1. **Test Case 1: `1 2 3 4`**
   - This is a simple case where the array has no duplicates and no three consecutive elements form a stable set. The only way to partition this array is to split it into single-element subsegments, which does not satisfy the requirement of having at least two partitions.

2. **Test Case 2: `1 1 1`**
   - This is a case with all elements being the same. The array is trivially stable, but the only way to partition it is to split it into single-element subsegments, which does not satisfy the requirement of having at least two partitions.

3. **Test Case 3: `10 10 10 10 10`**
   - This is a case with all elements being the same. Similar to the previous case, the array is trivially stable, but the only way to partition it is to split it into single-element subsegments, which does not satisfy the requirement of having at least two partitions.

4. **Test Case 4: `1 2 2 3`**
   - This case includes duplicates and a stable set of three consecutive elements. The array can be partitioned in multiple ways, such as `[1], [2, 2, 3]` and `[1, 2, 2], [3]`, satisfying the requirement of having at least two partitions.

5. **Test Case 5: `100 200`**
   - This is a simple case with two elements. The array can be partitioned in multiple ways, such as `[100], [200]`, satisfying the requirement of having at least two partitions.

These test cases are designed to expose the bug in the buggy code by focusing on edge cases and scenarios where the stability check and partitioning logic might fail.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and a MEX Problem (Hard Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: The two versions are different problems. In this version of the problem, you can't choose the same integer twice or more. You can make hacks only if both versions are solved.

One day, Turtle was playing with $n$ sequences. Let the length of the $i$-th sequence be $l_i$. Then the $i$-th sequence was $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$.

Piggy gave Turtle a problem to solve when Turtle was playing. The statement of the problem was:

  * There was a non-negative integer $x$ at first. Turtle would perform an arbitrary number (possibly zero) of operations on the integer.    * In each operation, Turtle could choose an integer $i$ such that $1 \le i \le n$ and $i$ wasn't chosen before, and set $x$ to $\text{mex}^{\dagger}(x, a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i})$.    * Turtle was asked to find the answer, which was the maximum value of $x$ after performing an arbitrary number of operations.

Turtle solved the above problem without difficulty. He defined $f(k)$ as the answer to the above problem when the initial value of $x$ was $k$.

Then Piggy gave Turtle a non-negative integer $m$ and asked Turtle to find the value of $\sum\limits_{i = 0}^m f(i)$ (i.e., the value of $f(0) + f(1) + \ldots + f(m)$). Unfortunately, he couldn't solve this problem. Please help him!

$^{\dagger}\text{mex}(c_1, c_2, \ldots, c_k)$ is defined as the smallest non-negative integer $x$ which does not occur in the sequence $c$. For example, $\text{mex}(2, 2, 0, 3)$ is $1$, $\text{mex}(1, 2)$ is $0$.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($1 \le n \le 2 \cdot 10^5, 0 \le m \le 10^9$).

Each of the following $n$ lines contains several integers. The first integer $l_i$ ($1 \le l_i \le 2 \cdot 10^5$) represents the length of the $i$-th sequence, and the following $l_i$ integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$ ($0 \le a_{i, j} \le 10^9$) represent the elements of the $i$-th sequence.

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

Output: For each test case, output a single integer — the value of $\sum\limits_{i = 0}^m f(i)$.

Examples:
input:
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
output:
16
18
1281
4
6556785365
1842836177961

Note: In the first test case, when $x$ is initially $2$, Turtle can choose $i = 3$ and set $x$ to $\text{mex}(x, a_{3, 1}, a_{3, 2}, a_{3, 3}, a_{3, 4}) = \text{mex}(2, 7, 0, 1, 5) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(2) = 3$.

It can be seen that $f(0) = 3$, $f(1) = 3$, $f(2) = 3$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 3 + 3 + 3 + 3 + 4 = 16$.

In the second test case, when $x$ is initially $1$, Turtle can choose $i = 1$ and set $x$ to $\text{mex}(x, a_{1, 1}, a_{1, 2}, a_{1, 3}, a_{1, 4}, a_{1, 5}) = \text{mex}(1, 0, 2, 0, 4, 11) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(1) = 3$.

