**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

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

#define all(__v) __v.begin(), __v.end()
#define rall(__v) __v.rbegin(), __v.rend()

using namespace std;

using ll = long long;
using iii = tuple<ll, ll, ll>;
using ii = pair<ll, ll>;

template<int MOD>
struct _mint {

  long long value;

  _mint(long long value = 0) : value(value % MOD) {
    if (value < 0) value += MOD;
  }

  _mint operator + (_mint const& b) const {
    return _mint(value + b.value);
  }

  _mint operator - (_mint const& b) const {
    return _mint(value - b.value);
  }

  _mint operator * (_mint const& b) const {
    return _mint(value * b.value);
  }

  _mint& operator += (_mint const& b) {
    *this = *this + b;
    return *this;
  }

  _mint& operator -= (_mint const& b) {
    *this = *this - b;
    return *this;
  }

  _mint& operator *= (_mint const& b) {
    *this = *this * b;
    return *this;
  }

  bool operator == (_mint const& b) const {
    return value == b.value;
  }

  bool operator != (_mint const& b) const {
    return value != b.value;
  }

  friend std::ostream& operator<<(std::ostream& os, _mint const& a) {
      return os << a.value;
  }

  _mint exp(long long b) const {
    auto fast_pow = [](auto&& self, long long a, long long b) -> long long {
        if (b == 0)
            return 1;
        long long x = self(self, a, b / 2);
        x = (x * x) % MOD;
        return b & 1 ? (a * x) % MOD : x;
    };
    return _mint(fast_pow(fast_pow, value, b));
  }

  _mint inv() const {
    if (value == 0) {
        throw std::invalid_argument("Inverse of zero is not defined");
    }
    return exp(MOD - 2);
  }
};

using mint = _mint<1'000'000'007>;

void solve(){
  ll n;
  cin >> n;

  vector<ll> v(n);
  for ( auto&i : v)
    cin >>i;

  ll s = accumulate(all(v), 0ll);

  mint res(0);
  for ( auto e : v)
    res += e*(s-e);

  cout << res * mint(((n) * (n-1))).inv() << endl;

}

signed main() {

  ios_base::sync_with_stdio(false and true);
  cin.tie(nullptr);

  int t;
  cin >> t;

  while (t--)
    solve();

}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, let's analyze the algorithm and the potential issues:

### Algorithm Analysis
The code calculates the expected value of the product of two elements from the array. The formula used is:

\[ \text{res} = \sum_{i=1}^{n} \left( a_i \cdot (s - a_i) \right) \]

where \( s \) is the sum of all elements in the array. The result is then divided by the number of ways to choose 2 elements from \( n \) elements, which is \( \frac{n(n-1)}{2} \).

### Potential Issues
1. **Overflow**: The sum of products can be very large, and the code might not handle large numbers correctly.
2. **Modulo Operations**: The code uses modulo operations to handle large numbers, but the order of operations might be incorrect.
3. **Division by Zero**: The inverse function might throw an error if the value is zero, but the code should handle this correctly.

### Challenging Test Input
To expose these issues, we can create a test input with:
- A large sum of elements to test overflow.
- A small number of elements to ensure the input is small-scale.
- Elements that are zero to test division by zero handling.

Here is a small-scale test input that includes these elements:

```plaintext
3
3
1000000000 1000000000 1000000000
4
0 0 0 0
5
1 2 3 4 5
```

### Explanation
1. **First Test Case**:
   - Array: [1000000000, 1000000000, 1000000000]
   - Sum: 3000000000
   - Products: 1000000000 * 2000000000, 1000000000 * 2000000000, 1000000000 * 2000000000
   - Expected value: \(\frac{6000000000000000000}{3} = 2000000000000000000 \mod 1000000007\)

2. **Second Test Case**:
   - Array: [0, 0, 0, 0]
   - Sum: 0
   - Products: All zero
   - Expected value: \(\frac{0}{6} = 0 \mod 1000000007\)

3. **Third Test Case**:
   - Array: [1, 2, 3, 4, 5]
   - Sum: 15
   - Products: 2, 3, 4, 6, 8, 12, 15, 20, 24, 30
   - Expected value: \(\frac{120}{10} = 12 \mod 1000000007\)

This test input should help expose any issues in the provided code, especially related to large numbers and zero handling.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

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

public class b {
    static class InputReader {
        BufferedReader reader;
        StringTokenizer tokenizer;
        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }
        String next() { // reads in the next string
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }
        public int nextInt() { // reads in the next int
            return Integer.parseInt(next());
        }
        public long nextLong() { // reads in the next long
            return Long.parseLong(next());
        }
        public double nextDouble() { // reads in the next double
            return Double.parseDouble(next());
        }
    }
    static InputReader sc = new InputReader(System.in);
    static PrintWriter pw = new PrintWriter(System.out);

    static long mod = (long)1e9+7;

    static long binpow(long a,long b)
    {
        if(b==0)
        {
            return 1;
        }
        if(b%2>0)
        {
            return (a*binpow(a,b-1))%mod;
        }
        return binpow((a*a)%mod,b/2);
    }

    static long findProductSum(int A[], int n)
    {
        long array_sum = 0;
        for (int i = 0; i < n; i++)
            array_sum += A[i];
            array_sum%= mod;

        long array_sum_square = ((array_sum%mod) * (array_sum%mod))%mod;

        int individual_square_sum = 0;
        for (int i = 0; i < n; i++)
            individual_square_sum += ((A[i]%mod) * (A[i]%mod))%mod;
            individual_square_sum%= mod;

        long temp =  (((array_sum_square%mod) - (individual_square_sum%mod)+mod)%mod);
        return (temp*binpow(2,mod-2))%mod;

    }

    public static void main(String[] args) {
        int t = sc.nextInt();
        while(t-->0){
            int n = sc.nextInt();
            int[] a = new int[n];
            for(int i=0;i<n;i++){a[i] = sc.nextInt();}
            long sum = findProductSum(a,n);
            long cnt=n*(n-1)/2;cnt%=mod;
            System.out.println((sum%mod)*binpow(cnt,mod-2)%mod);
        }
        pw.close(); // flushes the output once printing is done
    }
}

**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 or scenarios that might cause it to fail.

### Analysis of the Buggy Code

1. **Array Sum Calculation**:
   - The code calculates the sum of the array elements and squares it.
   - It also calculates the sum of the squares of the individual elements.

2. **Expected Value Calculation**:
   - The expected value is calculated as:
     \[
     \text{Expected Value} = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} (a_i \cdot a_j) - \sum_{i=1}^{n} (a_i^2)}{n(n-1)}
     \]
   - The code simplifies this to:
     \[
     \text{Expected Value} = \frac{(\sum_{i=1}^{n} a_i)^2 - \sum_{i=1}^{n} a_i^2}{n(n-1)}
     \]
   - The code then multiplies by the modular inverse of \(2\) to account for the division by \(2\) in the expected value formula.

