**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Complex Segments (Hard Version)
time_limit_ms: 13000
memory_limit_mb: 256
Description: [Ken Arai - COMPLEX](https://soundcloud.com/diatomichail2/complex)

This is the hard version of the problem. In this version, the constraints on $n$ and the time limit are higher. You can make hacks only if both versions of the problem are solved.

A set of (closed) segments is complex if it can be partitioned into some subsets such that

  * all the subsets have the same size; and    * a pair of segments intersects if and only if the two segments are in the same subset.

You are given $n$ segments $[l_1, r_1], [l_2, r_2], \ldots, [l_n, r_n]$. Find the maximum size of a complex subset of these segments.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 3 \cdot 10^5$) — the number of segments.

The second line of each test case contains $n$ integers $l_1, l_2, \ldots, l_n$ ($1 \le l_i \le 2n$) — the left endpoints of the segments.

The third line of each test case contains $n$ integers $r_1, r_2, \ldots, r_n$ ($l_i \leq r_i \le 2n$) — the right endpoints of the segments.

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.

Output: For each test case, output a single integer: the maximum size of a complex subset of the given segments.

Examples:
input:
3
3
1 2 3
5 4 6
5
1 2 3 6 8
5 4 7 9 10
5
3 1 4 1 5
7 2 6 5 10
output:
3
4
4

Note: In the first test case, all pairs of segments intersect, therefore it is optimal to form a single group containing all of the three segments.

In the second test case, there is no valid partition for all of the five segments. A valid partition with four segments is the following: $\\{\\{ [1, 5], [2, 4] \\}, \\{ [6, 9], [8, 10] \\}\\}$.

In the third test case, it is optimal to make a single group containing all the segments except the second.

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

public class Main{
    static final int limite = 600000;
    static int totalPairs;
    static Par[] segmentos = new Par[limite + 5];
    static int[] subsetSize = new int[limite + 5];
    static int[] parent = new int[limite + 5];
    static int[] subsetCounter = new int[limite + 5];

    static class Par {
        int primero, segundo;
        Par(int primero, int segundo) {
            this.primero = primero;
            this.segundo = segundo;
        }
    }

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

        while (nCasos-- > 0) {
            totalPairs = sc.nextInt();
            for (int i = 1; i <= totalPairs; i++) {
                int first = sc.nextInt();
                segmentos[i] = new Par(first, 0);
            }
            for (int i = 1; i <= totalPairs; i++) {
                segmentos[i].segundo = sc.nextInt();
            }
            Arrays.sort(segmentos, 1, totalPairs + 1, Comparator.comparingInt(o -> o.segundo));
            Arrays.fill(subsetSize, 1, totalPairs + 1, -1);
            solveRange(1, totalPairs);
            int result = 0;
            for (int i = 1; i <= totalPairs; i++) {
                result = Math.max(result, i * subsetSize[i]);
            }
            System.out.println(result);
        }

        sc.close();
    }

    static int findParent(int x) {
        if (parent[x] == x) return x;
        return parent[x] = findParent(parent[x]);
    }

    static int check(int x) {
        if (subsetSize[x] != -1) return subsetSize[x];
        for (int i = 1; i <= 2 * totalPairs; i++) {
            parent[i] = i;
            subsetCounter[i] = 0;
        }
        int activeCount = 0, current = 0, limit = 0;
        subsetSize[x] = 0;

        for (int i = 1; i <= totalPairs; i++) {
            if (segmentos[i].primero > limit) {
                for (int j = current + 1; j < segmentos[i].segundo; j++) {
                    parent[j] = current;
                }
                subsetCounter[segmentos[i].segundo]++;
                activeCount++;
                int to = findParent(segmentos[i].primero - 1);
                if (to != 0) {
                    if (to > limit) activeCount--;
                    subsetCounter[to]--;
                    if (subsetCounter[to] == 0) {
                        parent[to] = findParent(to - 1);
                    }
                }
                current = segmentos[i].segundo;
                if (activeCount == x) {
                    subsetSize[x]++;
                    activeCount = 0;
                    limit = current;
                }
            }
        }
        return subsetSize[x];
    }

    static void solveRange(int left, int right) {
        if (left == right) {
            subsetSize[left] = check(left);
            return;
        }
        if (check(left) == check(right)) {
            for (int i = left; i <= right; i++) {
                subsetSize[i] = check(left);
            }
            return;
        }
        int mid = (left + right) / 2;
        if (check(mid) != check(left)) solveRange(left, mid);
        else {
            for (int i = left; i <= mid; i++) {
                subsetSize[i] = check(left);
            }
        }
        if (check(mid + 1) != check(right)) solveRange(mid + 1, right);
        else {
            for (int i = mid + 1; i <= right; i++) {
                subsetSize[i] = check(right);
            }
        }
    }
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Complex Segments (Hard Version)
time_limit_ms: 13000
memory_limit_mb: 256
Description: [Ken Arai - COMPLEX](https://soundcloud.com/diatomichail2/complex)

This is the hard version of the problem. In this version, the constraints on $n$ and the time limit are higher. You can make hacks only if both versions of the problem are solved.

A set of (closed) segments is complex if it can be partitioned into some subsets such that

  * all the subsets have the same size; and    * a pair of segments intersects if and only if the two segments are in the same subset.

You are given $n$ segments $[l_1, r_1], [l_2, r_2], \ldots, [l_n, r_n]$. Find the maximum size of a complex subset of these segments.

Input: Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^3$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($1 \le n \le 3 \cdot 10^5$) — the number of segments.

The second line of each test case contains $n$ integers $l_1, l_2, \ldots, l_n$ ($1 \le l_i \le 2n$) — the left endpoints of the segments.

The third line of each test case contains $n$ integers $r_1, r_2, \ldots, r_n$ ($l_i \leq r_i \le 2n$) — the right endpoints of the segments.

It is guaranteed that the sum of $n$ over all test cases does not exceed $3 \cdot 10^5$.

Output: For each test case, output a single integer: the maximum size of a complex subset of the given segments.

Examples:
input:
3
3
1 2 3
5 4 6
5
1 2 3 6 8
5 4 7 9 10
5
3 1 4 1 5
7 2 6 5 10
output:
3
4
4

Note: In the first test case, all pairs of segments intersect, therefore it is optimal to form a single group containing all of the three segments.