It can be seen that $f(0) = 4$, $f(1) = 3$, $f(2) = 4$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 4 + 3 + 4 + 3 + 4 = 18$.

In the fourth test case, it can be seen that $f(0) = 3$ and $f(1) = 1$. So $f(0) + f(1) = 3 + 1 = 4$.

Buggy Code:
#include <algorithm>
#include <iostream>
#include <vector>
using ll = long long;
using std::cin, std::ios, std::string, std::max, std::cout, std::max,
    std::vector;

const int L = 200003;
int f[L]{0};

void solve() {
  std::fill(f, f + L, 0);
  int n, m;
  cin >> n >> m;

  vector<int> graph[L];
  int maxl = 0;

  while (n--) {
    ll l;
    cin >> l;

    maxl = max((int)l, maxl);
    vector<int> a(l + 1);
    for (int i = 0; i < l; i++)
      cin >> a[i];

    std::sort(a.begin(), a.end() - 1);
    auto s = std::unique(a.begin(), a.end() - 1) - a.begin();
    a[s] = a[s - 1];
    s++;

    ll miss = 0, v = -1, u = 0;

    for (int i = 0; i < s; i++) {
      while (miss < 2 && a[i] != i + miss) {
        if (miss == 1) {
          v = miss + i;
          if (u < L)
            graph[u].push_back(v);
          break;
        }
        u = miss + i;
        miss++;
      }
      if (v != -1)
        break;
    }
    a.clear();
  }
  ll mex2 = -1;

  for (int i = maxl; i >= 0; i--) {
    for (int val : graph[i]) {
      f[i] = max(val, f[i]);
      if (val < L)
        f[i] = max(f[i], f[val]);
    }
    if (graph[i].size() >= 2) {
      mex2 = max((ll)f[i], mex2);
    }
    if (graph[i].size() >= 1) {
      mex2 = max((ll)i, mex2);
    }
  }

  ll ans = 0;
  for (int i = 0; i <= std::min(maxl, m); i++) {
    ll val1 = max(mex2, (ll)f[i]);
    ans += max(val1, (ll)i);
  }
  if (m > maxl) {
    ans += (m - (ll)maxl) * ((ll)(maxl + 1) + m) / 2LL;
  }
  cout << ans << "\n";
}

int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  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 4
2 0 2
3 2 3 3
4 7 0 1 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and a MEX Problem (Hard Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: The two versions are different problems. In this version of the problem, you can't choose the same integer twice or more. You can make hacks only if both versions are solved.

One day, Turtle was playing with $n$ sequences. Let the length of the $i$-th sequence be $l_i$. Then the $i$-th sequence was $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$.

Piggy gave Turtle a problem to solve when Turtle was playing. The statement of the problem was:

  * There was a non-negative integer $x$ at first. Turtle would perform an arbitrary number (possibly zero) of operations on the integer.    * In each operation, Turtle could choose an integer $i$ such that $1 \le i \le n$ and $i$ wasn't chosen before, and set $x$ to $\text{mex}^{\dagger}(x, a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i})$.    * Turtle was asked to find the answer, which was the maximum value of $x$ after performing an arbitrary number of operations.

Turtle solved the above problem without difficulty. He defined $f(k)$ as the answer to the above problem when the initial value of $x$ was $k$.

Then Piggy gave Turtle a non-negative integer $m$ and asked Turtle to find the value of $\sum\limits_{i = 0}^m f(i)$ (i.e., the value of $f(0) + f(1) + \ldots + f(m)$). Unfortunately, he couldn't solve this problem. Please help him!

$^{\dagger}\text{mex}(c_1, c_2, \ldots, c_k)$ is defined as the smallest non-negative integer $x$ which does not occur in the sequence $c$. For example, $\text{mex}(2, 2, 0, 3)$ is $1$, $\text{mex}(1, 2)$ is $0$.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($1 \le n \le 2 \cdot 10^5, 0 \le m \le 10^9$).

Each of the following $n$ lines contains several integers. The first integer $l_i$ ($1 \le l_i \le 2 \cdot 10^5$) represents the length of the $i$-th sequence, and the following $l_i$ integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$ ($0 \le a_{i, j} \le 10^9$) represent the elements of the $i$-th sequence.