3. **Potential Bugs**:
   - **Modulo Arithmetic**: The code uses modulo operations to handle large numbers, but it might not handle edge cases correctly.
   - **Division by Zero**: The code assumes \(n(n-1)\) is not zero, but it should handle \(n = 1\) or \(n = 0\) (though the problem constraints ensure \(2 \le n \le 2 \cdot 10^5\)).
   - **Overflow**: The code might overflow when calculating large sums or products, even though it uses modulo operations.

### Challenging Test Input

To expose the bug, we can use a test case with a small \(n\) and values that are likely to cause issues in the modulo arithmetic. Specifically, we can use values that are close to the modulo limit \(10^9 + 7\).

### Test Case

```plaintext
1
3
1000000000 1000000000 1000000000
```

### Explanation

- **Array**: `[1000000000, 1000000000, 1000000000]`
- **Sum**: \(1000000000 + 1000000000 + 1000000000 = 3000000000\)
- **Sum of Squares**: \(1000000000^2 + 1000000000^2 + 1000000000^2 = 3000000000000000000\)

When these values are taken modulo \(10^9 + 7\):
- \(3000000000 \mod (10^9 + 7) = 2999999993\)
- \(3000000000000000000 \mod (10^9 + 7) = 2999999993000000000 \mod (10^9 + 7) = 2999999993000000000 - 2999999993 \cdot 1000000000 = 2999999993000000000 - 2999999993000000000 = 0\)

This test case should highlight any issues with the modulo arithmetic and the calculation of the expected value.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.*;

public class Main {

    public static void main(String args[]) throws IOException {
        BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
        PrintWriter out = new PrintWriter(System.out);
        int t = Integer.parseInt(in.readLine());
        while (t-- > 0) {
            StringTokenizer st1 = new StringTokenizer(in.readLine());
            StringTokenizer st2 = new StringTokenizer(in.readLine());
            int n = Integer.parseInt(st1.nextToken());
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = Integer.parseInt(st2.nextToken());

            long[] suffixSum = new long[n];
            suffixSum[n-1] = a[n-1];
            for (int i=n-1; i>=0; i--)
                suffixSum[i] = a[i] + (i+1 < n ? suffixSum[i+1] : 0);

            long numerator = 0;
            for (int i = 0; i < n-1; i++)
                numerator += a[i] * (suffixSum[i+1]);
            numerator %= mod;

            long denominator = (((long)n*(n-1)) / 2) % mod;

            long ans = ModularArithmetic.modDivide(numerator, denominator, mod);
            out.println(ans);
        }
        in.close();
        out.close();
    }

    private static int mod = 1000000007;

    public static class ModularArithmetic {

        private static long gcd(long a, long b) {
            return b == 0 ? a : gcd(b, a % b);
        }

        private static long modInverse(long b, long m){
            long g = gcd(b, m) ;
            if (g != 1)
                return -1;
            else
                return power(b, m-2, m);
        }

        /**
         * (a / b) % mod
         * Source: https://www.geeksforgeeks.org/modular-division/
         */
        public static long modDivide(long a, long b, int mod) {
            a = a % mod;
            long inv = modInverse(b,mod);
            if(inv == -1) {
                throw new RuntimeException("Division not defined");
            }
            else {
                return ((inv*a) % mod);
            }
        }

        public static long power(long a, long b, long mod) {
            long res = 1;
            a = a % mod; // Update a if it is more than or equal to mod
            if (a == 0)
                return 0; // In case a is divisible by mod;
            while (b > 0) {
                if ((b & 1) != 0)
                    res = (res * a) % mod;
                b = b >> 1; // b = b/2
                a = (a * a) % mod;
            }
            return (int) res;
        }

    }

}

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

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

Think step by step.

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

1. **Zero Values**: The code should handle arrays with zero values correctly. The expected value calculation should not be affected by zeros.
2. **Large Values**: Test with large values to ensure the modulo operations are handled correctly.
3. **Duplicate Values**: Test with arrays containing many duplicate values to ensure the suffix sum and product calculations are correct.
4. **Small Arrays**: Test with the smallest possible array size to ensure the code handles edge cases.
5. **Large Arrays**: Test with the largest possible array size to ensure the code handles large inputs efficiently.

Given these considerations, let's create a test input that includes a mix of these scenarios:

```plaintext
3
2
0 1
5
1 1 1 1 1
10
1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000 1000000000
```

This test input includes:
1. A small array with a zero value.
2. A small array with all elements being 1.
3. A larger array with all elements being the maximum possible value (10^9).

These scenarios should help in exposing any issues in the code, particularly in handling edge cases and large values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
import math
mod = 10**9+7
t=int(input())
def inv(x):
    return pow(x,mod-2,mod)
for j in range(t):
    n=int(input())
    abcdefg=input()
    bitchestemp=abcdefg.split()
    bitchsqd=0
    bitches=[]
    for i in range(n):
        bitches.append(int(bitchestemp[i]))
        temp = int(bitchestemp[i])**2
        bitchsqd = bitchsqd+temp
    totalsum = int((sum(bitches)**2 - bitchsqd)/2)
    gcdyay=math.gcd(totalsum,int(n*(n-1)*0.5))
    P=int(totalsum/gcdyay)
    Q=int((n*(n-1)*0.5)/gcdyay)
    print((P*inv(Q))%mod)

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

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

Think step by step.

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

1. **Edge Cases**: Test cases with small values, large values, and values that might cause overflow or precision issues.
2. **Special Cases**: Test cases with repeated values, all values being zero, and values that are powers of the modulus.
3. **Large Input**: Test cases with the maximum number of elements to stress the algorithm.
4. **Modulo Properties**: Test cases that exploit the properties of the modulo operation.

Given the constraints and the nature of the problem, a good test input should include a mix of these elements. Here is a small-scale test input that is complete and valid:

```plaintext
3
3
1 0 1
4
1000000000 1000000000 1000000000 1000000000
5
1 2 3 4 5
```

### Explanation:
1. **Test Case 1**: `3 1 0 1`
   - This test case includes a zero value, which can cause issues if not handled correctly. The expected value should be 0 because any product involving zero is zero.
2. **Test Case 2**: `4 1000000000 1000000000 1000000000 1000000000`
   - This test case includes large values that are close to the upper limit of the value range. This can cause overflow if not handled correctly. The expected value should be calculated correctly using modulo operations.
3. **Test Case 3**: `5 1 2 3 4 5`
   - This test case is a standard case with distinct values. It should help verify the correctness of the algorithm.

This test input should help expose any issues in the buggy code, particularly with handling zeros, large values, and the modulo operation.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
def mod_inverse(a, mod):
    # Fermat's Little Theorem to compute the modular inverse
    return pow(a, mod - 2, mod)

MOD = int(1e9 + 7)
for _ in range(int(input())):
	n=int(input())
	l=[int(x) for x in input().split()]
	totalsum=sum(l)
	pro_sum=0
	for i in range(len(l)):
		pro_sum+=l[i]*totalsum-l[i]**2
		pro_sum%=MOD
	n=len(l)
	deno=n*(n-1)//2
	pro_sum//=2
	deno_inverse = mod_inverse(deno, MOD)

	# Calculate final result
	result = (pro_sum * deno_inverse) % MOD
	print(result)

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

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

Think step by step.

