**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Klee's SUPER DUPER LARGE Array!!!
time_limit_ms: 2000
memory_limit_mb: 256
Description: Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.

Output the minimum possible value of $$$x$$$.

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

Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.

Output: For each test case, output the minimum value of $$$x$$$ on a new line.

Examples:
input:
4
2 2
7 2
5 3
1000000000 1000000000
output:
1
5
1
347369930

Note: In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

Buggy Code:
#include<iostream>
using namespace std;
#include<vector>
#include<set>
#include<algorithm>
#include<map>
#include<string>
#include<stack>
#include<unordered_set>
#include<cmath>
#include <unordered_map>
// Include for std::fixed and std::setprecision
#include <iomanip>
#include<queue>

//bool nodublicate(string x, int start, int end)
//{
//	set<char>s;
//	for (int i = start; i < end; i++)
//	{
//		s.insert(x[i]);
//	}
//	if (s.size() < end - start)
//		return false;
//	else return true;
//}
//void lengthOfLongestSubstring2(string s, int start, int end, int& max) {
//	if ((start == s.size() && end == s.size()) || start > end)
//		return;
//	if (nodublicate(s, start, end))
//	{
//		if ((end - start) > max)
//			max = end - start;
//	}
//	if (start != s.size())
//	{
//		lengthOfLongestSubstring2(s, start + 1, end, max);
//	}
//	if (end != s.size())
//	{
//		lengthOfLongestSubstring2(s, start, end + 1, max);
//	}
//
//}
//int lengthOfLongestSubstring(string s)
//{
//	int start = 0;
//	int end = 0;
//	int max = 0;
//	lengthOfLongestSubstring2(s, start, end, max);
//	return max;
//}
//int fun5(string x, int& ffirst, int& eend)
//{
//	int max = 0;
//	unordered_set<char>s;
//	int first = 0;
//	int end = 0;
//	vector<char>v;
//	for (int i = 0; i < x.size(); i++)
//	{
//		s = { x[i] };
//		v = { x[i] };
//		first = i;
//		for (int j = i + 1; j < x.size(); j++)
//		{
//			end = j;
//			if (s.find(x[j]) == s.end())
//				break;
//			else
//			{
//				v.push_back(x[j]);
//				s.insert(x[j]);
//			}
//		}
//		if (v.size() > max)
//		{
//			ffirst = first;
//			eend = end - 1;
//			max = v.size();
//		}
//	}
//	return max;
//}
//int characterReplacement(string s, int k) {
//	int start = 0;
//	int end = 0;
//	fun5(s, start, end);
//	int ssart = start;
//	int eend = end;
//	for (int i = start - 1; i > -1; i--)
//	{
//
//		if (s[i] == s[start])
//		{
//			ssart--;
//			continue;
//		}
//		if (k == 0)
//			break;
//		s[i] = s[start];
//		k--;
//		ssart--;
//	}
//	for (int i = end + 1; i < s.size(); i++)
//	{
//
//		if (s[i] == s[start])
//		{
//			eend++;
//			continue;
//		}
//		if (k == 0)
//			break;
//		s[i] = s[start];
//		k--;
//		eend++;
//	}
//
//	return (eend - ssart) + 1;
//}
//int fun4(string x, int k)
//{
//	int maxstart = 0;
//	int maxend = 0;
//	int start = 0;
//	int end = 0;
//	int f = k;
//	for (int i = 1; i < x.size(); i++)
//	{
//		if (x[i] == x[start])
//		{
//			end++;
//		}
//		else
//		{
//			f--;
//			if (f < 0)
//			{
//				if (maxend - maxstart < end - start)
//				{
//					maxstart = start;
//					maxend = end;
//				}
//				f = k;
//				i = start+1;
//				end = start;
//				continue;
//			}
//			end++;
//		}
//	}
//	return maxend - maxstart;
//}
