**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

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

#define ll long long int
#define endl "\n"
const int N = 1e5 + 10;
#define unique s.erase(unique(s.begin(), s.end()), s.end())

// Function to find the most significant bit
int __MSB(int x){
    return (int) log2(x);
}

// Policy-based data structures
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/tree_policy.hpp>
using namespace __gnu_pbds;
#define os tree<int,null_type,less<int>,rb_tree_tag,tree_order_statistics_node_update>

// Priority queue with custom comparator
#define pq priority_queue<ll,vector<ll>,greater<ll>>pq

// Function to convert integer to binary string
string binary(int n) {
    string k="";
    while(n>0) {
        if(n%2==0) k+="0";
        else k+="1";
        n/=2;
    }
    reverse(k.begin(),k.end());
    return k;
}

// Function to find the longest common subsequence using dynamic programming
int lcs(int m, int n, string &a, string &b) {
    vector<vector<int>> dp(m + 1, vector<int>(n + 1, 0));
    for (int i = 1; i <= m; i++) {
        for (int j = 1; j <= n; j++) {
            if (a[i - 1] == b[j - 1]) {
                dp[i][j] = dp[i - 1][j - 1] + 1;
            } else {
                dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
            }
        }
    }
    return dp[m][n];
}

int main()
{

int t;
cin>>t;
while(t--){
    ll n;
    cin>>n;
    vector<ll>v(n);
    ll sum=0;
    for(int i=0;i<n;i++)cin>>v[i];
    string s;
    cin>>s;
    ll l=0,r=n-1;
    while(l<=r){
            ll j=l;
    while(s[j]!='L'){
        j++;
    }
    l=j;
    ll k=r;
    while(s[k]!='R'){
        k--;
    }
    r=k;
    for(int i=l;i<=r;i++){
        sum+=v[i];
    }
    l++;
    r--;
    }
    cout<<sum<<endl;
}

    return 0;
}

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

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

Think step by step.

Response:
```plaintext
1
8
1 1 1 1 1 1 1 1
LRLRLRLR
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

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

void solve()
{
  ll n; cin>>n;
  vector<ll> a(n);
    for(ll i=0; i<n; i++) cin>>a[i];
  string s; cin>>s;

  ll i=0, j=n-1;
  ll ans=0;

  while(i<j){
    while(s[i]!='L')i++;
    while(s[j]!='R')j--;

    for(ll k=i;k<=j;k++){
        ans+=a[k];
    }
    i++;j--;
  }
  cout<<ans<<endl;
}

int main()
{
    ll 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 code, we need to focus on scenarios where the code might incorrectly skip over valid 'L' and 'R' pairs or incorrectly calculate the score. The code uses a while loop to find the next 'L' and 'R' and then adds the sum of the elements between them. However, it does not handle cases where there are multiple 'L' and 'R' pairs that could be selected in a more optimal way.

### Key Issues to Target:
1. **Multiple 'L' and 'R' Pairs**: The code might miss optimal pairs if it only processes the first 'L' and 'R' it finds.
2. **Overlapping Pairs**: The code might incorrectly handle overlapping 'L' and 'R' pairs, leading to incorrect scores.
3. **Edge Cases**: The code might fail for edge cases where the strip has only a few elements or all elements are 'L' or 'R'.

### Test Input:
Let's create a test input that includes multiple 'L' and 'R' pairs and ensures the code must handle them correctly.

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

### Explanation:
- **n = 8**: The strip has 8 cells.
- **a = [1, 2, 3, 4, 5, 6, 7, 8]**: The numbers on the strip.
- **s = "LRLRRLRR"**: The directions on the strip.

### Expected Output:
- **28**: The maximum possible score.

### Steps to Verify:
1. **First Pair (1, 4)**:
   - `i = 0, j = 3`
   - Add `1 + 2 + 3 + 4 = 10`
   - `s` becomes `....RLRR`
   - `i = 4, j = 3` (not valid, so move to the next pair)
2. **Second Pair (4, 7)**:
   - `i = 4, j = 7`
   - Add `5 + 6 + 7 + 8 = 26`
   - `s` becomes `........`
   - Final score: `10 + 18 = 28`

This test input should expose the bug in the code where it might not handle multiple 'L' and 'R' pairs correctly, leading to an incorrect score.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

Buggy Code:
def readint():
    return(int(input()))

def readintlist():
    return(list(map(int,input().split())))

def readstring():
    s = input().strip()
    return s

t = readint()
for _ in range(t):
    N = readint()
    A = readintlist()
    S = [ch for ch in readstring()]

    total = 0
    done = False
    while not done:
        try:
            start = S.index('L')
            S[start] = '.'
        except Exception as e:
            done = True
            continue
        end = -1
        for end in range(N-1, start-1, -1):
            if S[end] == 'R':
                S[end] = '.'
                break
        if end <= start:
            done = True
            continue
        total += sum(A[start:end+1])
    print(total)

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

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

Think step by step.

Response:
```plaintext
3
5
1 2 3 4 5
LRLRR
4
1 1 1 1
LLLL
4
1 1 1 1
RRRR
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

Buggy Code:
def solve(x):
    while i[0]<i[1]:
        # print(i,'a')
        if x[i[0]-1]=='R':i[0]+=1
        else:break
    while i[0]<i[1]:
        # print(i,'b')
        if x[i[1]-1]=='L':i[1]-=1
        else:break

    if i[0]<i[1]:
        ans[0]+=prefix[i[1]]-prefix[i[0]-1]
        i[0]+=1
        i[1]-=1
        solve(x)

    # pass
# file = open('El3b.txt','r')
for _ in range(int(input().strip())):
    n = int(input().strip())
    a = list(map(int, input().split()))
    x =  input().strip()
    prefix = [0]
    for i in range(n):
        prefix.append(prefix[-1]+a[i])
    # print(prefix)
    i = [1,n]
    ans = [0]
    solve(x)
    print(*ans)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios that can cause the code to fail. The buggy code has issues with how it handles the indices and the conditions for the operations.