Response:
```plaintext
1
2
0 0
```

### Explanation:
This test case is designed to expose potential issues in the buggy code by using the smallest possible valid input size (two elements) and setting both elements to zero.

1. **Small Scale**: The test case uses the minimum possible number of elements, which is 2, to ensure that the input is small and manageable.
2. **Edge Case**: Using zero values can highlight issues with the calculation of the product and the expected value, especially when the denominator in the expected value calculation could be problematic.
3. **Valid Input**: The input is valid according to the problem constraints, as it fits within the range of allowed values for `n` and the elements of the array.

### Expected Behavior:
- The total sum of the elements is 0.
- The product sum calculation should handle the zero values correctly.
- The denominator for the expected value should be 1 (since \( n = 2 \), and \( \frac{2 \cdot (2 - 1)}{2} = 1 \)).
- The modular inverse of 1 is 1, so the final result should be 0.

This test case will help identify if the code correctly handles zero values and small input sizes.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
#include <bits/stdc++.h>
#define forr(i, a, n) for(int i = a; i < n; i++)
#define forn(i, n) for(int i = 0; i < n; i++)
#define dforn(i, n) for(int i = n - 1; i >= 0; i--)
#define forall(it, v) for(auto it = v.begin(); it != v.end(); it++)
#define pb push_back
#define sz(a) ((int)a.size())
#define all(x) (x).begin(),(x).end()
#define rall(x) (x).rbegin(),(x).rend()
#define dbg(x) cout << #x << " = " << (x) << endl
#define vdbg(x) cout << '['; for(auto &i : x) cout << i << ", "; cout << "]\n";
#define fst first
#define snd second

using namespace std;

typedef long long ll;
typedef pair<int, int> ii;

const ll MOD = 1e9 + 7;

ll modpow(ll x, ll n, ll m = MOD){
    if(n == 0) return 1;

    ll u = modpow(x, n / 2, m);
    return (((u * u) % m) * (n % 2 ? x : 1)) % m;
}

int main(){
    ios::sync_with_stdio(0);
    cin.tie(0);
    #ifdef fran
        freopen("input.in", "r", stdin);
        freopen("output.out", "w", stdout);
    #endif

    ll t, n;
    cin >> t;
    while(t--){
        cin >> n;
        ll rta = 0, sum = 0;
        vector<ll> v(n);
        forn(i, n) cin >> v[i];
        forn(i, n) sum = (sum + v[i]) % MOD;
        forn(i, n) rta = (rta + (sum - v[i] + MOD) * v[i]) % MOD;

        ll inv2 = modpow(2, MOD - 2);
        ll invq = modpow((n * (n - 1)) / 2, MOD - 2);

        rta = (rta * inv2) % MOD;
        rta = (rta * invq) % MOD;
        cout << rta << '\n';
    }

    return 0;
}

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases or scenarios that can maximize the failure probability. The code calculates the expected value of the product of two elements from the array, and it uses modular arithmetic to handle large numbers.

Let's consider the following edge cases:
1. **Array with all elements being the same**: This can cause issues with the modulo operations and the expected value calculation.
2. **Array with a large number of elements**: This can test the efficiency and correctness of the modular inverse calculations.
3. **Array with elements that are multiples of each other**: This can test the correctness of the product calculations.

Given the constraints, a small-scale test input that can expose the bug is an array with all elements being the same. This will test the correctness of the modular inverse and the sum calculations.

Here is a small-scale test input that fits the criteria:

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

This test case has:
- 1 test case
- 5 elements in the array
- All elements are 3

This input will help verify if the code correctly handles the case where all elements are the same. The expected value of the product of any two elements in this array is 9, and the number of pairs is 10. The expected value is \(\frac{90}{10} = 9\). The output should be \(9 \mod (10^9 + 7)\).

Let's see if the provided code handles this input correctly.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

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

using namespace std;

#define fi first
#define se second
#define int long long
#define alls(x) x.begin(), x.end()
#define pb push_back
#define end1n " \n"[i == n]
#define endn1 " \n"[i == 1]
#define endl '\n'

typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;
typedef pair<pii, int> piii;
typedef vector<int> vi;
typedef vector<pii> vpii;
typedef vector<vector<int>> vii;
const int N = 1e6 + 10;
const int mod = 1e9 + 7;
const int MAX = 1e18;
int a[N];

int qmi(int a, int b, int k) {
	int res = 1;
	while(b) {
		if(b & 1) res = res * a % k;
		a = a * a % k;
		b >>= 1;
	}

	return res;
}

void solve() {
	int n;
	cin >> n;
	int q = n * (n - 1) / 2;
	q %= mod;
	q = qmi(q, mod - 2, mod);
	for(int i = 1; i <= n; i ++) cin >> a[i];

	int res = 0;
	for(int i = 1; i <= n; i ++) {
		for(int j = i + 1; j <= n; j ++) {
			res += a[i] * a[j] % mod;
			res %= mod;
		}
	}
//	cout << res << endl;
	res = res * q % mod;

	cout << res << endl;
}

signed main() {
    ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
    int t;
    cin >> t;
//    t = 1;
    while(t --) {
        solve();
    }
    return 0;
}

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

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

Think step by step.

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

1. **Large Input Size**: The code should handle large input sizes within the given constraints.
2. **Zero Values**: The presence of zero values can affect the expected value calculation.
3. **Duplicate Values**: The code should handle arrays with many duplicate values correctly.
4. **Modulo Arithmetic**: The code should correctly handle modulo operations, especially when dealing with large numbers and inverses.

Given these points, let's construct a test input that includes these scenarios:

- A large number of elements to test the performance and correctness of the modulo operations.
- Zero values to test how the code handles them.
- Duplicate values to test the expected value calculation.

Here is the test input:

```plaintext
1
10
0 1 2 3 4 5 6 7 8 9
```

This test input includes:
- A single test case with 10 elements.
- The elements include a zero value and a sequence of integers from 1 to 9.

This input will test:
- The handling of zero values in the product calculation.
- The correct calculation of the expected value with a mix of unique and sequential values.
- The performance and correctness of the modulo operations.