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

Output: For each test case, output a single integer — the value of $\sum\limits_{i = 0}^m f(i)$.

Examples:
input:
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
output:
16
18
1281
4
6556785365
1842836177961

Note: In the first test case, when $x$ is initially $2$, Turtle can choose $i = 3$ and set $x$ to $\text{mex}(x, a_{3, 1}, a_{3, 2}, a_{3, 3}, a_{3, 4}) = \text{mex}(2, 7, 0, 1, 5) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(2) = 3$.

It can be seen that $f(0) = 3$, $f(1) = 3$, $f(2) = 3$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 3 + 3 + 3 + 3 + 4 = 16$.

In the second test case, when $x$ is initially $1$, Turtle can choose $i = 1$ and set $x$ to $\text{mex}(x, a_{1, 1}, a_{1, 2}, a_{1, 3}, a_{1, 4}, a_{1, 5}) = \text{mex}(1, 0, 2, 0, 4, 11) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(1) = 3$.

It can be seen that $f(0) = 4$, $f(1) = 3$, $f(2) = 4$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 4 + 3 + 4 + 3 + 4 = 18$.

In the fourth test case, it can be seen that $f(0) = 3$ and $f(1) = 1$. So $f(0) + f(1) = 3 + 1 = 4$.

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

typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tiii;

const int MAXN = 200010;
bool vis[MAXN];
vector<int> graph[MAXN];
int MX;
int val[MAXN];

void DFS(int u)
{
    val[u] = u;
    for(int v : graph[u])
    {
        if(val[v] == -1) DFS(v);
        val[u] = max(val[u], val[v]);
    }

    if(graph[u].size() > 1) MX = max(MX, val[u]);
}

void solve()
{
    int n, m; cin >> n >> m;
    memset(val, -1, sizeof(val));
    MX = 0;

    int mxV = 0;
    for(int i=1; i<=n; i++)
    {
        ll l; cin >> l;

        fill(vis, vis + l + 5, false);
        for(int j=1; j<=l; j++)
        {
            ll aij; cin >> aij;
            if(aij < l + 5) vis[aij] = true;
        }

        bool chk = false;
        int u, v;
        for(int i=0; i<l+5; i++)
        {
            if(vis[i]) continue;
            if(chk)
            {
                v = i;
                break;
            }
            u = i;
            chk = true;
        }

        graph[u].push_back(v);
        mxV = max(mxV, v);
    }

    for(int i=0; i<=mxV; i++) DFS(i);

    ll ans = 0;
    for(int i=0; i<=min(m, mxV); i++)
        ans += max(MX, val[i]);
    if(m > mxV) ans += ll(m)*(m+1)/2 - mxV*(mxV+1)/2;

    cout << ans << "\n";

    for(int i=0; i<=mxV; i++) graph[i].clear();
}

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

	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
1
5 5
2 0 1
2 1 2
2 2 3
2 3 4
2 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and a MEX Problem (Hard Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: The two versions are different problems. In this version of the problem, you can't choose the same integer twice or more. You can make hacks only if both versions are solved.

One day, Turtle was playing with $n$ sequences. Let the length of the $i$-th sequence be $l_i$. Then the $i$-th sequence was $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$.

Piggy gave Turtle a problem to solve when Turtle was playing. The statement of the problem was:

  * There was a non-negative integer $x$ at first. Turtle would perform an arbitrary number (possibly zero) of operations on the integer.    * In each operation, Turtle could choose an integer $i$ such that $1 \le i \le n$ and $i$ wasn't chosen before, and set $x$ to $\text{mex}^{\dagger}(x, a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i})$.    * Turtle was asked to find the answer, which was the maximum value of $x$ after performing an arbitrary number of operations.

Turtle solved the above problem without difficulty. He defined $f(k)$ as the answer to the above problem when the initial value of $x$ was $k$.

Then Piggy gave Turtle a non-negative integer $m$ and asked Turtle to find the value of $\sum\limits_{i = 0}^m f(i)$ (i.e., the value of $f(0) + f(1) + \ldots + f(m)$). Unfortunately, he couldn't solve this problem. Please help him!