Here are some key points to consider:
1. **Initial 'L' and 'R' positions**: The code might fail if there are no valid 'L' and 'R' pairs at the beginning or end of the strip.
2. **Single 'L' or 'R' in the middle**: The code might not handle cases where there is a single 'L' or 'R' without a matching pair.
3. **Multiple 'L' and 'R' pairs**: The code might not correctly handle multiple valid 'L' and 'R' pairs that can be chosen.

Given these points, let's construct a small-scale test input that can expose these issues:

```plaintext
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
```

This test input includes:
1. A strip with multiple 'L' and 'R' pairs (`LRLLLR`).
2. A strip with a single 'L' and 'R' (`LR`).
3. A strip with a single 'R' and 'L' (`RL`).
4. A strip with multiple 'L' and 'R' pairs, including a middle 'L' (`LRLRR`).

These cases should help identify any issues in the buggy code, particularly with how it handles the indices and the conditions for the operations.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

Buggy Code:
import java.util.*;
public class scalee
{
    public static void main(String args[])
    {
        int n,i,j;
        Scanner sc = new Scanner(System.in);
        n=sc.nextInt();
        for(i=0;i<n;i++)
        {
            int a,c=1,l=0,r=0,cl=0,cr=0;
            long summ=0;
            a=sc.nextInt();
            int arr[]=new int[a];
            for(j=0;j<a;j++)
            {
                arr[j]=sc.nextInt();
            }
            String s;
            s=sc.next();
            while(c!=0)
            {
                c=0;
                cl=0;
                cr=0;
                for(j=0;j<a;j++)
                {
                    if(s.charAt(j)=='L')
                    {
                        cl++;
                        l=j;
                        break;
                    }
                }
                if(cl==0)
                {
                    break;
                }
                for(j=a-1;j>=0;j--)
                {
                    if(s.charAt(j)=='R')
                    {
                        cr++;
                        r=j;
                        break;
                    }
                }
                if(cr==0)
                {
                    break;
                }
                c=cl+cr;
                if(l<r)
                {
                    int sum=0;
                    for(j=l;j<=r;j++)
                    {
                        sum+=arr[j];
                    }
                    summ+=sum;
                    String st,str,stri;
                    st=s.substring(0,l);
                    str=s.substring(l+1,r);
                    stri=s.substring(r+1);
                    s=st+"."+str+"."+stri;
                }
                else
                {
                    c=0;
                }
            }
            System.out.println(summ);
        }
    }
}

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

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

Think step by step.

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

1. **Single 'L' and 'R'**: This is a simple case that should work correctly.
2. **Multiple 'L' and 'R' in a sequence**: This can help identify issues with the logic for selecting the indices.
3. **No 'L' or 'R'**: This should result in a score of 0.
4. **'L' and 'R' interleaved**: This can test the logic for multiple operations.
5. **'L' and 'R' at the boundaries**: This can test edge cases where the first or last characters are 'L' or 'R'.

Given these scenarios, let's create a test input that combines multiple edge cases to maximize the failure probability in the faulty solution.

```plaintext
5
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
4
1 1 1 1
LRLR
```

This test input includes:
1. The first test case from the example, which should score 18.
2. A simple case with 'L' and 'R' at the boundaries, which should score 10.
3. A case with 'R' and 'L' at the boundaries, which should score 0.
4. A case with 'L' and 'R' interleaved, which should score 22.
5. A case with multiple 'L' and 'R' in a sequence, which should score 4.

This input should help identify any issues with the logic for selecting indices and performing operations in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

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

// http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2016/p0200r0.html
template<class Fun> class y_combinator_result {
    Fun fun_;
public:
    template<class T> explicit y_combinator_result(T &&fun): fun_(std::forward<T>(fun)) {}
    template<class ...Args> decltype(auto) operator()(Args &&...args) { return fun_(std::ref(*this), std::forward<Args>(args)...); }
};
template<class Fun> decltype(auto) y_combinator(Fun &&fun) { return y_combinator_result<std::decay_t<Fun>>(std::forward<Fun>(fun)); }

template<typename A, typename B> ostream& operator<<(ostream &os, const pair<A, B> &p) { return os << '(' << p.first << ", " << p.second << ')'; }
template<typename T_container, typename T = typename enable_if<!is_same<T_container, string>::value, typename T_container::value_type>::type> ostream& operator<<(ostream &os, const T_container &v) { os << '{'; string sep; for (const T &x : v) os << sep << x, sep = ", "; return os << '}'; }

void dbg_out() { cerr << endl; }
template<typename Head, typename... Tail> void dbg_out(Head H, Tail... T) { cerr << ' ' << H; dbg_out(T...); }
#ifdef NEAL_DEBUG
#define dbg(...) cerr << '[' << __FILE__ << ':' << __LINE__ << "] (" << #__VA_ARGS__ << "):", dbg_out(__VA_ARGS__)
#else
#define dbg(...)
#endif

void run_case() {
    // read in n
    int n;
    // read in a
    cin >> n;
    vector<int> a(n);
    for (auto &x : a)
        cin >> x;

    // read in s
    string s;
    cin >> s;
    // prefix aray
    vector<int> pref(n + 1);
    pref[0] = 0;
    for (int i = 0; i < n; i++) {
        pref[i + 1] = pref[i] + a[i];
    }

    // be greedy

    int l = 0, r = n - 1;

        ll res = 0;

        while(true){
            // find next l
            while(l < n && s[l] != 'L'){
                l++;
            }
            // find next r
            while(r >= 0 && s[r] != 'R'){
                r--;
            }

            if (l >= r){
                break;
            } else {
                res += pref[r + 1] - pref[l];
                l++;
                r--;
            }
        }
        cout << res << '\n';

}

int main() {
    ios::sync_with_stdio(false);
#ifndef NEAL_DEBUG
    cin.tie(nullptr);
#endif

    int tests;
    cin >> tests;
    while (tests-- > 0)
        run_case();
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases and scenarios where the greedy approach might fail. The code assumes that it can always find the next 'L' and 'R' and add the sum of the segment between them. However, this approach might not always yield the maximum score, especially when there are multiple 'L' and 'R' pairs that can be chosen in different orders.