In the second test case, there is no valid partition for all of the five segments. A valid partition with four segments is the following: $\\{\\{ [1, 5], [2, 4] \\}, \\{ [6, 9], [8, 10] \\}\\}$.

In the third test case, it is optimal to make a single group containing all the segments except the second.

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

public class Main {
    static final int limite = 600000;
    static int nCasos, totalPairs;
    static Par[] segmentos = new Par[limite + 5];
    static int[] subsetSize = new int[limite + 5];
    static int[] parent = new int[limite + 5];
    static int[] subsetCounter = new int[limite + 5];

    static class Par {
        int primero, segundo;
        Par(int primero, int segundo) {
            this.primero = primero;
            this.segundo = segundo;
        }
    }

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        nCasos = sc.nextInt();

        for(int i=0;i<nCasos;i++) {
            mainTask(sc);
        }
        sc.close();
    }

    static int findParent(int x) {
        if (parent[x] == x) return x;
        return parent[x] = findParent(parent[x]);
    }

    static int check(int x) {
        if (subsetSize[x] != -1) return subsetSize[x];
        for (int i = 1; i <= 2 * totalPairs; i++) {
            parent[i] = i;
            subsetCounter[i] = 0;
        }
        int activeCount = 0, current = 0, limit = 0;
        subsetSize[x] = 0;

        for (int i = 1; i <= totalPairs; i++) {
            if (segmentos[i].primero > limit) {
                for (int j = current + 1; j < segmentos[i].segundo; j++) {
                    parent[j] = current;
                }
                subsetCounter[segmentos[i].segundo]++;
                activeCount++;
                int to = findParent(segmentos[i].primero - 1);
                if (to != 0) {
                    if (to > limit) activeCount--;
                    subsetCounter[to]--;
                    if (subsetCounter[to] == 0) {
                        parent[to] = findParent(to - 1);
                    }
                }
                current = segmentos[i].segundo;
                if (activeCount == x) {
                    subsetSize[x]++;
                    activeCount = 0;
                    limit = current;
                }
            }
        }
        return subsetSize[x];
    }

    static void solveRange(int left, int right) {
        if (left == right) {
            subsetSize[left] = check(left);
            return;
        }
        if (check(left) == check(right)) {
            for (int i = left; i <= right; i++) {
                subsetSize[i] = check(left);
            }
            return;
        }
        int mid = (left + right) / 2;
        if (check(mid) != check(left)) solveRange(left, mid);
        else {
            for (int i = left; i <= mid; i++) {
                subsetSize[i] = check(left);
            }
        }
        if (check(mid + 1) != check(right)) solveRange(mid + 1, right);
        else {
            for (int i = mid + 1; i <= right; i++) {
                subsetSize[i] = check(right);
            }
        }
    }

    static void mainTask(Scanner sc) {
        totalPairs = sc.nextInt();
        for (int i = 1; i <= totalPairs; i++) {
            int first = sc.nextInt();
            segmentos[i] = new Par(first, 0);
        }
        for (int i = 1; i <= totalPairs; i++) {
            segmentos[i].segundo = sc.nextInt();
        }

        Arrays.sort(segmentos, 1, totalPairs + 1, Comparator.comparingInt(o -> o.segundo));
        Arrays.fill(subsetSize, 1, totalPairs + 1, -1);
        solveRange(1, totalPairs);

        int result = 0;
        for (int i = 1; i <= totalPairs; i++) {
            result = Math.max(result, i * subsetSize[i]);
        }
        System.out.println(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
4
1 2 3 4
4 3 2 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
/*
ID: akram
TASK: friday
PROG: friday
LANG: C++
*/
/* LANG can be C++11 or C++14 for those more recent releases */
#include <bits/stdc++.h>
#define FAST ios::sync_with_stdio(0), cin.tie(0) //, cout.tie(0)
#define ll long long
#define ld long double
#define endl "\n"
#define pb push_back
using namespace std;
const int MOD = 1e9 + 7;
const int N = 5e5 + 5;
const ll INF = 1e18 + 1;
const ll MIN = -1e18;
const int prime = 2;
#ifndef ONLINE_JUDGE
#include "template.cpp"
#else
#define dbg(...)
#define debugArr(...)
#define show(...)
#endif
bool compare(const std::pair<ll, int> &left, const std::pair<ll, int> &right)
{
    return left.first < right.first;
}
bool compare2(const std::pair<ll, int> &left, const std::pair<ll, int> &right)
{
    return left.first > right.first;
}
void solve()
{
    int a[501][501];
    int n;
    cin >> n;
    ll s = 0;
    vector<pair<int, int>> p;
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            cin >> a[i][j];
            if (a[i][j] < 0)
            {
                p.pb({i, j});
            }
        }
    }

    for (int i = 0; i < p.size(); i++)
    {
        for (int j = i + 1; j < p.size(); j++)
        {
            int left = p[i].first - p[j].first;
            int right = p[i].second - p[j].second;

            if (left == right && p[i].second < p[j].second)
            {
                int x = min(abs(a[p[i].first][p[i].second]), abs(a[p[j].first][p[j].second]));
                x = abs(x);
                s += x;
                a[p[i].first][p[i].second] += x;
                a[p[j].first][p[j].second] += x;
            }
            if (!a[p[i].first][p[i].second])
                break;
        }

        int c = a[p[i].first][p[i].second];
        dbg(p[i], c,s);
        s += abs(a[p[i].first][p[i].second]);
        a[p[i].first][p[i].second] = 0 ;

    }

    cout << s << endl;
}
int main()
{
#ifndef ONLINE_JUDGE
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w", stdout);
    freopen("err.txt", "w", stderr);
#endif
    FAST;
    int t;
    cin >> t;
    while (t--)
    {
        solve();
    }
    return 0;
}
/* stuff you should look for
 * int overflow, array bounds
 * special cases (n=1?)
 * do smth instead of nothing and stay organized
 * WRITE STUFF DOWN
 * DON'T GET STUCK ON ONE APPROACH
 * think backwards
 */
/* stuff you should look for
 * int overflow, array bounds
 * special cases (n=1?)
 * do smth instead of nothing and stay organized
 * WRITE STUFF DOWN
 * DON'T GET STUCK ON ONE APPROACH
 * think backwards
 */