$^{\dagger}\text{mex}(c_1, c_2, \ldots, c_k)$ is defined as the smallest non-negative integer $x$ which does not occur in the sequence $c$. For example, $\text{mex}(2, 2, 0, 3)$ is $1$, $\text{mex}(1, 2)$ is $0$.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($1 \le n \le 2 \cdot 10^5, 0 \le m \le 10^9$).

Each of the following $n$ lines contains several integers. The first integer $l_i$ ($1 \le l_i \le 2 \cdot 10^5$) represents the length of the $i$-th sequence, and the following $l_i$ integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$ ($0 \le a_{i, j} \le 10^9$) represent the elements of the $i$-th sequence.

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

Output: For each test case, output a single integer — the value of $\sum\limits_{i = 0}^m f(i)$.

Examples:
input:
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
output:
16
18
1281
4
6556785365
1842836177961

Note: In the first test case, when $x$ is initially $2$, Turtle can choose $i = 3$ and set $x$ to $\text{mex}(x, a_{3, 1}, a_{3, 2}, a_{3, 3}, a_{3, 4}) = \text{mex}(2, 7, 0, 1, 5) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(2) = 3$.

It can be seen that $f(0) = 3$, $f(1) = 3$, $f(2) = 3$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 3 + 3 + 3 + 3 + 4 = 16$.

In the second test case, when $x$ is initially $1$, Turtle can choose $i = 1$ and set $x$ to $\text{mex}(x, a_{1, 1}, a_{1, 2}, a_{1, 3}, a_{1, 4}, a_{1, 5}) = \text{mex}(1, 0, 2, 0, 4, 11) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(1) = 3$.

It can be seen that $f(0) = 4$, $f(1) = 3$, $f(2) = 4$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 4 + 3 + 4 + 3 + 4 = 18$.

In the fourth test case, it can be seen that $f(0) = 3$ and $f(1) = 1$. So $f(0) + f(1) = 3 + 1 = 4$.

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

typedef long long ll;
const ll N = 3e5+5, ANY = 2e5+12;
ll n, m;
ll a[N];
ll lll[N];
ll m1[N], m2[N];
ll in[N];
ll cnt[N];
ll f[N];

vector <int> v[N];

vector <int> edge[N];

void solve () {
    map <pair<int,int>, bool> cb;
    cin >> n >> m;
    ll mxll=0;
    for (ll i=1; i<=n; i++) {
        v[i].clear();
        cin >> lll[i];
        for (ll j=0; j<lll[i]; j++) {
            ll q;
            cin >> q;
            v[i].push_back (q);
        }
        mxll = max(mxll, lll[i]);
    }
    for (ll i=0; i<=mxll+2; i++) {
        edge[i].clear();
        in[i] = 0;
        cnt[i] = 0;
        f[i] = 0;
    }
    for (ll i=1; i<=n; i++) {
        ll l = lll[i];
        for (ll j=0; j<l; j++) {
            a[j] = v[i][j];
        }
        sort (a, a+l);
        l = unique (a, a+l)-a;

        ll mex1 = l;
        for (ll i=0; i<=l; i++) {
            if (a[i] != i) {
                mex1 = i;
                break;
            }
        }
        a[l++] = mex1;
        sort (a, a+l);
        ll mex2 = l;
        for (ll i=0; i<l; i ++) {
            if (a[i] != i) {
                mex2 = i;
                break;
            }
        }
        m1[i] = mex1, m2[i] = mex2;
        cnt[m1[i]] ++;
        edge[m1[i]].push_back (m2[i]);
    }
    ll mx=0;
    for (int i=mxll+2; i>=0; i--) {
        f[i] = i;
        for (int j=0; j<edge[i].size(); j ++) {
            int to = edge[i][j];
            f[i]= max(f[i], f[to]);
        }
        if (cnt[i] > 1)
            mx = max(mx, f[i]);
    }
    ll ans=0;
    for (ll i=0; i<=m && i <= mxll+2; i++) {
        ans += max(f[i], mx);
    }
    if (m > mxll+2) {
        ll s = mxll+3, e = m;
        ans += (s+e)*(e-s+1ll)/2ll;
    }
    cout << ans << '\n';
}

int main () {
    ios::sync_with_stdio(false);
    ll 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
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Turtle and a MEX Problem (Hard Version)
time_limit_ms: 2000
memory_limit_mb: 256
Description: The two versions are different problems. In this version of the problem, you can't choose the same integer twice or more. You can make hacks only if both versions are solved.