Let's consider a test case where the greedy approach might not find the optimal solution:

1. **Case with Multiple 'L' and 'R' Pairs:**
   - Suppose we have a strip where there are multiple 'L' and 'R' pairs, and the optimal solution involves choosing a specific pair first, but the greedy approach might choose a different pair.

2. **Case with Overlapping Pairs:**
   - Consider a strip where choosing one pair might affect the ability to choose another optimal pair.

Let's construct a small-scale test input that exposes this issue:

```plaintext
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
```

This input includes the example provided in the problem statement, which is already known to work correctly. Let's add a new test case that could expose the bug:

```plaintext
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
4
1 2 3 4
LRLR
```

### Explanation:
- **First Test Case (6 cells):**
  - `3 5 1 4 3 2`
  - `LRLLLR`
  - Optimal score: 18 (as described in the problem statement)

- **Second Test Case (2 cells):**
  - `2 8`
  - `LR`
  - Optimal score: 10

- **Third Test Case (2 cells):**
  - `3 9`
  - `RL`
  - Optimal score: 0 (no valid 'L' 'R' pair)

- **Fourth Test Case (5 cells):**
  - `1 2 3 4 5`
  - `LRLRR`
  - Optimal score: 22

- **Fifth Test Case (4 cells):**
  - `1 2 3 4`
  - `LRLR`
  - Optimal score: 10 (choosing the pairs (1, 2) and (3, 4))

The fifth test case is particularly interesting because the greedy approach might not choose the optimal pairs first. The optimal solution is to choose the pairs (1, 2) and (3, 4), which gives a score of 10. However, if the code is not careful, it might choose (1, 4) first, which is not optimal.

This test input should help expose any issues with the greedy approach in the provided code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
typedef vector<int> vi;
typedef vector<long long> vll;
typedef vector<char> vc;
typedef long long ll;
#define sq(a) (a)*(a)
#define pb push_back
#define pob pop_back
#define MOD 1000000007
#define sz(x) (int)x.size()
#define int long long
//remember pow() always returns double type, so like compares like
void odd_even(int num)
{
    if(num & 1) cout << "odd";//last bit is 1
    if(~num & 1) cout << "even";//last bit is 0
}
int gcd(int a, int b) {
    if(b==0) return a;
    return gcd(b, a%b);
}
void solve(int ind){

    int n;
    cin >> n;
    vi v(n);
    for(int i = 0;i < n;i++) {
        cin >> v[i];
        if(i > 0) v[i] += v[i-1];
    }

    string s;
    cin >> s;
    int r = n-1;
    while(r >= 0 && s[r] != 'R') {
        r--;
    }
    int l = 0;
    while(l < n && s[l] != 'L') {
        l++;
    }

    if(r < l) {
        cout << 0;
    }
    else {
        int sum = 0;
        sum += l-1 >= 0 ? (v[r] - v[l-1]) : v[r];
        l++;r--;
        while(l < r) {
            if(s[l] == 'L' && s[r] == 'R') {
                sum += l-1 >= 0 ? (v[r] - v[l-1]) : v[r];
                l++;r--;
            }
            else if(s[l] == 'L') {
                r--;
            }
            else if(s[r] == 'R') {
                l++;
            }
            else {
                l++;r--;
            }

        }

        cout << sum;
    }

}
int32_t main()
{
    ios::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
    int tt=1, k = 1;
    cin >> tt;
    while(tt--) {
        solve(k);
        k++;
        cout << "\n";
    }

    /*while(scanf("%d %d", &a, &b)!=EOF){
            //fill it in
    }*/
    //solve(0);
    //exception_handle();

}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, let's analyze the code and identify potential edge cases and scenarios where it might fail.

### Analysis of the Buggy Code
1. **Prefix Sum Calculation**:
   - The code calculates the prefix sum of the array `v` to make it easier to compute the sum of subarrays.
   - However, it does not handle the case where the prefix sum array is not properly initialized or updated.

2. **Boundary Conditions**:
   - The code uses two pointers, `l` and `r`, to find the first 'L' and the last 'R' in the string `s`.
   - It then iterates through the string to find pairs of 'L' and 'R' and calculates the sum of the subarray between them.
   - The code might fail if there are no 'L' or 'R' in the string, or if the string is very short.

3. **Sum Calculation**:
   - The sum calculation might fail if the indices `l` and `r` are not properly updated, leading to incorrect subarray sums.

### Challenging Test Input
To expose the bug, we can create a test input that includes:
- A string with no 'L' or 'R'.
- A string with a single 'L' or 'R'.
- A string with multiple 'L' and 'R' but in a configuration that might cause the pointers to miss valid pairs.
- A string with a very short length to test boundary conditions.

Here is a small-scale test input that covers these scenarios:

```plaintext
4
2
1 1
LL
2
1 1
RR
4
1 2 3 4
LRLR
6
1 2 3 4 5 6
LRLRRL
```

### Explanation
1. **Test Case 1**:
   - `n = 2`
   - `a = [1, 1]`
   - `s = "LL"`
   - No 'R' in the string, so the expected output is `0`.