Let's break down the expected output for this test input:

1. **Calculation of the Product of Pairs**:
   - The pairs and their products are:
     - (0, 1) = 0
     - (0, 2) = 0
     - (0, 3) = 0
     - (0, 4) = 0
     - (0, 5) = 0
     - (0, 6) = 0
     - (0, 7) = 0
     - (0, 8) = 0
     - (0, 9) = 0
     - (1, 2) = 2
     - (1, 3) = 3
     - (1, 4) = 4
     - (1, 5) = 5
     - (1, 6) = 6
     - (1, 7) = 7
     - (1, 8) = 8
     - (1, 9) = 9
     - (2, 3) = 6
     - (2, 4) = 8
     - (2, 5) = 10
     - (2, 6) = 12
     - (2, 7) = 14
     - (2, 8) = 16
     - (2, 9) = 18
     - (3, 4) = 12
     - (3, 5) = 15
     - (3, 6) = 18
     - (3, 7) = 21
     - (3, 8) = 24
     - (3, 9) = 27
     - (4, 5) = 20
     - (4, 6) = 24
     - (4, 7) = 28
     - (4, 8) = 32
     - (4, 9) = 36
     - (5, 6) = 30
     - (5, 7) = 35
     - (5, 8) = 40
     - (5, 9) = 45
     - (6, 7) = 42
     - (6, 8) = 48
     - (6, 9) = 54
     - (7, 8) = 56
     - (7, 9) = 63
     - (8, 9) = 72

2. **Sum of the Products**:
   - Sum = 0 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 12 + 15 + 18 + 21 + 24 + 27 + 20 + 24 + 28 + 32 + 36 + 30 + 35 + 40 + 45 + 42 + 48 + 54 + 56 + 63 + 72 = 567

3. **Number of Pairs**:
   - Number of pairs = 10 * (10 - 1) / 2 = 45

4. **Expected Value**:
   - Expected value = 567 / 45 = 12.6

5. **Modulo Operation**:
   - P = 567
   - Q = 45
   - Q inverse = qmi(45, mod - 2, mod)
   - Expected value = (567 * Q inverse) % mod

This test input should expose any issues in the buggy code, particularly in the handling of zero values and large numbers.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

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

public class sub {

	/*
	 * /\_/\ (= ._.) / > \>
	 */
	static int mod = 1000000007;

	public static void main(String args[]) {
		int t = sc.nextInt();
		while (t-- > 0) {
			int n = sc.nextInt();
			int a[] = new int[n];
			long sum = 0;
			long sumSq = 0;
			for(int i = 0;i<n;i++) {
				a[i]=sc.nextInt();
				sum+=a[i];
				sumSq=(sumSq+(a[i]*(a[i]+0L))%mod)%mod;
			}
			sum%=mod;
			long div = (n-1L)*n;
			//find the modular inverse
			long ans = (pow(div, mod-2 ,mod)*(((sum*sum)%mod-sumSq%mod+mod)%mod))%mod;
			out.println(ans);
		}
		out.close();
	}

	/*
	 * public static long bincoef(int n, int k) { if(n < k) return 0L; return (p[n]
	 * * pow((p[n - k] * p[k]) % mod, mod - 2,mod))%mod; }
	 */
	static long pow(long a, long b, long mod) {
		if (b == 0) {
			return 1L;
		}
		long val = pow(a, b / 2, mod);
		if (b % 2 == 0) {
			return val * val % mod;
		}
		return val * val % mod * a % mod;
	}

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

	static int lcm(int a, int b) {
		return a * b / gcd(a, b);
	}

	static Kattio sc = new Kattio();
	static PrintWriter out = new PrintWriter(System.out);

	static class Kattio {

		static BufferedReader r;
		static StringTokenizer st;

		public Kattio() {
			r = new BufferedReader(new InputStreamReader(System.in));
		}

		public String next() {
			try {
				while (st == null || !st.hasMoreTokens()) {
					st = new StringTokenizer(r.readLine());
				}
				return st.nextToken();
			} catch (Exception e) {
				return null;
			}
		}

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

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

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

}

**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 test input that exposes the bug in the provided code, we need to focus on edge cases or scenarios where the code might fail. Here are some considerations:

1. **Zero Values**: The code should handle cases where some or all values in the array are zero.
2. **Large Values**: The code should handle large values that could cause overflow or incorrect modular arithmetic.
3. **Repeated Values**: The code should handle cases where many values in the array are the same.
4. **Small Arrays**: The code should handle small arrays to ensure basic functionality.
5. **Large Arrays**: The code should handle large arrays to test performance and correctness.

Given the constraints and the nature of the problem, let's focus on a scenario that might expose the bug in the modular arithmetic and the calculation of the expected value.

### Test Input
```plaintext
1
4
1000000000 1000000000 1000000000 1000000000
```

### Explanation
- **Test Case**: 1
- **Array Size**: 4
- **Array Values**: [1000000000, 1000000000, 1000000000, 1000000000]

### Reasoning
- **Large Values**: Using the maximum value (1000000000) in the array ensures that the sums and products are large, which can help expose issues with modular arithmetic.
- **Repeated Values**: All values in the array are the same, which simplifies the expected value calculation. The expected value for the product of two elements from this array should be:
  - Sum of elements: \(4 \times 1000000000 = 4000000000\)
  - Sum of squares: \(4 \times (1000000000^2) = 4000000000000000000\)
  - Number of pairs: \(\frac{4 \times 3}{2} = 6\)
  - Expected value: \(\frac{4000000000^2 - 4000000000000000000}{6} = \frac{16000000000000000000 - 4000000000000000000}{6} = \frac{12000000000000000000}{6} = 2000000000000000000\)
  - Modular inverse of 6: \(6^{-1} \mod 1000000007\)
  - Final answer: \((2000000000000000000 \times 6^{-1}) \mod 1000000007\)

This test input should help expose any issues with the modular arithmetic and the calculation of the expected value 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: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
import java.io.BufferedReader;
import java.io.BufferedWriter;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.OutputStreamWriter;
import java.io.PrintWriter;
import java.util.*;

public class f {
    static FastReader fr = new FastReader();
    static FastWriter fw = new FastWriter();
    static int mod = (int)1e9+7;
    static long power(long a, long b) {
        if(a==0) return 0;
        long ans = 1;
        while (b>0) {
            if ((b & 1)!=0) ans = (ans * a) % mod;
            a = (a * a) % mod;
            b >>= 1;
        }
        return (int)ans;
    }
    static long INV(long a){ return power(a, mod-2);}
    public static void solve(){
        int n = fr.nextInt();
        int a[] = fr.nextIntArray();

        long sum = a[0];
        long ans = 0;
        long cnt = n*(n-1)/2;
        long val = INV(cnt)%mod;
        for(int i=1;i<n;i++){
            ans = (ans + (sum*a[i])%mod*val)%mod;
            sum  = (sum+a[i])%mod;
        }
        fw.println(ans);
    }
    public static void main(String[] args) throws NumberFormatException, IOException{
        //long start = System.nanoTime();
        int t = fr.nextInt();
        //System.out.println(solve());
        //System.out.println(solve()?"YES":"NO");
        while(t-->0){
            solve();
            //System.out.println(solve());
            //System.out.println(solve()?"YES":"NO");
        }
        //System.out.println(((long)System.nanoTime()-start)/1000000);
    }
}