One day, Turtle was playing with $n$ sequences. Let the length of the $i$-th sequence be $l_i$. Then the $i$-th sequence was $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$.

Piggy gave Turtle a problem to solve when Turtle was playing. The statement of the problem was:

  * There was a non-negative integer $x$ at first. Turtle would perform an arbitrary number (possibly zero) of operations on the integer.    * In each operation, Turtle could choose an integer $i$ such that $1 \le i \le n$ and $i$ wasn't chosen before, and set $x$ to $\text{mex}^{\dagger}(x, a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i})$.    * Turtle was asked to find the answer, which was the maximum value of $x$ after performing an arbitrary number of operations.

Turtle solved the above problem without difficulty. He defined $f(k)$ as the answer to the above problem when the initial value of $x$ was $k$.

Then Piggy gave Turtle a non-negative integer $m$ and asked Turtle to find the value of $\sum\limits_{i = 0}^m f(i)$ (i.e., the value of $f(0) + f(1) + \ldots + f(m)$). Unfortunately, he couldn't solve this problem. Please help him!

$^{\dagger}\text{mex}(c_1, c_2, \ldots, c_k)$ is defined as the smallest non-negative integer $x$ which does not occur in the sequence $c$. For example, $\text{mex}(2, 2, 0, 3)$ is $1$, $\text{mex}(1, 2)$ is $0$.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n, m$ ($1 \le n \le 2 \cdot 10^5, 0 \le m \le 10^9$).

Each of the following $n$ lines contains several integers. The first integer $l_i$ ($1 \le l_i \le 2 \cdot 10^5$) represents the length of the $i$-th sequence, and the following $l_i$ integers $a_{i, 1}, a_{i, 2}, \ldots, a_{i, l_i}$ ($0 \le a_{i, j} \le 10^9$) represent the elements of the $i$-th sequence.

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

Output: For each test case, output a single integer — the value of $\sum\limits_{i = 0}^m f(i)$.

Examples:
input:
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
output:
16
18
1281
4
6556785365
1842836177961

Note: In the first test case, when $x$ is initially $2$, Turtle can choose $i = 3$ and set $x$ to $\text{mex}(x, a_{3, 1}, a_{3, 2}, a_{3, 3}, a_{3, 4}) = \text{mex}(2, 7, 0, 1, 5) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(2) = 3$.

It can be seen that $f(0) = 3$, $f(1) = 3$, $f(2) = 3$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 3 + 3 + 3 + 3 + 4 = 16$.

In the second test case, when $x$ is initially $1$, Turtle can choose $i = 1$ and set $x$ to $\text{mex}(x, a_{1, 1}, a_{1, 2}, a_{1, 3}, a_{1, 4}, a_{1, 5}) = \text{mex}(1, 0, 2, 0, 4, 11) = 3$. It can be proved that Turtle can't make the value of $x$ greater than $3$, so $f(1) = 3$.

It can be seen that $f(0) = 4$, $f(1) = 3$, $f(2) = 4$, $f(3) = 3$, and $f(4) = 4$. So $f(0) + f(1) + f(2) + f(3) + f(4) = 4 + 3 + 4 + 3 + 4 = 18$.

In the fourth test case, it can be seen that $f(0) = 3$ and $f(1) = 1$. So $f(0) + f(1) = 3 + 1 = 4$.

Buggy Code:
import sys
import os
from io import BytesIO, IOBase

# region fastio
BUFSIZE = 8192

class FastIO(IOBase):
    newlines = 0

    def __init__(self, file):
        self._fd = file.fileno()
        self.buffer = BytesIO()
        self.writable = "x" in file.mode or "r" not in file.mode
        self.write = self.buffer.write if self.writable else None

    def read(self):
        while True:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            if not b:
                break
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines = 0
        return self.buffer.read()

    def readline(self):
        while self.newlines == 0:
            b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
            self.newlines = b.count(b"\n") + (not b)
            ptr = self.buffer.tell()
            self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
        self.newlines -= 1
        return self.buffer.readline()