2. **Test Case 2**:
   - `n = 2`
   - `a = [1, 1]`
   - `s = "RR"`
   - No 'L' in the string, so the expected output is `0`.

3. **Test Case 3**:
   - `n = 4`
   - `a = [1, 2, 3, 4]`
   - `s = "LRLR"`
   - Valid pairs: (1, 2) and (3, 4)
   - Expected output: `1 + 2 + 3 + 4 = 10`

4. **Test Case 4**:
   - `n = 6`
   - `a = [1, 2, 3, 4, 5, 6]`
   - `s = "LRLRRL"`
   - Valid pairs: (1, 2), (3, 4), and (5, 6)
   - Expected output: `1 + 2 + 3 + 4 + 5 + 6 = 21`

This test input should help identify any issues with the boundary conditions and sum calculations in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Right Left Wrong
time_limit_ms: 2000
memory_limit_mb: 256
Description: Vlad found a strip of $n$ cells, numbered from left to right from $1$ to $n$. In the $i$-th cell, there is a positive integer $a_i$ and a letter $s_i$, where all $s_i$ are either 'L' or 'R'.

Vlad invites you to try to score the maximum possible points by performing any (possibly zero) number of operations.

In one operation, you can choose two indices $l$ and $r$ ($1 \le l < r \le n$) such that $s_l$ = 'L' and $s_r$ = 'R' and do the following:

  * add $a_l + a_{l + 1} + \dots + a_{r - 1} + a_r$ points to the current score;    * replace $s_i$ with '.' for all $l \le i \le r$, meaning you can no longer choose these indices.

For example, consider the following strip:

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   L| R| L| L| L| R      You can first choose $l = 1$, $r = 2$ and add $3 + 5 = 8$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| L| L| L| R      Then choose $l = 3$, $r = 6$ and add $1 + 4 + 3 + 2 = 10$ to your score.

$3$| $5$| $1$| $4$| $3$| $2$   ---|---|---|---|---|---   .| .| .| .| .| .      As a result, it is impossible to perform another operation, and the final score is $18$.

What is the maximum score that can be achieved?

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

The first line of each test case contains one integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the length of the strip.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^5$) — the numbers written on the strip.

The third line of each test case contains a string $s$ of $n$ characters 'L' and 'R'.

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

Output: For each test case, output one integer — the maximum possible number of points that can be scored.

Examples:
input:
4
6
3 5 1 4 3 2
LRLLLR
2
2 8
LR
2
3 9
RL
5
1 2 3 4 5
LRLRR
output:
18
10
0
22

Note:

Buggy Code:
import java.io.*;
import java.util.*;
public class Main {
    static class Scanner {
        BufferedReader br;
        StringTokenizer st = null;
        PrintWriter pw;
        BufferedWriter bw;

        public Scanner() {
            br = new BufferedReader(
                    new InputStreamReader(System.in));
            pw = new PrintWriter(
                    new OutputStreamWriter(System.out));
            bw = new BufferedWriter(
                    new OutputStreamWriter(System.out));
        }

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

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

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

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

        String nextLine() {
            String str = "";
            try {
                if (st != null && st.hasMoreTokens()) {
                    str = st.nextToken("\n");
                } else {
                    str = br.readLine();
                }
            } catch (IOException e) {
                e.printStackTrace();
            }
            return str;
        }

        int[] nextArr(int n) {
            int arr[] = new int[n];
            for (int i = 0; i < n; i++) {
                arr[i] = nextInt();
            }
            return arr;
        }
    }
    static  long mod = (long)1e9+7;
    public static void main(String args[] ) throws Exception {
        Scanner sc = new Scanner();
        int t = sc.nextInt();
        out:
        while(t-->0)
        {
            //
            int n = sc.nextInt();
            int[] arr = sc.nextArr(n);
            for(int i=1;i<n;i++)
            {
                arr[i] += arr[i-1];
            }
            char[] ch = sc.nextLine().toCharArray();
            long sum = 0;
            int left = 0, right = n-1;
            while(left<right)
            {
                while(left<right && ch[left] == 'R') left++;
                while(left<right && ch[right] == 'L') right--;
                if(left<right)
                {
                    sum += 1l*arr[right];
                    if(left>0)
                    {
                        sum -= 1l* arr[left-1];
                    }
                }
                left++;
                right--;
            }
            sc.pw.println(sum);
        }
        sc.pw.close();
    }

    public static int reverse(int num)
    {
        int rev = 0;
        while(num>0)
        {
            rev = (rev*10)+(num%10);
            num /= 10;
        }
        return rev;
    }

    public static long gcd(long a,long b)
    {
        if(b == 0)
        {
            return a;
        }
        return gcd(b,a%b);
    }

    public static int[] pre()
    {
        int sieve[] = new int[10000000];
        for(int i=2;i*i< sieve.length;i++)
        {
            if(sieve[i] == 0)
            {
                for(int j = 2*i;j< sieve.length;j+=i)
                {
                    sieve[j] = 1;
                }
            }
        }
        return sieve;
    }

    public static void swap(int[] arr,int i,int j)
    {
        int temp = arr[i];
        arr[i] = arr[j];
        arr[j] = temp;
    }

}

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

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

Think step by step.

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

1. **Edge Cases**: Cases where the strip has very few elements, or where the strip is entirely 'L' or 'R'.
2. **Complex Patterns**: Cases where the strip has a mix of 'L' and 'R' in a way that the algorithm might not handle correctly.
3. **Boundary Conditions**: Cases where the indices `left` and `right` might not be correctly managed.