class FastReader{
    BufferedReader br;
    StringTokenizer st;
    public FastReader(){
        br = new BufferedReader(new InputStreamReader(System.in));
    }
    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{
            str = br.readLine();
        }
        catch (IOException e){
            e.printStackTrace();
        }
        return str;
    }
    int[] nextIntArray() {
        String line = nextLine();
        String[] strNums = line.split("\\s+");
        int[] arr = new int[strNums.length];
        for (int i = 0; i < strNums.length; i++) {
            arr[i] = Integer.parseInt(strNums[i]);
        }
        return arr;
    }
    long[] nextLongArray() {
        String line = nextLine();
        String[] strNums = line.split("\\s+");
        long[] arr = new long[strNums.length];
        for (int i = 0; i < strNums.length; i++) {
            arr[i] = Long.parseLong(strNums[i]);
        }
        return arr;
    }
    char[] nextCharArray() {
        String line = nextLine();
        String[] strNums = line.split("\\s+");
        char[] arr = new char[strNums.length];
        for (int i = 0; i < strNums.length; i++) {
            arr[i] = strNums[i].charAt(0);
        }
        return arr;
    }
    char[] StringtoChar() {
        String line = nextLine();
        return line.toCharArray();
    }
}

class FastWriter {
    private final PrintWriter writer;

    public FastWriter() {
        writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)), true);
    }

    public void print(Object object) {
        writer.print(object);
    }

    public void println(Object object) {
        writer.println(object);
    }

    public void flush() {
        writer.flush();
    }
}

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

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

Think step by step.

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

1. **Understand the Bug**: The buggy code seems to calculate the expected value of the product of two elements from the array. The main issue is in the calculation of the sum and the product, and how the modulo operation is handled.

2. **Edge Cases**:
   - **Small Arrays**: Test with the smallest possible array size (2 elements).
   - **Large Arrays**: Test with the maximum possible array size (200,000 elements).
   - **Zero and Large Values**: Test with arrays containing zeros and very large values (up to 10^9).
   - **Duplicate Values**: Test with arrays containing many duplicate values.
   - **Special Values**: Test with values that are likely to cause overflow or modulo issues.

3. **Challenging Input**:
   - **Small Array with Zeros**: This can test how the code handles zeros.
   - **Array with All Same Values**: This can test how the code handles duplicate values.
   - **Array with Large Values**: This can test how the code handles large numbers and modulo operations.

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

```plaintext
3
2
0 1000000000
3
1 1 1
4
1000000000 1000000000 1000000000 1000000000
```

This test input includes:
- A small array with a zero and a large value.
- An array with all the same values.
- An array with large values to test modulo operations.

This should help expose any issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

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

const int MOD = 1e9 + 7;

// Function to perform modular exponentiation
long long mod_exp(long long base, long long exp, long long mod) {
    long long result = 1;
    while (exp > 0) {
        if (exp % 2 == 1)
            result = (result * base) % mod;
        base = (base * base) % mod;
        exp /= 2;
    }
    return result;
}

// Function to find the modular inverse
long long mod_inv(long long x, long long mod) {
    return mod_exp(x, mod - 2, mod);
}

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

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

        vector<long long> a(n);
        long long sum = 0, sum_of_squares = 0;

        // Read the array and compute sum and sum of squares
        for (int i = 0; i < n; ++i) {
            cin >> a[i];
            sum = (sum + a[i]) % MOD;
            sum_of_squares = (sum_of_squares + a[i] * a[i]) % MOD;
        }

        // Calculate the sum of all products
        long long total_pairs = (n * (n - 1) / 2) % MOD;
        long long sum_of_products = (sum * sum % MOD - sum_of_squares + MOD) % MOD;
        sum_of_products = (sum_of_products * mod_inv(2, MOD)) % MOD;

        // Calculate the modular inverse of total_pairs
        long long total_pairs_inv = mod_inv(total_pairs, MOD);

        // Calculate the final result
        long long result = (sum_of_products * total_pairs_inv) % MOD;

        cout << result << 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 edge cases or scenarios where the code might fail. The code calculates the expected value of the product of two elements from the array. The key steps in the code include:

1. Calculating the sum of all elements.
2. Calculating the sum of the squares of all elements.
3. Calculating the total number of pairs.
4. Using these to compute the sum of all products.
5. Using modular arithmetic to ensure the result is within the limits.

### Potential Issues:
1. **Modulo Arithmetic**: The code uses modulo operations extensively, which can introduce errors if not handled correctly.
2. **Total Pairs Calculation**: The calculation of `total_pairs` might overflow if `n` is large.
3. **Modular Inverse**: The modular inverse function might not handle certain edge cases correctly.

### Edge Cases:
1. **All Elements are Zero**: This should result in an expected value of 0.
2. **All Elements are the Same**: This should result in a specific expected value.
3. **Large Values**: Ensure the code handles large values without overflow.
4. **Small Values**: Test with small values to ensure correctness.

### Chosen Test Input:
Let's create a test input that includes a small number of elements, but with values that can expose potential issues in the modulo arithmetic and total pairs calculation.

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

### Explanation:
1. **First Test Case**: All elements are zero. The expected value should be 0.
2. **Second Test Case**: All elements are the same (1). The expected value should be 1.
3. **Third Test Case**: A mix of small values. The expected value should be 500000012, as given in the example.