    def flush(self):
        if self.writable:
            os.write(self._fd, self.buffer.getvalue())
            self.buffer.truncate(0), self.buffer.seek(0)

class IOWrapper(IOBase):
    def __init__(self, file):
        self.buffer = FastIO(file)
        self.flush = self.buffer.flush
        self.writable = self.buffer.writable
        self.write = lambda s: self.buffer.write(s.encode("ascii"))
        self.read = lambda: self.buffer.read().decode("ascii")
        self.readline = lambda: self.buffer.readline().decode("ascii")

if sys.version_info[0] < 3:
    sys.stdin = BytesIO(os.read(0, os.fstat(0).st_size))
else:
    sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)

file = sys.stdin
if os.environ.get('USER') == "loic":
    file = open("data.in")

line = lambda: file.readline().split()
ui = lambda: int(line()[0])
mi = lambda: map(int,line())
li = lambda: list(mi())
lsi = lambda: list(map(int,line()[0]))
#######################################################################

def solve():

    ref = 0

    tmp = []

    for _ in range(N):
        A = li()
        s = list(set(A[1:]))
        s.sort()

        m = 0
        while m < len(s) and s[m] == m:
            m += 1

        mex = m

        m += 1
        while m - 1 < len(s) and s[m-1] == m:
            m += 1

        ref = max(ref, m)
        tmp.append((mex, m))

    tmp.sort(key = lambda e: e[1], reverse = True)

    p = [-1] * (ref + 1)
    c = [0] * (ref + 1)
    for a,b in tmp:
        if p[b] == -1:
            p[b] = b
        p[a] = max(p[a], p[b])
        c[a] += 1

    res = 0
    mx = -1
    multi = False
    for i in range(min(M +1, ref + 1)):
        if c[i] >= 2:
            multi = True
            mx = max(mx, p[i])

        if multi:
            res += mx
        else:
            if c[i] == 1:
                res += p[i]
            else:
                res += i

    if M > ref:
        res += M * (M + 1) // 2 - ref * (ref + 1) // 2

    return str(res)

for test in range(1,ui()+1):
    N,M = mi()

    print(solve())

file.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
6
3 4
2 0 2
3 2 3 3
4 7 0 1 5
3 4
5 0 2 0 4 11
1 1
5 1 3 0 3 3
2 50
2 1 2
2 1 2
1 1
7 1 2 4 1 4 9 5
4 114514
2 2 2
5 7 3 6 0 3
3 0 1 1
5 0 9 2 1 5
5 1919810
1 2
2 324003 0
3 1416324 2 1460728
4 1312631 2 0 1415195
5 1223554 192248 2 1492515 725556
```

### Explanation:
1. **Test Case 1:**
   - **N = 3, M = 4**
   - Sequences: [0, 2], [2, 3, 3], [7, 0, 1, 5]
   - This test case checks the basic functionality and the ability to handle small sequences with simple values.

2. **Test Case 2:**
   - **N = 3, M = 4**
   - Sequences: [0, 2, 0, 4, 11], [1]
   - This test case includes a sequence with repeated values and a sequence with a single value.

3. **Test Case 3:**
   - **N = 1, M = 1**
   - Sequence: [1, 3, 0, 3, 3]
   - This test case checks the ability to handle a single sequence with repeated values.

4. **Test Case 4:**
   - **N = 2, M = 50**
   - Sequences: [1, 2], [1, 2]
   - This test case checks the ability to handle repeated sequences and a large value of M.

5. **Test Case 5:**
   - **N = 4, M = 114514**
   - Sequences: [2, 2, 2], [7, 3, 6, 0, 3], [0, 1, 1], [0, 9, 2, 1, 5]
   - This test case checks the ability to handle larger values of N and M, and sequences with varied values.

6. **Test Case 6:**
   - **N = 5, M = 1919810**
   - Sequences: [2, 324003, 0], [1416324, 2, 1460728], [1312631, 2, 0, 1415195], [1223554, 192248, 2, 1492515, 725556]
   - This test case checks the ability to handle very large values of M and sequences with large values.

These test cases are designed to cover a range of edge cases and scenarios that can expose bugs in the code, such as handling repeated values, large values of M, and sequences with varied values.