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

1. **Case 1**: A strip with only 'L' characters.
2. **Case 2**: A strip with only 'R' characters.
3. **Case 3**: A strip with a mix of 'L' and 'R' characters where the bug might be more apparent.
4. **Case 4**: A strip with a pattern that might cause the algorithm to miss some valid operations.

Here is the test input:

```plaintext
4
5
1 2 3 4 5
LLLLL
5
1 2 3 4 5
RRRRR
10
1 2 3 4 5 6 7 8 9 10
LRLRLRLRLR
10
1 2 3 4 5 6 7 8 9 10
RLRLRLRLRL
```

Explanation:
1. **Case 1**: A strip with all 'L' characters. The expected output is 0 because no 'L' and 'R' pairs exist.
2. **Case 2**: A strip with all 'R' characters. The expected output is 0 because no 'L' and 'R' pairs exist.
3. **Case 3**: A strip with alternating 'L' and 'R' characters. The expected output is the sum of all elements because each 'L' and 'R' pair can be used to score points.
4. **Case 4**: A strip with alternating 'R' and 'L' characters. The expected output is the sum of all elements because each 'R' and 'L' pair can be used to score points.

This test input should help expose any issues in the buggy code, especially related to the management of indices and the handling of different patterns of 'L' and 'R' characters.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

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

using namespace std;
// #include <ext/pb_ds/assoc_container.hpp>
// #include <ext/pb_ds/tree_policy.hpp>
// using namespace __gnu_pbds;

// #define ordered_set tree<int, null_type,less<int>, rb_tree_tag,tree_order_statistics_node_update>

#define f0r(a, b) for (long long a = 0; a < (b); ++a)
#define f1r(a, b, c) for (long long a = (b); a < (c); ++a)
#define f0rd(a, b) for (long long a = (b); a >= 0; --a)
#define f1rd(a, b, c) for (long long a = (b); a >= (c); --a)
#define ms(arr, v) memset(arr, v, sizeof(arr))
#define send {ios_base::sync_with_stdio(false);}
#define help {cin.tie(NULL); cout.tie(NULL);}
#define fix(prec) {cout << setprecision(prec) << fixed;}
#define mp make_pair
#define all(v) v.begin(), v.end()
#define getunique(v) {sort(all(v)); v.erase(unique(all(v)), v.end());}
#define readgraph(list, edges) for (int i = 0; i < edges; i++) {int a, b; cin >> a >> b; a--; b--; list[a].pb(b); list[b].pb(a);}
#define vsz(x) ((long long) x.size())
typedef long long ll;
typedef long double lld;
typedef unsigned long long ull;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef vector<pii> vpi;
typedef vector<pll> vpl;
typedef long long ll;
typedef unsigned long long ull;
typedef long double lld;

void __print(int x) {cerr << x;}
void __print(long x) {cerr << x;}
void __print(long long x) {cerr << x;}
void __print(unsigned x) {cerr << x;}
void __print(unsigned long x) {cerr << x;}
void __print(unsigned long long x) {cerr << x;}
void __print(float x) {cerr << x;}
void __print(double x) {cerr << x;}
void __print(long double x) {cerr << x;}
void __print(char x) {cerr << '\'' << x << '\'';}
void __print(const char *x) {cerr << '"' << x << '"';}
void __print(const string &x) {cerr << '"' << x << '"';}
void __print(bool x) {cerr << (x ? "true" : "false");}

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

void solve(int tst) {
	// TODO: Add your code here
	int n;
	cin>>n;
	vi arr(n);
	for(int i=0;i<n;i++){
		cin>>arr[i];
	}
	sort(arr.begin(), arr.end());
	int prev = arr[n-1];
	int tail = 0;
	int head = n-2;
	int ans = 0;
	while(tail <= head){
		sort(arr.begin() + tail, arr.begin() + head + 1);
		int avg = (arr[tail] + arr[head])/2;
		debug(tail, head, arr[tail], arr[head], avg);
		ans = avg;
		arr[head] = avg;
		tail++;
	}

	cout << (ans + prev)/2 << '\n';
}

int main() {
	#ifndef ONLINE_JUDGE
	freopen("Error.txt", "w", stderr);
	#endif
	send help;
	int _t=1;
	cin>>_t;
	for(int i=1;i<=_t;i++)
	solve(i);
	return 0;
}

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

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

Think step by step.

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

1. **Edge Cases**: Test the boundaries of the input constraints.
2. **Optimal Operations**: Ensure the test case requires the optimal sequence of operations to achieve the maximum possible value.
3. **Sorting and Averaging**: The bug in the code might be related to how the array is sorted and how the averaging is performed.

### Analysis of the Buggy Code
The buggy code sorts the array and repeatedly calculates the average of the smallest and largest elements, updating the array. The final result is calculated as the average of the last remaining element and the initial maximum element. This approach might not always yield the correct maximum possible value of \( x \).

### Challenging Test Input
To expose the bug, we can create a test case where the optimal sequence of operations is not simply the averaging of the smallest and largest elements. Consider an array where the optimal sequence involves more complex operations.

### Test Case
Let's create a test case with an array of length 5, where the elements are such that the optimal sequence of operations is not straightforward:

```plaintext
1
5
1 3 5 7 9
```

### Explanation
- **Initial Array**: \( [1, 3, 5, 7, 9] \)
- **Optimal Sequence**:
  1. Pick \( i = 1 \) and \( j = 5 \): \( \lfloor \frac{1 + 9}{2} \rfloor = 5 \). Array becomes \( [3, 5, 7, 5] \).
  2. Pick \( i = 1 \) and \( j = 4 \): \( \lfloor \frac{3 + 5}{2} \rfloor = 4 \). Array becomes \( [5, 7, 4] \).
  3. Pick \( i = 1 \) and \( j = 3 \): \( \lfloor \frac{5 + 4}{2} \rfloor = 4 \). Array becomes \( [7, 4] \).
  4. Pick \( i = 1 \) and \( j = 2 \): \( \lfloor \frac{7 + 4}{2} \rfloor = 5 \). Array becomes \( [5] \).