This test input should help in verifying the correctness of the code and potentially exposing any bugs in the modulo arithmetic or total pairs calculation.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
#include <bits/stdc++.h>
using namespace std;
#define ll long long
const int MOD=1e9+7;
#define pb push_back
//vector<int>v;
//string ret;
//unordered_map<int,int>mp;
ll modinverse(ll a, ll m) {
    ll res = 1;
    ll e = m - 2;
    while(e){
        if(e % 2){
            res=res*a%m;
        }
        a=a*a%m;
        e/=2;
    }
    return res;
}
int main()
{
    int t;
    cin >> t;
    while (t--)
    {
        int n;
        cin>>n;
        ll sum=0,ss=0;
        vector<ll>v(n);
        for(int i=0;i<n;i++){
            cin>>v[i];
            sum=(sum+v[i])%MOD;
            ss=(ss+v[i]*v[i])%MOD;
        }
        ll ttsum=(sum*sum-ss+MOD)%MOD;
        ll np=(n*(n-1)/2)%MOD;
        ll invp=modinverse(np*2%MOD,MOD);
        ll ev=(ttsum*invp)%MOD;
        cout<<ev<<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
2
0 0
```

### Explanation:
This test case is designed to expose potential issues in the buggy code. Here's the reasoning:

1. **Input Constraints**: The input is within the given constraints: \(1 \le t \le 10^4\) and \(2 \le n \le 2 \cdot 10^5\).

2. **Edge Case**: The input contains two zeros. This is a critical edge case because:
   - The product of any two zeros is zero.
   - The expected value should be zero, as all possible pairs will have a product of zero.

3. **Buggy Code Analysis**:
   - The code calculates the sum of all elements and the sum of squares of all elements.
   - It then computes the expected value using the formula \(\frac{\sum_{i=1}^{n} a_i \cdot \sum_{j=1}^{n} a_j - \sum_{i=1}^{n} a_i^2}{n(n-1)/2}\).
   - For the input \([0, 0]\):
     - \( \sum_{i=1}^{n} a_i = 0 \)
     - \( \sum_{i=1}^{n} a_i^2 = 0 \)
     - The expected value should be \(\frac{0 \cdot 0 - 0}{1} = 0\).

4. **Potential Issues**:
   - If the code does not handle the case where all elements are zero correctly, it might produce an incorrect result.
   - The code should handle the division and modular inverse correctly, especially when the numerator is zero.

This test case is small and valid, and it should help in identifying any issues in the buggy code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
MOD = 10**9 + 7
MAXN = 2 * 10**5 + 5

# Precompute factorial and inverse factorial
factorial = [1] * MAXN
inverse_factorial = [1] * MAXN
for i in range(1, MAXN):
    factorial[i] = (factorial[i - 1] * i) % MOD
inverse_factorial[-1] = pow(factorial[-1], MOD - 2, MOD)
for i in reversed(range(1, MAXN)):
    inverse_factorial[i - 1] = (inverse_factorial[i] * i) % MOD

def binom(n, k):
    if k > n or k < 0:
        return 0
    return (factorial[n] * inverse_factorial[k] * inverse_factorial[n - k]) % MOD

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

        # Sum of pairwise products
        sum_products = sum(arr[i] * arr[j] for i in range(n) for j in range(i + 1, n))

        # Denominator (C(n, 2))
        denom = binom(n, 2)

        # Expected value (sum / denom) mod MOD
        exp_value = (sum_products * pow(denom, MOD - 2, MOD)) % MOD

        print(exp_value)

if __name__ == "__main__":
    main()

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

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

Think step by step.

Response:
To expose the bug in the provided code, we need to focus on edge cases that might cause the code to fail. Here are some potential edge cases to consider:

1. **All elements are the same**: This can test the code's handling of repeated values.
2. **Large values**: This can test the code's ability to handle large numbers without overflow.
3. **Zero values**: This can test how the code handles zero values, which might affect the product calculation.
4. **Mixed values**: This can test the code's ability to handle a mix of small and large values.

Given the constraints and the need for a small-scale test input, let's create a test case that includes all elements being the same and a mix of values, including zero.

Here is a small-scale test input that includes these edge cases:

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

This test input includes:
1. A case where all elements are the same (1, 1, 1).
2. A case where all elements are zero (0, 0, 0, 0).
3. A case with a mix of values (1, 2, 3, 4, 5).

These cases should help expose any bugs in the code, particularly in handling repeated values and zero values.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
def solve():

    t = int(input())  # number of test cases
    for _ in range(t):
        n=int(input())
        balls= list(map(int, input().split()))
        p=0
        q=n*(n-1)//2

        for i in range(n):
            for j in range(i+1,n):

                p+=balls[i]*balls[j]

        m=int(1e9)+7

        if p%q==0:

            print((p//q)%m)

        else:

            c=0
            h=1
            while h!=0:
                c+=1
                h=(c*m+1)%q

            x=int((c*m+1)/q)

            expected_value=int((p*x)%m)

            print(expected_value)

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 given code, we need to consider edge cases and scenarios that can maximize the failure probability. Here are the steps to think through this:

1. **Edge Cases**:
   - **Single Test Case with Minimum Number of Balls**: This helps in checking the basic functionality.
   - **Maximum Number of Balls**: This can help in identifying performance issues and boundary conditions.
   - **All Balls with the Same Value**: This can test the handling of duplicate values.
   - **Balls with Zero Values**: This can test how the code handles zero values.
   - **Balls with Large Values**: This can test the handling of large numbers and potential overflow issues.

2. **Scenarios**:
   - **Large Number of Test Cases**: This can test the overall performance and correctness of the solution.
   - **Mixed Values**: This can test the code's ability to handle a variety of values.

Given the constraints and the need for a small-scale test input, let's focus on a few specific cases:

- A test case with a small number of balls to check basic functionality.
- A test case with all balls having the same value to test duplicate handling.
- A test case with a zero value to test zero handling.
- A test case with a large value to test large number handling.

Here is the test input:

```plaintext
4
3
1 2 3
4
2 2 2 2
2
0 1
2
1000000000 1000000000
```

Explanation:
- **Test Case 1**: 3 balls with values 1, 2, 3. This is a basic test case to check the expected value calculation.
- **Test Case 2**: 4 balls with the same value 2. This tests the handling of duplicate values.
- **Test Case 3**: 2 balls with values 0 and 1. This tests the handling of zero values.
- **Test Case 4**: 2 balls with large values 1000000000. This tests the handling of large numbers.

This test input should help in identifying potential bugs in the given code.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
import java.util.Scanner;

public class ProblemSet {

    static final long MOD = 1000000007L;

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while (t-- > 0) {
            int n = sc.nextInt();
            long[] a = new long[n];
            for (int i = 0; i < n; i++) {
                a[i] = sc.nextLong();
            }
            long ans = 0;
            long s = 0;

            // Calcula a soma total dos elementos
            for (int i = 0; i < n; i++) {
                s = (s + a[i]) % MOD;
            }

            // Calcula o resultado
            for (int i = 0; i < n; i++) {
                s = (s - a[i] + MOD) % MOD; // Mantém o valor não negativo
                ans = (ans + a[i] * s) % MOD;
            }

            // Calcula o número total de pares
            long totalPairs = n * (n - 1) / 2;
            // Calcula a inversa modular de totalPairs
            long invTotalPairs = binpow(totalPairs, MOD - 2, MOD);

            // Calcula o resultado final
            ans = (ans * invTotalPairs) % MOD;

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

    // Exponenciação modular rápida
    private static long binpow(long a, long b, long mod) {
        long res = 1;
        while (b > 0) {
            if ((b & 1) == 1) {
                res = (res * a) % mod;
            }
            a = (a * a) % mod;
            b >>= 1;
        }
        return res;
    }
}

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

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

Think step by step.

Response:
```plaintext
1
2
0 0
```

This test case is designed to expose potential issues in the buggy code by using the smallest possible valid input size with all values being zero. This scenario can help identify problems with the code's handling of zero values and the calculation of the expected value, especially in the modular arithmetic and the calculation of the modular inverse.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako's Box
time_limit_ms: 2000
memory_limit_mb: 256
Description: Sakurako has a box with $n$ balls. Each ball has it's value. She wants to bet with her friend that if the friend randomly picks two balls from the box (it could be two distinct balls, but they may have the same value), the product of their values will be the same as the number that Sakurako guessed.

Since Sakurako has a PhD in probability, she knows that the best number to pick is [the expected value](http://tiny.cc/matozh_en), but she forgot how to calculate it. Help Sakurako and find the expected value of the product of two elements from the array.

It can be shown that the expected value has the form $\frac{P}{Q}$, where $P$ and $Q$ are non-negative integers, and $Q \ne 0$. Report the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

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

The first line of each test case contains a single integer $n$ ($2\le n\le 2\cdot 10^5$)  — the number of elements in the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0\le a_i\le 10^9$)  — the elements of the array.

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

Output: For each test case, output the value of $P \cdot Q^{-1}(\bmod 10^9+7)$.

Examples:
input:
3
3
3 2 3
4
2 2 2 4
5
1 2 3 4 5
output:
7
6
500000012

Note: For the first test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_2,a_3)$. Their products equal to $3\cdot 2=6$ , $3\cdot 3=9$ , $3\cdot 2=6$ respectively, so the expected value is $\frac{6+9+6}{3}=7$.

For the second test, Sakurako's friend can pick these pairs of balls: $(a_1,a_2)$, $(a_1,a_3)$, $(a_1,a_4)$, $(a_2,a_3)$, $(a_2,a_4)$, $(a_3,a_4)$. Their products equal to $2\cdot 2=4$ , $2\cdot 2=4$ , $2\cdot 4=8$, $2\cdot 2=4$, $2\cdot 4=8$, $2\cdot 4=8$ respectively, so the expected value is $\frac{4+4+8+4+8+8}{6}=\frac{36}{6}=6$.

Buggy Code:
import java.util.Scanner;

public class ExpectedValueCalculation {

    private static final int MOD = 1_000_000_007;

    private static long modExp(long base, long exponent, int mod) {
        long result = 1;
        base = base % mod;
        while (exponent > 0) {
            if ((exponent & 1) == 1) {
                result = (result * base) % mod;
            }
            exponent >>= 1;
            base = (base * base) % mod;
        }
        return result;
    }

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

        while (t-- > 0) {
            int n = scanner.nextInt();
            long[] a = new long[n];
            long sum = 0;
            long sumOfSquares = 0;

            for (int i = 0; i < n; i++) {
                a[i] = scanner.nextLong();
                sum = (sum + a[i]) % MOD;
                sumOfSquares = (sumOfSquares + a[i] * a[i]) % MOD;
            }

            // Numerator: (sum^2 - sum_of_squares) / 2
            long numerator = (sum * sum % MOD - sumOfSquares + MOD) % MOD;
            if (numerator % 2 != 0) {
                numerator += MOD;
            }
            numerator = numerator / 2 % MOD;

            // Denominator: n * (n - 1) / 2
            long denominator = n * (n - 1) / 2 % MOD;

            // Calculate modular inverse of denominator
            long denominatorInv = modExp(denominator, MOD - 2, MOD);

            // Result = (numerator * denominatorInv) % MOD
            long result = (numerator * denominatorInv) % MOD;

            System.out.println(result);
        }

        scanner.close();
    }
}

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

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

Think step by step.

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

1. **All elements are zero**: This can cause issues with the calculation of the modular inverse.
2. **All elements are the same**: This can simplify the expected value calculation, but might still expose issues in the code.
3. **Large values**: Using large values can test the limits of the modulo operations.
4. **Small values**: Using small values can help in verifying the correctness of the basic logic.
5. **Mix of large and small values**: This can test the robustness of the code in handling a variety of values.

Let's generate a test input that includes a mix of these edge cases:

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

This test input includes:
- A case where all elements are zero.
- A case where all elements are the same.
- A case with a mix of small and large values.

This should help in exposing any bugs in the code, particularly in the handling of zero values and the calculation of the modular inverse.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Microwavable Subsequence
time_limit_ms: 1000
memory_limit_mb: 1024
Description: You are given an array of $$$N$$$ integers: $$$[A_1, A_2, \dots, A_N]$$$.

A subsequence can be derived from an array by removing zero or more elements without changing the order of the remaining elements. For example, $$$[2, 1, 2]$$$, $$$[3, 3]$$$, $$$[1]$$$, and $$$[3, 2, 1, 3, 2]$$$ are subsequences of array $$$[3, 2, 1, 3, 2]$$$, while $$$[1, 2, 3]$$$ is not a subsequence of array $$$[3, 2, 1, 3, 2]$$$.

A subsequence is microwavable if the subsequence consists of at most two distinct values and each element differs from its adjacent elements. For example, $$$[2, 1, 2]$$$, $$$[3, 2, 3, 2]$$$, and $$$[1]$$$ are microwavable, while $$$[3, 3]$$$ and $$$[3, 2, 1, 3, 2]$$$ are not microwavable.

Denote a function $$$f(x, y)$$$ as the length of the longest microwavable subsequence of array $$$A$$$ such that each element within the subsequence is either $$$x$$$ or $$$y$$$. Find the sum of $$$f(x, y)$$$ for all $$$1 \leq x < y \leq M$$$.

Input: The first line consists of two integers $$$N$$$ $$$M$$$ ($$$1 \leq N, M \leq 300\,000$$$).

The second line consists of $$$N$$$ integers $$$A_i$$$ ($$$1 \leq A_i \leq M$$$).

Output: Output a single integer representing the sum of $$$f(x, y)$$$ for all $$$1 \leq x < y \leq M$$$.

Examples:
input:
5 4
3 2 1 3 2
output:
13

input:
3 3
1 1 1
output:
2

Note: Explanation for the sample input/output #1

The value of $$$f(1, 2)$$$ is $$$3$$$, taken from the subsequence $$$[2, 1, 2]$$$ that can be obtained by removing $$$A_1$$$ and $$$A_4$$$. The value of $$$f(1, 3)$$$ is $$$3$$$, taken from the subsequence $$$[3, 1, 3]$$$ that can be obtained by removing $$$A_2$$$ and $$$A_5$$$. The value of $$$f(2, 3)$$$ is $$$4$$$, taken from the subsequence $$$[3, 2, 3, 2]$$$ that can be obtained by removing $$$A_3$$$. The value of $$$f(1, 4)$$$, $$$f(2, 4)$$$, and $$$f(3, 4)$$$ are all $$$1$$$.

Explanation for the sample input/output #2

The value of $$$f(1, 2)$$$ and $$$f(1, 3)$$$ are both $$$1$$$, while the value of $$$f(2, 3)$$$ is $$$0$$$.

Buggy Code:
#include <iostream>
#include <vector>
using namespace std;

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

    int N, M;
    cin >> N >> M;

    vector<int> A(N);
    for (int i = 0; i < N; ++i) {
        cin >> A[i];
    }

    // To store the positions of occurrence of each number
    vector<vector<int>> positions(M + 1);
    for (int i = 0; i < N; ++i) {
        positions[A[i]].push_back(i);
    }

    long long answer = 0;
    // Iterate through all pairs (x, y)
    for (int x = 1; x <= M; ++x) {
        for (int y = x + 1; y <= M; ++y) {
            int i = 0, j = 0;
            int len_x = positions[x].size(), len_y = positions[y].size();
            int count = 0;
            bool current = true;

            while (i < len_x && j < len_y) {
                if (current) {
                    if (positions[x][i] < positions[y][j]) {
                        ++i;
                    } else {
                        ++count;
                        ++j;
                        current = !current;
                    }
                } else {
                    if (positions[y][j] < positions[x][i]) {
                        ++j;
                    } else {
                        ++count;
                        ++i;
                        current = !current;
                    }
                }
            }

            answer += count + 1;
        }
    }

    cout << answer << endl;
    return 0;
}

// rF9uJ4c1h0kTpwYjfgs

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

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

Think step by step.

Response:
To create a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases that might cause the algorithm to fail. Here are some steps to think through this:

1. **Identify the Bug**: The buggy code seems to have an issue with how it handles the alternating subsequence. Specifically, it might not correctly handle cases where the positions of `x` and `y` are not strictly alternating.

2. **Edge Cases**:
   - **Sequences with repeated elements**: The code might not handle sequences where `x` or `y` appear multiple times in a row correctly.
   - **Sequences with no valid microwavable subsequences**: The code might not handle cases where no valid microwavable subsequences can be formed.
   - **Sequences with very few elements**: The code might not handle small input sizes correctly.

3. **Constructing the Test Input**:
   - We can create a sequence where `x` and `y` appear multiple times in a row to see if the code correctly handles the non-alternating subsequence.
   - We can also create a sequence where no valid microwavable subsequence can be formed to see if the code handles such cases correctly.

Let's construct a test input that includes these edge cases:

```plaintext
6 3
1 1 2 2 3 3
```

Explanation:
- **Input**: `6 3` (N = 6, M = 3)
- **Array**: `1 1 2 2 3 3`
- **Expected Output**: The expected output should be `0` because there are no valid microwavable subsequences that can be formed with the given constraints.

This test input will help expose the bug in the buggy code by testing its ability to handle sequences with repeated elements and no valid microwavable subsequences.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Microwavable Subsequence
time_limit_ms: 1000
memory_limit_mb: 1024
Description: You are given an array of $$$N$$$ integers: $$$[A_1, A_2, \dots, A_N]$$$.

A subsequence can be derived from an array by removing zero or more elements without changing the order of the remaining elements. For example, $$$[2, 1, 2]$$$, $$$[3, 3]$$$, $$$[1]$$$, and $$$[3, 2, 1, 3, 2]$$$ are subsequences of array $$$[3, 2, 1, 3, 2]$$$, while $$$[1, 2, 3]$$$ is not a subsequence of array $$$[3, 2, 1, 3, 2]$$$.

A subsequence is microwavable if the subsequence consists of at most two distinct values and each element differs from its adjacent elements. For example, $$$[2, 1, 2]$$$, $$$[3, 2, 3, 2]$$$, and $$$[1]$$$ are microwavable, while $$$[3, 3]$$$ and $$$[3, 2, 1, 3, 2]$$$ are not microwavable.

Denote a function $$$f(x, y)$$$ as the length of the longest microwavable subsequence of array $$$A$$$ such that each element within the subsequence is either $$$x$$$ or $$$y$$$. Find the sum of $$$f(x, y)$$$ for all $$$1 \leq x < y \leq M$$$.

Input: The first line consists of two integers $$$N$$$ $$$M$$$ ($$$1 \leq N, M \leq 300\,000$$$).

The second line consists of $$$N$$$ integers $$$A_i$$$ ($$$1 \leq A_i \leq M$$$).

Output: Output a single integer representing the sum of $$$f(x, y)$$$ for all $$$1 \leq x < y \leq M$$$.

Examples:
input:
5 4
3 2 1 3 2
output:
13

input:
3 3
1 1 1
output:
2

Note: Explanation for the sample input/output #1

The value of $$$f(1, 2)$$$ is $$$3$$$, taken from the subsequence $$$[2, 1, 2]$$$ that can be obtained by removing $$$A_1$$$ and $$$A_4$$$. The value of $$$f(1, 3)$$$ is $$$3$$$, taken from the subsequence $$$[3, 1, 3]$$$ that can be obtained by removing $$$A_2$$$ and $$$A_5$$$. The value of $$$f(2, 3)$$$ is $$$4$$$, taken from the subsequence $$$[3, 2, 3, 2]$$$ that can be obtained by removing $$$A_3$$$. The value of $$$f(1, 4)$$$, $$$f(2, 4)$$$, and $$$f(3, 4)$$$ are all $$$1$$$.

Explanation for the sample input/output #2

The value of $$$f(1, 2)$$$ and $$$f(1, 3)$$$ are both $$$1$$$, while the value of $$$f(2, 3)$$$ is $$$0$$$.

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

signed main() {
    int n, m; int c = 0;
    cin >> n >> m;
    int arr[n + 1];
    int cnt[m + 1];
    fill(cnt, cnt + n + 1, 0);
    for (int i = 1; i <= n; i++){
    	cin >> arr[i];
    	cnt[arr[i]]++;
	}
	for (int i = 1; i <= m; i++){
    	if (cnt[i] != 0) c++;
	}
	map<int, int> ma; int ans = 0;
	for (int i = 1; i <= n; i++){
		int x = i + 1;
		ma.clear();
    	while(arr[i] != arr[x] && x <= n){
    		ma[arr[x]] = 1;
    		x++;
		}
		ans += ma.size();
	}
	cout << ans + c + c * (m - c);
}

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

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

Think step by step.

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

### Explanation:
This test case is designed to expose the bug in the buggy code. Here’s why:

1. **Input Values**:
   - `N = 4` (length of the array)
   - `M = 3` (range of values in the array)
   - Array: `[1, 2, 2, 3]`

2. **Expected Behavior**:
   - The function `f(x, y)` should find the longest microwavable subsequence for each pair of distinct values in the array.
   - For `f(1, 2)`, the longest microwavable subsequence is `[1, 2, 1]` (length 3).
   - For `f(1, 3)`, the longest microwavable subsequence is `[1, 3]` (length 2).
   - For `f(2, 3)`, the longest microwavable subsequence is `[2, 3]` (length 2).

3. **Sum of `f(x, y)`**:
   - `f(1, 2) = 3`
   - `f(1, 3) = 2`
   - `f(2, 3) = 2`
   - Sum: `3 + 2 + 2 = 7`

4. **Buggy Code Behavior**:
   - The buggy code incorrectly counts the size of the map `ma` for each element in the array.
   - It does not correctly handle the microwavable subsequence condition, especially when consecutive elements are the same.
   - The code also adds `c` and `c * (m - c)` to the final answer, which is incorrect for this problem.

5. **Expected Output**:
   - The correct output for this test case should be `7`.

By using this test case, the buggy code will likely produce an incorrect result, exposing the flaw in its logic.