**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
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

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

int main()
{
   ios_base::sync_with_stdio(false);
   cin.tie(NULL);
   int ara [500][500];
   int t;
   cin >> t;
   while (t--) {
      int n;
      int sum = 0, sum1 = 0, sum2 = 0;
      int min =INT_MAX;
      cin >> n;
      for (int i = 0; i < n; i++)
      {
         for (int j = 0; j < n; j++)
         {
            cin >> ara [i][j];
            if (i == j)
            {
               if (ara[i][j] < 0 && ara[i][j] < min)
               {
                  min = ara[i][j];
                  sum = min;
               }
            }
         }
      }
      min =INT_MAX;
      for (int i = 0, j= 1; j < n-1; j++)
      {
         for (int k = i, l = j; l < n; k++, l++)
         {
            if ( ara [k][l] < 0 && ara[k][l] < min)
            {
               min = ara[k][l];
               sum1 = min;
            }
         }
         // cout<< sum1 << "\n";
         sum += sum1;
         sum1 = 0;
         min = INT_MAX;

      }
      // cout << "\n";
      min =INT_MAX;
      for (int j = 0, i= 1; i < n-1; i++)
      {
         for (int k = i, l = j; k < n; k++, l++)
         {
            if ( ara [k][l] < 0 && ara[k][l] < min)
            {
               min = ara[k][l];
               sum2 = min;
            }
         }
         // cout<< sum2 << "\n";
         sum += sum2;
         sum2 = 0;
         min = INT_MAX;
      }
      if (ara[0][n-1] < 0) sum += (ara[0][n-1]);
      if (ara[n-1][0] < 0) sum += (ara[n-1][0]);

      cout << abs(sum) << endl;
   }
}

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

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

Think step by step.

Response:
```plaintext
1
3
-1 2 3
-2 -1 -1
-1 0 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

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

int main() {
	int t;
	cin>>t;
	while(t--){
		int n;
		cin>>n;
		int a[n][n];
		for(int i = 0;i<n;i++){
			for(int j = 0;j<n;j++){
				cin>>a[i][j];
			}
		}

		int ans = 0;
		int i = n;
		int j = 0;
		while(i>=0 && j<n){
			int l = i;
			int k = j;
			int temp = a[l][k];
			while(k<n && i<n){
				temp = min(temp,a[l][k]);
				k++;
				l++;
			}
			if(temp<0){
				temp = temp*-1;
				ans+= temp;
			}

			i--;
		}

		i=0;
		j = 1;
		while(i>=0 && i<n && j>=0 && j<n){
			int l = i;
			int k = j;
			int temp = a[l][k];
			while(k<n && l<n){
				temp = min(temp,a[l][k]);
				k++;
				l++;
			}
			if(temp<0){
				temp = temp*-1;
				ans+= temp;
			}
			j++;
		}
		cout<<ans<<endl;
	}
}

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

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

Think step by step.

Response:
```plaintext
1
3
-1 2 3
-2 -1 -1
0 0 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
t = int(input())
for _ in range(t):
    n = int(input())
    matrix = []
    for _ in range(n): matrix.append(list(map(int, input().split())))

    dp = [0] * (n*1000000)

    for i in range(n):
        for j in range(n):
            dp[i-j] = min(dp[(i-j)], matrix[i][j])
            #print(i,j, i-j)
    print(-sum(dp))
    #print()

**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 given buggy code, we need to focus on edge cases that can highlight the issues in the algorithm. The buggy code has a few key problems:

1. **Incorrect Use of `dp` Array**: The `dp` array is used incorrectly. It should be a 2D array to store the minimum heights of the diagonals, but it is defined as a 1D array.
2. **Incorrect Indexing**: The indexing in the `dp` array is incorrect. The code uses `dp[i-j]` which can lead to negative indices and out-of-bounds errors.
3. **Incorrect Logic for Minimum Heights**: The logic to update the `dp` array is incorrect. It should be updating the minimum heights of the diagonals, but it is not doing so correctly.

To expose these issues, we can create a test case where the valley has a specific pattern that causes the incorrect indexing and logic to fail. A good test case would be a small matrix with negative values on the main diagonal and positive values elsewhere.

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

```plaintext
1
3
-1 2 3
4 -1 5
6 7 -1
```

### Explanation:
- **Matrix**:
  - The main diagonal has negative values: `[-1, -1, -1]`.
  - All other values are positive.
- **Expected Output**:
  - Sakurako needs to use her magic 3 times to increase the height of each mountain on the main diagonal by 1, making all values non-negative.

This test case should highlight the issues with the `dp` array and the incorrect logic used 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 and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
t = int(input())
for _ in range(t):
    n = int(input())

    matrix = []
    for _ in range(n):
        row = list(map(int, input().split()))
        matrix.append(row)

    answer = 0
    for i in range(0, n):
        for j in range(0, n):
            if matrix[i][j] < 0:
                add = abs(matrix[i][j])
                answer += add
                if i >= n-j:
                    matrix[i][j] += add
                else:
                    for k in range(i, n-j):
                        matrix[k][j+k-i] += add

    print(answer)

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
def trav(matrix):
    ans = 0
    n = len(matrix)

    for start_row in range(n-2, -1, -1):
        row, col = start_row, 0
        diagonal = []
        while row < n and col < n:
            diagonal.append(matrix[row][col])
            row += 1
            col += 1
        x = min(diagonal)
        if x<0:
            ans += abs(x)

    for start_col in range(1, n-1):
        row, col = 0, start_col
        diagonal = []
        while row < n and col < n:
            diagonal.append(matrix[row][col])
            row += 1
            col += 1
        x = min(diagonal)
        if x<0:
            ans += abs(x)

    return ans

for _ in range(int(input())):
    m = int(input())
    mat = []
    for _ in range(m):
        mat.append(list(map(int,input().split())))

    print(trav(mat))

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

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

Think step by step.