The maximum possible value of \( x \) is \( 5 \).

### Output
```plaintext
1
5
1 3 5 7 9
```

This test case should expose the bug in the buggy code, as the optimal sequence of operations is not simply the averaging of the smallest and largest elements.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
#include <bits/stdc++.h>
#define ll long long
#define ld long double
#define F first
#define S second
#define PB push_back
#define MP make_pair
#define rep(i, a, b) for (ll i = a; i <= b; i++)
#define fast                          \
    ios_base::sync_with_stdio(false); \
    cin.tie(NULL);
#define pi 3.141592653589793238462643383279502884197
using namespace std;

void solve()
{
    int n;
    cin >> n;
    vector<ll> v(n);
    rep(i, 0, n - 1)
    {
        cin >> v[i];
    }
    while (v.size() > 2)
    {
        sort(v.begin(), v.end());
        ll avg = floor((v[v.size() - 2] + v[0]) / 2);
        v.erase(v.begin() + v.size() - 2);
        v.erase(v.begin());
        v.push_back(avg);
    }
    cout << (*v.begin() + *v.rbegin()) / 2 << endl;
}

signed main()
{
    fast
        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:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on scenarios where the code's logic might fail. The buggy code sorts the array and calculates the average of the smallest and the second-largest elements, which is not the optimal way to maximize the final value.

### Key Points to Consider:
1. **Edge Cases:**
   - Arrays with identical elements.
   - Arrays with a mix of very large and very small elements.
   - Arrays with a small number of elements (e.g., 2 or 3).

2. **Optimal Strategy:**
   - The optimal strategy involves always picking the two largest elements to maximize the average.

### Challenging Test Input:
Let's create a test input that includes an array with a mix of very large and very small elements, which should expose the bug in the code's logic.

```plaintext
3
5
1 1000000000 1000000000 1000000000 1000000000
4
1 1000000000 1000000000 1000000000
3
1 2 3
```

### Explanation:
1. **Test Case 1:**
   - Array: `[1, 1000000000, 1000000000, 1000000000, 1000000000]`
   - The optimal strategy is to always pick the two largest elements, which will result in a final value of `1000000000`.

2. **Test Case 2:**
   - Array: `[1, 1000000000, 1000000000, 1000000000]`
   - Similarly, the optimal strategy will result in a final value of `1000000000`.

3. **Test Case 3:**
   - Array: `[1, 2, 3]`
   - The optimal strategy will result in a final value of `2`.

### Expected Output:
```plaintext
1000000000
1000000000
2
```

This test input will help expose the bug in the buggy code, which incorrectly calculates the average of the smallest and second-largest elements instead of always picking the two largest elements.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
#ifdef Sabbir
#include <E:\CodeBox\TempBox\Debug\debug.h>
#include <E:\CodeBox\TempBox\Basic\db.h>
#else
#define dbg(x...)
#endif
#define ll                 long long
#define all(x)             (x).begin(), (x).end()
#define rev(x)              reverse(all(x))
#define in(v)               for(auto &b:v)cin>>b;
#define pb                  push_back
#define vi                 vector<int>
vector<int> primes;
void solve(){
 int n;
 cin>>n;
 vi v(n);
 in(v);
 int mx=*max_element(all(v));
 ll sum=accumulate(all(v),0ll);
 sum-=mx;
 sum/=(n-1);

 cout<<((mx+sum)/2)<<endl;
}

int32_t main() {

 #ifdef Sabbir
      freopen("Error.txt", "w", stderr);
  #endif
    int t = 1;
    cin>>t;
    while (t--) {
        solve();
    }
    return 0;
}
//যদ্দপী, আমি তাহাতেই মুগ্ধ হইয়া গিয়াছি

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on scenarios where the code's logic fails to produce the correct maximum possible value of \( x \). The code's approach is to find the maximum element, subtract it from the sum of the array, divide the remaining sum by the number of elements minus one, and then take the floor of the average of this result and the maximum element. This approach is flawed because it does not consider the optimal way to combine elements to maximize the final value.

### Step-by-Step Analysis:
1. **Maximum Element and Sum Calculation**:
   - The code correctly identifies the maximum element and the sum of the array.
   - It then subtracts the maximum element from the sum and divides by the number of elements minus one.

2. **Average Calculation**:
   - The code calculates the average of the maximum element and the result of the division.
   - This approach does not account for the optimal sequence of operations to maximize the final value.

### Challenging Test Case:
To challenge the code, we can use an array where the optimal sequence of operations to maximize the final value is not straightforward. For example, an array with elements that are close to each other but not identical.

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

### Explanation:
1. **Test Case 1: [1, 2, 3, 4]**
   - The maximum element is 4.
   - The sum of the array is 10.
   - The remaining sum after removing the maximum element is 6.
   - The code will calculate \(\left\lfloor \frac{4 + \frac{6}{3}}{2} \right\rfloor = \left\lfloor \frac{4 + 2}{2} \right\rfloor = \left\lfloor 3 \right\rfloor = 3\).
   - The correct answer should be 3, but the optimal sequence of operations can achieve a higher value.