Response:
```plaintext
1
3
-1 0 0
0 -1 0
0 0 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

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

/**
 * B_Sakurako_and_Water
 */
public class B_Sakurako_and_Water {

    // end of fast i/o code
    public static void main(String[] args) {
        FastReader sc = new FastReader();
        int t = sc.nextInt();
        while (t-- > 0) {
            int n = sc.nextInt();
            int res[][] = new int[n][n];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    res[i][j] = sc.nextInt();
                }
            }
            int ans[] = new int[2 * (n - 1) + 1];
            int m = n - 1;
            for (int i = 0; i < n; i++) {
                ans[(ans.length - 1) / 2] = Math.min(ans[(ans.length - 1) / 2], res[i][i]);
            }
            for (int x = 1; x < ans.length / 2; x++) {
                for (int i = 0; i < n - 1 && i + x < n; i++) {
                    ans[ans.length / 2 - x] = Math.min(ans[ans.length / 2 - x], res[i + x][i]);
                    ans[ans.length / 2 + x] = Math.min(ans[ans.length / 2 + x], res[i][i + x]);
                }
            }
            long sum = 0;
            for (int i = 0; i < ans.length; i++) {
                sum += Math.abs(ans[i]);
            }
            System.out.println(sum);

        }

    }

    // For fast input output
    static class FastReader {
        BufferedReader br;
        StringTokenizer st;

        public FastReader() {
            try {
                br = new BufferedReader(
                        new FileReader("input.txt"));
                PrintStream out = new PrintStream(new FileOutputStream("output.txt"));
                System.setOut(out);
            } catch (Exception e) {
                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;
        }
    }

}

**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, we need to focus on edge cases or scenarios that are likely to cause the algorithm to fail. Here are the steps to think through this:

1. **Understand the Bug**: The code seems to be incorrectly calculating the minimum number of times Sakurako needs to use her magic. It uses an array `ans` to store the minimum values of the diagonals, but it does not correctly account for the changes made to the matrix.

2. **Identify Edge Cases**:
   - **Single Element Matrix**: A matrix with a single element.
   - **Matrix with All Negative Values**: A matrix where all elements are negative.
   - **Matrix with a Single Lake**: A matrix with a single lake in a position that requires the maximum number of operations to fix.
   - **Matrix with Multiple Lakes**: A matrix with multiple lakes in strategic positions that require careful operations to fix.

3. **Create a Test Input**:
   - **Single Element Matrix**: This is trivial and should work correctly.
   - **Matrix with All Negative Values**: This will test the ability to handle all negative values.
   - **Matrix with a Single Lake**: This will test the ability to handle a single lake in a specific position.
   - **Matrix with Multiple Lakes**: This will test the ability to handle multiple lakes and ensure the correct number of operations is calculated.

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

```plaintext
4
1
-1
2
-2 -1
-1 -2
3
-1 -2 -3
-2 -3 -4
-3 -4 -5
4
-1 -2 -3 -4
-2 -3 -4 -5
-3 -4 -5 -6
-4 -5 -6 -7
```

This test input includes:
- A single element matrix with a negative value.
- A 2x2 matrix with all negative values.
- A 3x3 matrix with all negative values.
- A 4x4 matrix with all negative values.

These scenarios are designed to challenge the algorithm's ability to handle negative values and multiple lakes, which should expose the bug 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 and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
import java.util.Scanner;

public class Solution {

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

        int setCount = scanner.nextInt();

        for (int i = 0; i < setCount; ++i) {

            int x = scanner.nextInt();

            int[][] matrix = new int[x][x];
            for (int r = 0; r < x; ++r) {
                for (int c = 0; c < x; ++c) {
                    matrix[r][c] = scanner.nextInt();
                }
            }
            System.out.println(printMatrix(matrix));
        }
        scanner.close();
    }

    private static int printMatrix(int[][] matrix) {
        int result = 0;
        for (int i = 0; i < matrix.length; ++i) {
            for (int j = 0; j < matrix[0].length; ++j) {
                if (matrix[i][j] < 0) {
                    result += bfs(matrix, i, j, 0);
                }
            }
        }
        return result;
    }

    private static int bfs(int[][] matrix, int i, int j, int result) {
            if (i < 0 || j < 0 || i >= matrix.length || j >= matrix[0].length || matrix[i][j] >= 0) return 0;
            int currentValue = 0;
            if (matrix[i][j] < 0) {
                currentValue = Math.abs(matrix[i][j]);
                matrix[i][j] = 0;
            }
            bfs(matrix, i + 1, j + 1, currentValue);
            bfs(matrix, i - 1, j - 1, currentValue);
            return Math.max(currentValue != 0 ? currentValue : 0, 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
3
-1 2 3
-2 1 -1
0 0 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
/*
    author: Rishi Mohan Jha
    created_at: 24-10-2024  08:26 PM
*/

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

#ifdef LOCAL
#include "debug.h"
#else
#define debug(...)
#endif

#define ll     long long
#define ff     first
#define ss     second
#define pb     push_back
#define sz(x)  ((int)(x).size())
#define all(a) (a).begin(),(a).end()
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
ll getRandomNumber(ll l, ll r) {return uniform_int_distribution<ll>(l, r)(rng);}

void Solve() {
	int n;
	cin >> n;

	int maxi = n;

	vector<vector<int>>v(maxi, vector<int>(maxi, 0));

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

	// debug(v);

	ll ans = 0;
	for (int j = maxi - 2; j >= 0; j--) {
		int res = 0, k = j;
		for (int i = 0; i < maxi - j; i++) {
			// debug(i,k);
			if (v[i][k] < 0)
				res = max(res, abs(v[i][k]));
			k += 1;
		}
		// cout << endl;
		ans += res;

	}

	for (int j = maxi - 2; j > 0; j--) { // start
		int res = 0, k = j;
		for (int i = 0; i < maxi - j; i++) {
			// debug(k,i);
			if (v[k][i] < 0)
				res = max(res, abs(v[k][i]));
			k += 1;
		}
		// cout << endl;
		ans += res;
	}
	cout << ans << endl;
}

int main() {
	ios_base::sync_with_stdio(false);
	cin.tie(NULL);  int T = 1;
	cin >> T;
	while (T--) Solve();
}

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

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

Think step by step.

Response:
To expose the bug in the given code, we need to focus on edge cases and scenarios that might cause the code to fail. The code seems to have a logical issue in how it handles the main diagonals of the submatrices. Specifically, it only considers the main diagonal from the top-left to the bottom-right of the matrix and its shifted versions, but it doesn't handle all possible submatrices correctly.

Let's consider a small-scale test input that can expose this issue:

1. **A matrix with a single negative value on a diagonal that is not the main diagonal.**
2. **A matrix with multiple negative values on different diagonals.**

Here is a test input that should expose the bug:

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

### Explanation:
- **Matrix size:** 4x4
- **Values:** The matrix has two negative values, `-1` at position (1, 1) and `-2` at position (2, 2) when considering 0-based indexing.
- **Expected behavior:** The code should correctly identify and handle these negative values by increasing the heights on the appropriate diagonals.

This test input is small enough to be manageable and should highlight the issue in the code's logic for handling diagonals. The code should output the minimum number of times Sakurako needs to use her magic to make all the values non-negative. The expected output for this test input is:

```plaintext
3
```

This is because:
- The value `-1` at (1, 1) needs to be increased by 1.
- The value `-2` at (2, 2) needs to be increased by 2.

Thus, the total number of operations is 1 + 2 = 3.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

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

int main()
{
    int tc;
    cin >> tc;

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

        int arr[500][500];
        for (int i = 0 ; i < n ; i++)
        {
            for (int j = 0 ; j < n ;j++) cin>>arr[i][j];
        }

        int ans = 0;

        for (int i = 0 ; i < n; i++)
        {
            int min = arr[0][i];
            for (int j = 1 ; j < n ; j++)
            {
                if (i + j >= n) break;
                else {if (arr[j][i+j] < min) min = arr[j][i+j];}
            }
            if (min < 0) ans = ans - min;
        }

        for (int i = 1 ; i < n; i++)
        {
            int min = arr[i][0];
            for (int j = 2 ; j < n ; j++)
            {
                if (i + j >= n) break;
                else {if (arr[i + j][j] < min) min = arr[i+j][j];}
            }
            if (min < 0) ans = ans - min;
        }

        cout<<ans<<endl;

    }

}

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided code, we need to focus on edge cases and scenarios where the code might fail. The code provided has a few potential issues:

1. **Incorrect Handling of Diagonals**: The code only considers the main diagonal starting from the first row and the first column. It does not correctly handle all possible diagonals in the matrix.
2. **Incorrect Minimum Calculation**: The code calculates the minimum value for each diagonal but does not account for all possible submatrices that can be used to increase the height of the mountains.

Given these issues, we can create a test case that has multiple negative values in different diagonals, ensuring that the code's logic of only considering certain diagonals will fail to correctly determine the minimum number of operations needed.

### Test Input
```plaintext
1
3
-1 2 3
1 -2 1
0 0 -1
```

### Explanation
- **Matrix Size**: 3x3
- **Initial Matrix**:
  ```
  -1 2 3
  1 -2 1
  0 0 -1
  ```

### Expected Output
- **Minimum Operations**: 4

### Reasoning
- The code will only consider the main diagonals starting from the first row and the first column, which are:
  - Diagonal starting from (0,0): [-1, -2, -1]
  - Diagonal starting from (0,1): [2, 1]
  - Diagonal starting from (0,2): [3]
  - Diagonal starting from (1,0): [1, 0]
  - Diagonal starting from (2,0): [0]

- The code will calculate the minimum value for these diagonals and add the necessary operations to make all values non-negative. However, it will miss the diagonal starting from (1,1) which is [-2], and the diagonal starting from (2,1) which is [0].

- The correct solution should consider all possible diagonals and determine the minimum number of operations needed to make all values non-negative.

This test case will expose the bug in the code by requiring more operations than the code will calculate, thus demonstrating the incorrect handling of diagonals.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
loop = int(input())

for k in range (loop):
    n = int(input())
    mounts = []
    for m in range(n):
        line = list(map(int, list(input().split(" "))))
        mounts.extend(line)
    mounts.reverse()
    mountain = 0
    magic = 0
    q = 0
    mn = 0
    while mountain != (len(mounts)):
        if mounts[mountain] < 0:
            q = -(mounts[mountain])
            magic = magic + q
            try:
                for z in range(n):
                    mounts[mountain + ((n + 1)*z)]+= q
            except:
                mn = 0
        mountain+= 1
    print(magic)

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to focus on edge cases and scenarios where the code might fail. The buggy code has several issues:

1. **Reversing the List**: The code reverses the list of all elements, which can lead to incorrect indexing when trying to update the heights.
2. **Incorrect Indexing**: The code uses a flat list and attempts to update the heights of the main diagonal of submatrices using a formula that might go out of bounds.
3. **Handling of Negative Heights**: The code might not correctly handle cases where multiple negative heights are present in the same submatrix.

Given these issues, a good test input should include:

- A small matrix with multiple negative heights.
- A matrix where the negative heights are on or near the main diagonal.
- A matrix where the negative heights are in a submatrix that should be updated multiple times.

Here is a small-scale test input that can expose these issues:

```plaintext
3
3
-1 2 3
0 -2 1
1 0 -1
4
-1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 -1
2
-1 1
1 -1
```

### Explanation:
1. **Test Case 1 (3x3 Matrix)**:
   - Contains multiple negative heights: -1, -2, -1.
   - The negative heights are on the main diagonal, which should be updated by the magic.

2. **Test Case 2 (4x4 Matrix)**:
   - Contains multiple negative heights: -1, -1, -1, -1.
   - All negative heights are on the main diagonal, which should be updated by the magic.

3. **Test Case 3 (2x2 Matrix)**:
   - Contains negative heights on the main diagonal: -1, -1.
   - This small matrix can help verify if the code correctly handles small inputs.

These test cases should help expose the issues in the buggy code, particularly the incorrect indexing and the handling of multiple negative heights in the same submatrix.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
def f(t, m):
    c = 0
    temp = [[i, i] for i in range(t)]
    g = min([m[x][y] for x, y in temp])
    if g < 1:
        c += g
    for i in range(1, t-1):
        for j in range(t):
            temp[j][0] += 1
            temp[j][1] += 0
        g = min([m[x][y] for x, y in temp[:t-i]])
        if g < 1:
            c += g
        g = min([m[y][x] for x, y in temp[:t-i]])
        if g < 1:
            c += g
    return -c

t = int(input())
for j in range(t):
    n = int(input())
    m = [[int(i) for i in input().split()] for j in range(n)]
    print(f(n, m))

**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 code and identify potential issues. The code is intended to find the minimum number of times Sakurako must use her magic to ensure there are no lakes (negative values) in the matrix.

### Analysis of the Buggy Code:
1. **Initial Main Diagonal Check:**
   - The code initializes `temp` as a list of coordinates for the main diagonal of the matrix.
   - It calculates the minimum value `g` on the main diagonal and increments the count `c` by `g` if `g < 1`.
   - This is incorrect because `g` is negative, and it should be incremented by the absolute value of `g`.

2. **Subsequent Diagonal Checks:**
   - The code iterates over the sub-diagonals by incrementing the rows and columns.
   - It calculates the minimum value `g` for each sub-diagonal and increments `c` by `g` if `g < 1`.
   - Similar to the initial check, this is incorrect because it should increment by the absolute value of `g`.

### Challenging Test Input:
To expose the bug, we need a test case where the initial main diagonal and some sub-diagonals have negative values that require multiple increments to become non-negative. Here is a small-scale test input that should highlight the issues:

```plaintext
1
3
-2 1 -1
1 -2 1
-1 1 -2
```

### Explanation:
- **Matrix:**
  ```
  -2 1 -1
  1 -2 1
  -1 1 -2
  ```
- **Main Diagonal:**
  - Values: `-2, -2, -2`
  - Minimum value: `-2`
  - Correct number of increments needed: `2` (since `-2 + 2 = 0`)

- **Sub-diagonals:**
  - Sub-diagonal from (0,1) to (2,3) (not applicable as it goes out of bounds).
  - Sub-diagonal from (1,0) to (2,1):
    - Values: `1, 1`
    - No negative values, no increments needed.
  - Sub-diagonal from (0,2) to (1,3) (not applicable as it goes out of bounds).
  - Sub-diagonal from (1,2) to (2,3) (not applicable as it goes out of bounds).
  - Sub-diagonal from (2,0) to (3,1) (not applicable as it goes out of bounds).

- **Correct Output:**
  - The minimum number of times Sakurako must use her magic: `6` (since each main diagonal element needs to be incremented by 2).

### Expected Output:
```plaintext
6
```

This test input should clearly expose the bug in the provided code, as the code will incorrectly count the number of increments needed.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
import java.util.*;
public class CF2 {
    static Scanner in = new Scanner(System.in);
    public static void main(String[] args) {

        int t = in.nextInt();
        while(t-- >0)
        {
            int n = in.nextInt();

            int matrix[][] = new int[n][n];
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    matrix[i][j] = in.nextInt();
                }
            }

            int ans = 0;
            int row = 0, col = 0;
            for (int i = 0; i < n-1; i++) {
                int k = 0;
                int mini = Integer.MAX_VALUE;
                while(k < n){
                    if(i+k < n){
                        mini = Math.min(matrix[row + k][i + k], mini);

                    }
                    k++;
                }
                if(mini < 0){
                    ans += Math.abs(mini);
                }
            }
            for (int i = 1; i < n-1; i++) {
                int k = 0;
                int mini = Integer.MAX_VALUE;
                while(k < n){
                    if(i+k < n){
                        mini = Math.min(matrix[i + k][col + k], mini);
                    }
                    k++;
                }
                if(mini < 0){
                    ans += Math.abs(mini);
                }
            }
            System.out.println(ans);

        }
    }
//    static final long MOD = 1000000007;
//    //  modular exponentiation for handling the large test cases and used in competitive programming
//    static long bexp(long x, long n) {
//        x %= MOD; // Reduce x modulo MOD
//        long res = 1;
//
//        while (n > 0) {
//            if (n % 2 == 1) { // If n is odd
//                res = (res * x) % MOD; // Multiply the result with x
//            }
//            x = (x * x) % MOD;
//            n /= 2;
//        }
//        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:
To expose the bug in the provided code, we need to focus on the logic that the code uses to find the minimum number of times Sakurako must use her magic. The code iterates over the diagonals of the matrix and finds the minimum value on each diagonal, then adds the absolute value of this minimum to the answer if it is negative. However, this approach is incorrect because it does not consider all possible submatrices and their diagonals.

### Key Points to Consider:
1. **Diagonal Calculation**: The code only checks the main diagonal and the diagonals starting from the first row and the first column. It misses diagonals that start from other positions.
2. **Submatrix Calculation**: The code does not correctly handle the submatrices and their diagonals.

### Test Input to Expose the Bug:
We need a test case where the minimum value on the diagonals is not the same as the minimum value in the entire matrix. This will show that the code is not considering all possible submatrices and their diagonals.

### Example Test Input:
```plaintext
1
4
1 -1 2 3
-1 1 -1 4
2 -1 1 5
3 4 5 6
```

### Explanation:
- The matrix is:
  ```
  1 -1 2 3
 -1 1 -1 4
  2 -1 1 5
  3 4 5 6
  ```
- The main diagonal is `1, 1, 1, 6`, and the minimum value is `1`, which is not negative.
- The diagonals starting from the first row and the first column are:
  - `1, -1, 1, 5` (minimum is `-1`)
  - `2, -1, 5` (minimum is `-1`)
  - `3, 4` (minimum is `3`)
- The diagonals starting from the first row and the second column are:
  - `-1, 1, 5` (minimum is `-1`)
  - `2, 4` (minimum is `2`)
- The diagonals starting from the second row and the first column are:
  - `-1, 1, 6` (minimum is `-1`)
  - `4, 5` (minimum is `4`)
- The diagonals starting from the second row and the second column are:
  - `1, 5` (minimum is `1`)
  - `4` (minimum is `4`)

The correct answer should be `3` because the minimum value `-1` appears three times, and each of these `-1` values needs to be increased by 1 to become non-negative.

However, the provided code will only check the main diagonals and a few other diagonals, missing the other submatrices and their diagonals. This will result in an incorrect answer.

### Expected Output:
```plaintext
3
```

This test input should expose the bug 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 and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

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

public class temi2 {

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

        while (tc-- > 0) {
            int n = scc.nextInt();
            int temp1 = 0;
            int arrM[][] = new int[n][n];
            for (int i = 0; i < n; i++) {
                for (int c = 0; c < n; c++) {
                    arrM[i][c] = scc.nextInt();
                }
            }
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    if (arrM[i][j] < 0) {
                        int temp2 = Math.abs(arrM[i][j]);
                        temp1 = temp1+ temp2;
                        for (int k = 0; k < n; k++) {
                            if (k + k < n && j + k < n) {
                                arrM[k + k][j + k] += temp2;
                            }
                        }
                    }
                }
            }
            System.out.println(temp1);
        }
    }
}

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

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

Think step by step.

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

**Explanation:**

The provided test input is designed to expose the bug in the buggy code. The matrix is small (3x3) to ensure it fits within the output constraints, but it is crafted to create a scenario where the buggy logic fails to correctly fill in the lakes.

- The matrix consists entirely of negative values, which means every cell is a lake that needs to be filled.
- The values are arranged in a way that the main diagonal and its immediate neighbors are also negative, which means the magic needs to be applied multiple times to these cells to bring them to a non-negative height.

The buggy code incorrectly updates the cells on the main diagonal of the submatrices, and this test case will highlight the issue because the main diagonal and its immediate neighbors will not be updated correctly, leading to an incorrect result.

The correct output for this test case should be 15, as each cell needs to be increased by its absolute value to become non-negative, and the total number of operations needed is the sum of the absolute values of all the cells. However, the buggy code will likely produce a different, incorrect result.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
using ll=long long;
void solve() {
   int n;cin>>n;
   int a[505][505]={0};
   for(int i=1;i<=n;i++)
   {
    for(int j=1;j<=n;j++)cin>>a[i][j];
   }

   //int j=n;
   int sum=0;
   for(int i=1;i<=n;i++)
   {
     if(i==1){
      int k=i,ans=0;
      for(int j=1;j<=n;j++){
        ans=min(a[k++][j],ans);
      }
      sum+=abs(ans);
     }
     else {
      int k=i,ans=0;
      for(int j=1;j<=n;j++){
         ans=min(ans,a[k++][j]);
      }
      sum+=abs(ans);
     }
   }
   for(int j=2;j<=n;j++){
    int k=j,ans=0;
    for(int i=1;i<=n;i++){
      ans=min(a[i][k++],ans);
    }
    sum+=abs(ans);
   }
   cout<<sum<<'\n';
}

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

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

     return 0;
}

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

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

Think step by step.

Response:
```plaintext
4
1
-1
2
-1 -2
-3 0
3
-1 -2 -3
-4 -5 -6
-7 -8 -9
4
-1 -2 -3 -4
-5 -6 -7 -8
-9 -10 -11 -12
-13 -14 -15 -16
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
#include<bits/stdc++.h>
using namespace std;
int a[1001]={0};
int main()
{
	int t,n;
	long long ans;
	long long temp2;
	scanf("%d",&t);
	for(int i=1;i<=t;i++)
	{
		memset(a,0,sizeof(a));
		ans=0;
		int temp;
		scanf("%d",&n);
		for(int i=1;i<=n;i++)
		{
			for(int j=1;j<=n;j++)
			{
				scanf("%d",&temp);
				if(temp+a[i-j+201]<0)
				{
					temp2=abs(temp+a[i-j+201]);
					ans+=temp2;
					a[i-j+201]+=temp2;
				}
			}
		}
		printf("%lld\n",ans);
	}
	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 given buggy code, we need to consider edge cases and scenarios that can maximize the failure probability. The buggy code seems to have issues with how it handles the main diagonal of the selected submatrices. Here are the steps to create such a test input:

1. **Test Case with a Single Lake**: This will check if the code correctly handles a single lake.
2. **Test Case with Multiple Lakes**: This will check if the code can handle multiple lakes efficiently.
3. **Test Case with a Lake on the Main Diagonal**: This will check if the code correctly handles lakes that are on the main diagonal.
4. **Test Case with a Lake in a Submatrix**: This will check if the code correctly handles lakes that are not on the main diagonal but can be covered by a submatrix.

Let's create a test input that includes these scenarios:

```plaintext
4
1
-1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
4
1 1 -1 2
-1 1 1 1
2 1 1 -1
1 1 1 1
```

### Explanation:
1. **First Test Case**:
   - A single 1x1 matrix with a lake at position (1,1).
   - Expected output: 1 (since the lake at (1,1) needs to be increased by 1).

2. **Second Test Case**:
   - A 2x2 matrix with a lake at position (1,1).
   - Expected output: 1 (since the lake at (1,1) needs to be increased by 1).

3. **Third Test Case**:
   - A 3x3 matrix with multiple lakes at positions (2,1), (2,3), and (3,3).
   - Expected output: 4 (since the lakes need to be increased by 1, 1, and 1 respectively).

4. **Fourth Test Case**:
   - A 4x4 matrix with lakes at positions (1,3) and (3,4).
   - Expected output: 3 (since the lakes need to be increased by 1, 1, and 1 respectively).

This test input should help expose any issues in the buggy code, particularly in how it handles the main diagonal and submatrices.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
import sys
input = sys.stdin.readline
t = int(input())
for _ in range(t):
    n = int(input())
    matrix = [list(map(int,input().split())) for _ in range(n)]
    ans = 0

    for size in range(n, 0, -1):
        for i in range(n - size + 1):
            for j in range(n - size + 1):
                min_value = min(matrix[i + k][j + k] for k in range(size))
                if min_value < 0:
                    for k in range(size):
                        matrix[i + k][j + k] -= min_value
                    ans += abs(min_value)

    print(ans)

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

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

Think step by step.

Response:
```plaintext
1
3
-1 -1 -1
-1 -2 -1
-1 -1 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
def solve(n, a):
    ans = 0
    while 1:
        mn = 0
        p = -1
        q = -1
        for i in range(n):
            for j in range(n):
                if a[i][j] < mn:
                    mn = a[i][j]
                    p = i
                    q = j
        if mn == 0:
            break
        ans += 1
        for i in range(n):
            for j in range(n):
                if i-p == j-q:
                    a[i][j] += 1
    return ans

t = int(input())
for i in range(t):
    n = int(input())
    a = []
    for j in range(n):
        r = list(map(int, input().split()))
        a.append(r)
    print(solve(n, a))

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

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

Think step by step.

Response:
```plaintext
1
3
-1 -1 -1
-1 -2 -1
-1 -1 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
import java.util.Comparator;
import java.util.PriorityQueue;
import java.util.Scanner;

public class Solution {

    public static void main(String[] args) {
        Scanner scanner = new Scanner(System.in);
        int t = scanner.nextInt();
        int[] ans = new int[t];
        for (int i = 0; i < t; i++) {
            int n = scanner.nextInt();
            int[][] inputs = new int[n][n];
            for (int j = 0; j < n; j++) {
                for (int k = 0; k < n; k++) {
                    inputs[j][k] = scanner.nextInt();
                }
            }
            ans[i] = solve(inputs);
        }
        for (int i = 0; i < t; i++) {
            System.out.println(ans[i]);
        }
        scanner.close();
    }

    public static int solve(int[][] inputs) {
        int n = inputs.length;
        int operations = 0;

        // Priority Queue to keep track of the most negative values
        PriorityQueue<int[]> pq = new PriorityQueue<>(Comparator.comparingInt(a -> a[2]));

        // Fill the priority queue with all negative values' positions
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (inputs[i][j] < 0) {
                    pq.offer(new int[]{i, j, inputs[i][j]});
                }
            }
        }

        // Perform operations until there are no negative values
        while (!pq.isEmpty()) {
            int[] minElement = pq.poll();
            int minI = minElement[0];
            int minJ = minElement[1];

            // Only process if the value is still negative (it might have been incremented before)
            if (inputs[minI][minJ] < 0) {
                // Increment the diagonal
                for (int k = 0; k < n; k++) {
                    int diagI = minI + k;
                    int diagJ = minJ + k;
                    if (diagI < n && diagJ < n) {
                        inputs[diagI][diagJ]++;
                    }
                }
                operations++;
            }

            // Rebuild the priority queue with remaining negative values
            pq.clear();
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    if (inputs[i][j] < 0) {
                        pq.offer(new int[]{i, j, inputs[i][j]});
                    }
                }
            }
        }

        return operations;
    }
}

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

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

Think step by step.

Response:
```plaintext
3
1
-1
2
-1 2
3 0
3
-1 -2 -3
-2 -1 -2
-3 -2 -1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Sakurako and Water
time_limit_ms: 2000
memory_limit_mb: 256
Description: During her journey with Kosuke, Sakurako and Kosuke found a valley that can be represented as a matrix of size $n \times n$, where at the intersection of the $i$-th row and the $j$-th column is a mountain with a height of $a_{i,j}$. If $a_{i,j} < 0$, then there is a lake there.

Kosuke is very afraid of water, so Sakurako needs to help him:

  * With her magic, she can select a square area of mountains and increase the height of each mountain on the main diagonal of that area by exactly one.

More formally, she can choose a submatrix with the upper left corner located at $(i, j)$ and the lower right corner at $(p, q)$, such that $p-i=q-j$. She can then add one to each element at the intersection of the $(i + k)$-th row and the $(j + k)$-th column, for all $k$ such that $0 \le k \le p-i$.

Determine the minimum number of times Sakurako must use her magic so that there are no lakes.

Input: The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases.

Each test case is described as follows:

  * The first line of each test case consists of a single number $n$ ($1 \le n \le 500$).    * Each of the following $n$ lines consists of $n$ integers separated by spaces, which correspond to the heights of the mountains in the valley $a$ ($-10^5 \le a_{i,j} \le 10^5$).

It is guaranteed that the sum of $n$ across all test cases does not exceed $1000$.

Output: For each test case, output the minimum number of times Sakurako will have to use her magic so that all lakes disappear.

Examples:
input:
4
1
1
2
-1 2
3 0
3
1 2 3
-2 1 -1
0 0 -1
5
1 1 -1 -1 3
-3 1 4 4 -4
-1 -1 3 0 -5
4 5 3 -3 -1
3 1 -3 -1 5
output:
0
1
4
19

Note:

Buggy Code:
import java.io.File;
import java.util.Scanner;

/**
 * codeforces
 */
public class codeforces {

    public static void main(String[] args) throws Exception{
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();
        while(t-->0){
            int n = sc.nextInt();
            int arr[][] = new int[n][n];
            for(int i = 0; i < n; i++)
                for(int j = 0; j < n; j++)
                    arr[i][j] = sc.nextInt();
            long ans = 0;
            for(int i = 0; i < n; i++)
                for(int j = 0; j < n; j++)
                    while(arr[i][j] < 0){
                        int x = i, y = j;
                        while (x < n && y < n) {
                            arr[x][y]++;
                            x++;
                            y++;
                        }
                        ans++;
                    }
            System.out.println(ans);
        }
    }
}

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

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

Think step by step.

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

1. **Edge Cases**:
   - A matrix with a single element.
   - A matrix with all elements being lakes.
   - A matrix with a diagonal that has negative values.
   - A matrix where the negative values are spread out and not on the diagonal.

2. **Maximizing Failure Probability**:
   - The code increments the values on the main diagonal of a selected submatrix. If the negative values are not on the diagonal, the code might not handle them correctly.

3. **Small-Scale Test Input**:
   - We need to create a small matrix (e.g., 3x3 or 4x4) that can clearly show the bug.

Let's create a 3x3 matrix where the negative values are not on the main diagonal:

```plaintext
1
3
-1 2 3
4 -2 5
6 7 -1
```

### Explanation:
- **Test Case**:
  - The matrix is 3x3.
  - The negative values are at positions (1,1), (2,2), and (3,3), but not on the main diagonal.
  - The main diagonal has values 2, -2, and 5.

### Expected Output:
- The code should increment the values on the main diagonal until all negative values are neutralized.
- However, since the negative values are not on the main diagonal, the code might not handle them correctly, leading to an incorrect result.

### Correct Solution:
- The correct solution would need to identify the negative values and increment the appropriate diagonals to neutralize them.

### Output:
```plaintext
1
3
-1 2 3
4 -2 5
6 7 -1
```