2. **Test Case 2: [1, 2, 3]**
   - The maximum element is 3.
   - The sum of the array is 6.
   - The remaining sum after removing the maximum element is 3.
   - The code will calculate \(\left\lfloor \frac{3 + \frac{3}{2}}{2} \right\rfloor = \left\lfloor \frac{3 + 1.5}{2} \right\rfloor = \left\lfloor 2.25 \right\rfloor = 2\).
   - The correct answer should be 2, but the optimal sequence of operations can achieve a higher value.

3. **Test Case 3: [1, 2, 3, 4, 5]**
   - The maximum element is 5.
   - The sum of the array is 15.
   - The remaining sum after removing the maximum element is 10.
   - The code will calculate \(\left\lfloor \frac{5 + \frac{10}{4}}{2} \right\rfloor = \left\lfloor \frac{5 + 2.5}{2} \right\rfloor = \left\lfloor 3.75 \right\rfloor = 3\).
   - The correct answer should be 4, but the optimal sequence of operations can achieve a higher value.

### Expected Output:
```plaintext
3
2
4
```

This test input will help expose the bug in the code by demonstrating that the current approach does not always yield the maximum possible value of \( x \).


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
def solution(n, arr):
    if n == 1:
        return arr[0]

    ans = 0
    def rec(arr):
        nonlocal ans
        if len(arr) == 2:
            ans = max(ans, sum(arr) // 2)
            return
        else:
            for i in range(len(arr) - 1):
                for j in range(i + 1, len(arr)):
                    arr_ = arr.copy()
                    avg = (arr_[i] + arr_[j]) // 2
                    arr_.pop(i)
                    arr_.pop(j - 1)
                    arr_.append(avg)
                    rec(arr_)
        return

    rec(arr)
    return ans

def main():
    t = int(input())
    for _ in range(t):
        n = int(input())
        arr = list(map(int, (input().split())))
        ans = solution(n, arr)
        print(ans)

if __name__ == "__main__":
    main()

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

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

Think step by step.

Response:
```plaintext
3
4
1 1 1 1
3
1000000000 1000000000 1000000000
5
1 2 3 4 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
from math import ceil
k = int(input())
otv = []
while k != 0:
    n = int(input())
    a = input().split()
    a = sorted([int(i) for i in a])

    while n != 1:
        i = a.index(max(a))
        j = a.index(min(a))
        a.append(ceil((max(a) + min(a)) / 2))
        a.pop(i)
        a.pop(j)
        n -= 1
    print(*a)
    k -= 1

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

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

Think step by step.

Response:
```plaintext
3
2
1 2
3
1 1 1
5
1 1000000000 1000000000 1000000000 1000000000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
from math import floor
for t in range(int(input())):
    n = int(input())
    a = [int(i) for i in input().split()]
    a.sort()
    mean = 2*a[0]
    for i in range(1, n):
        mean = mean/2 + a[i]
    print(floor(mean/2))

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

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

public class Main {
    static void solve() throws Exception {
        int i, j, n, m;
        n = in.nextInt();

        List<Integer> arr = new ArrayList<>();
        int mx = in.nextInt();

        for (i=1; i<n; i++) {
            int x = in.nextInt();

            if (mx < x) {
                arr.add(mx);
                mx = x;
            } else {
                arr.add(x);
            }
        }

        for (i=n-2; i>0; i--) {
            Collections.sort(arr);

            int x = arr.get(0);
            int y = arr.get(i);

            arr.remove(i);
            arr.remove(0);

            arr.add((x + y) >>1);
        }

        out.println((arr.get(0) + mx) >>1);
    }

    public static void main(String... args) {
        try { in = new FastReader(); out = new FastWriter();
            test();
            out.close();
        } catch (Exception e) { e.printStackTrace(); }
    }
    static void test() throws Exception {
        int t = in.nextInt();
        while (t-- > 0) { solve(); }
    } static FastReader in; static FastWriter out; final static int mod = (int) 1e9 +7;
    static class FastReader {
        BufferedReader br; StringTokenizer st;
        public FastReader() { br = new BufferedReader(new InputStreamReader(System.in)); }
        String next() {
            while (st == null || !st.hasMoreTokens()) {
                try { st = new StringTokenizer(br.readLine()); }
                catch (IOException e) { e.printStackTrace(); }
            } return st.nextToken();
        }
        int nextInt() { return Integer.parseInt(next()); }
        long nextLong() { return Long.parseLong(next()); }
        double nextDouble() { return Double.parseDouble(next()); }
        char nextChar() { return next().charAt(0); }
        String nextLine() {
            String str = "";
            try { str = br.readLine().trim(); }
            catch (Exception e) { e.printStackTrace(); }
            return str;
        }
    } static class FastWriter {
        private final BufferedWriter bw;
        public FastWriter() { this.bw = new BufferedWriter(new OutputStreamWriter(System.out)); }
        public void print(Object object) throws IOException { bw.append("" + object); }
        public void println(Object object) throws IOException { print(object); bw.append("\n"); }
        public void close() throws IOException { bw.close(); }
    }
    static long gcd(long a, long b) {
        if (b == 0) return a;
        return gcd(b, a % b);
    }
}

**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 1 1
4
1000000000 1000000000 1000000000 1000000000
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
import java.util.*;

public class main {
    public static void main(String args[]){
        Scanner sc=new Scanner(System.in);
        int n=sc.nextInt();
        for(int k=1;k<=n;k++){
        int l=sc.nextInt();
        int arr[]=new int[l];
        ArrayList <Integer> list=new ArrayList<Integer>();
        for(int i=0;i<l;i++){
           arr[i]=sc.nextInt();
           list.add(arr[i]);
        }

       Collections.sort(list);

        while(list.size()>=3){
            int res=list.get(0)+list.get((list.size())-2);
            list.remove((list.size())-2);
            list.remove(0);
            list.add(res/2);
            Collections.sort(list);
        }

        System.out.println((list.get(0)+list.get(1))/2);
    }}

}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

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

using namespace std;

void solve() {
	int n; cin >> n;
	vector<int> a(n);
	int s = 0, mx = -1e9;
	for (int i = 0; i < n; ++i) {
		cin >> a[i];
		s += a[i];
		mx = max(mx, a[i]);
	}
	cout << (int) floor((s - mx) / (n - 1) + mx >> 1) << '\n';
}

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

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

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

int main(){
    ios_base::sync_with_stdio(false);
    cin.tie(NULL);
    cout.tie(NULL);
    int t;
    cin>>t;

    while(t--){
        int n;
        cin>>n;

        vector<long long>v(n);

        for(int i=0;i<n;i++){
            cin>>v[i];
        }

        long long k=v[0];
        long long ans=LLONG_MIN;
        sort(v.begin(),v.end());

        for(int i=1;i<n;i++){
            k= floor((k+v[i])/2);
            ans=max(ans,k);
        }

        cout<< ans<<endl;

    }
    return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on the way the code handles the averaging and the order in which elements are processed. The buggy code always starts with the first element and iterates through the sorted array, which might not always yield the optimal result. Specifically, the code does not consider all possible pairs of elements, and it may prematurely reduce the values due to the floor division.

To create a test input that exposes this issue, we can use an array where the optimal solution involves a different order of operations than what the buggy code would perform. Here’s a step-by-step approach to creating such a test case:

1. **Identify the Bug**: The code always starts with the first element and iterates through the sorted array. This can lead to suboptimal results if the initial elements are too small.

2. **Create a Test Case**: We need an array where the optimal solution involves picking elements in a different order than the sorted order.

3. **Example Test Case**:
   - Consider an array where the largest elements are not at the beginning of the sorted array.
   - For instance, an array like `[1, 2, 1000000000, 1000000000]` can expose the bug. The optimal solution would involve picking the two largest elements first to maximize the result.

Let's generate the test input:

```plaintext
1
4
1 2 1000000000 1000000000
```

This test case will expose the bug because the buggy code will start with the smallest elements and reduce the values too quickly, leading to a suboptimal result. The optimal solution should involve picking the two largest elements first, which would result in a much higher final value.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
import math
n=int(input())
for i in range(n):
    len_array=int(input())
    str_input=input().split(' ')
    str_input=sorted(str_input,reverse=True)
    for i in range(len_array):
        str_input[i]=int(str_input[i])
    max_value=str_input[0]
    while len_array>1:
        mid=(str_input[-1]+str_input[-2])/2
        str_input.pop(len_array-1)
        str_input.pop(len_array-2)
        str_input.append(mid)
        len_array-=1
    print(math.floor(str_input[0]))

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

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

Think step by step.

Response:
```plaintext
3
2
1 1000000000
3
1 2 1000000000
5
1000000000 1000000000 1 1 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Meaning Mean
time_limit_ms: 1000
memory_limit_mb: 256
Description: Pak Chanek has an array $a$ of $n$ positive integers. Since he is currently learning how to calculate the floored average of two numbers, he wants to practice it on his array $a$.

While the array $a$ has at least two elements, Pak Chanek will perform the following three-step operation:

  1. Pick two different indices $i$ and $j$ ($1 \leq i, j \leq |a|$; $i \neq j$), note that $|a|$ denotes the current size of the array $a$.    2. Append $\lfloor \frac{a_i+a_j}{2} \rfloor$$^{\text{∗}}$ to the end of the array.    3. Remove elements $a_i$ and $a_j$ from the array and concatenate the remaining parts of the array.

For example, suppose that $a=[5,4,3,2,1,1]$. If we choose $i=1$ and $j=5$, the resulting array will be $a=[4,3,2,1,3]$. If we choose $i=4$ and $j=3$, the resulting array will be $a=[5,4,1,1,2]$.

After all operations, the array will consist of a single element $x$. Find the maximum possible value of $x$ if Pak Chanek performs the operations optimally.

$^{\text{∗}}$$\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer that is less than or equal to $x$. For example, $\lfloor 6 \rfloor = 6$, $\lfloor 2.5 \rfloor=2$, $\lfloor -3.6 \rfloor=-4$ and $\lfloor \pi \rfloor=3$

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 50$) — the length of the array $a$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^9$) — the elements of the array $a$.

Do note that the sum of $n$ over all test cases is not bounded.

Output: For each test case, output a single integer: the maximum possible value of $x$ after all numbers have been picked.

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

Note: In the first test case, the array is initially $a=[1,7,8,4,5]$. Pak Chanek will perform the following operations:

  1. Pick $i=1$ and $j=2$, then $a=[8,4,5,4]$.    2. Pick $i=3$ and $j=2$, then $a=[8,4,4]$.    3. Pick $i=2$ and $j=3$, then $a=[8,4]$.    4. Pick $i=1$ and $j=2$, then $a=[6]$.

After all the operations, the array consists of a single element $x=6$. It can be proven that there is no series of operations that results in $x$ greater than $6$ in the end.

Buggy Code:
def solve():
    N = int(input())
    val = sorted(map(int, input().split()), reverse=True)
    odd = []
    idx = 0
    while idx < len(val):
        if val[idx] & 1:
            odd.append(val.pop(idx))
        else:
            idx += 1
    while len(odd) > 2:
        a = odd.pop()
        b = odd.pop()
        c = (a + b) // 2
        if c & 1:
            odd.append(c)
        else:
            val.append(c)
    val.extend(odd)
    while len(val) > 1:
        a = val.pop()
        b = val.pop()
        val.append((a + b) // 2)
    print(val[0])

for _ in range(int(input())):
    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
3
1 2 3
```