////int Vanya_and_Fence(vector<int>v,int height)
////{
////	int count = 0;
////	for (int i = 0; i < v.size(); i++)
////	{
////		if (v[i] > height)
////			count += 2;
////		else count++;
////	}
////	return count;
////}
////string Anton_and_Danik(string x)
////{
////	int anton = 0;
////	int danik = 0;
////	for (int i = 0; i < x.size(); i++)
////	{
////		if (x[i] == 'A')
////			anton++;
////		else if (x[i] == 'D')
////			danik++;
////
////	}
////	if (anton > danik)
////		return "Anton";
////	else if (danik > anton)
////		return "Danik";
////	else return "Friendship";
////}
//int Bear_and_Big_Brother(int x, int y)
//{
//	float n = log(float(y) / x) / log(1.5);
//	return floor(n) + 1;
//}
//int Team(vector<vector<int>>v)
//{
//	int count = 0;
//	for (int i = 0; i < v.size(); i++)
//	{
//		int sum = 0;
//		for (int j = 0; j < 3; j++)
//		{
//			sum += v[i][j];
//		}
//		if (sum >= 2)
//			count++;
//	}
//	return count;
//}
//int Beautiful_Matrix(vector<vector<int>>v)
//{
//	int i;
//	int j;
//	int flag = false;
//	for (i = 0; i < 5; i++)
//	{
//		for (j = 0; j < 5; j++)
//		{
//			if (v[i][j] == 1)
//			{
//				flag = true;
//				break;
//			}
//		}
//		if (flag == true)
//			break;
//	}
//	return abs(2 - i) + abs(2 - j);
//}
//vector<int>Gravity_Flip(vector<int>v)
//{
//	sort(v.begin(), v.end());
//	return v;
//}
//int Petyaand_Strings(string x, string y)
//{
//	transform(x.begin(), x.end(), x.begin(), ::tolower);
//	transform(y.begin(), y.end(), y.begin(), ::tolower);
//	if (x > y)
//		return 1;
//	else if (x < y)
//		return -1;
//	else return 0;
//}
//string  Boy_or_Girl(string x)
//{
//	set<char>s;
//	for (int i = 0; i < x.size(); i++)
//	{
//		s.insert(x[i]);
//	}
//	if (s.size() % 2 == 0)
//		return "CHAT WITH HER!";
//	else return "IGNORE HIM!";
//}
//string Word(string x)
//{
//	int upper, lower;
//	upper = 0;
//	lower = 0;
//	for (int i = 0; i < x.size(); i++)
//	{
//		if (islower(x[i]))
//			lower++;
//		else upper++;
//	}
//	if (lower >= upper)
//		transform(x.begin(),x.end(),x.begin(),::tolower);
//	else
//		transform(x.begin(), x.end(), x.begin(), ::toupper);
//	return x;
//}
//int Magnets(vector<int>v)
//{
//	if (!v.empty())
//	{
//		int current = v[0];
//		int count = 0;
//		for (int i = 1; i < v.size(); i++)
//		{
//			if (v[i] != current)
//			{
//				current = v[i];
//				count++;
//			}
//		}
//		return count+1;
//	}
//	else return 0;
//}
//vector<int>Sereja_and_Dima(vector<int>v)
//{
//	int first = 0;
//	int second = 0;
//	int i = 0;
//	while (!v.empty())
//	{
//		if (v[0] >= v[v.size() - 1] && i % 2 == 0)
//		{
//			first += v[0];
//			v.erase(v.begin());
//			i++;
//		}
//
//		else if (v[0] < v[v.size() - 1] && i % 2 == 0)
//		{
//			first += v[v.size() - 1];
//			v.erase(v.end()-1);
//			i++;
//		}
//
//		else if(v[0] >= v[v.size() - 1] && i % 2 != 0)
//		{
//			second += v[0];
//			v.erase(v.begin());
//			i++;
//		}
//		else  if (v[0] < v[v.size() - 1] && i % 2 != 0)
//		{
//			second += v[v.size() - 1];
//			v.erase(v.end() - 1);
//			i++;
//		}
//
//	}
//	return { first,second };
//}
//int Stones_on_the_Table(string x)
//{
//	int count = 0;
//	for (int i = 1; i < x.size(); i++)
//	{
//		if (x[i] == x[i - 1])
//			count++;
//
//	}
//	return count;
//}
//int Police_Recruits(vector<int>v)
//{
//	int count = 0;
//	int num = 0;
//	for (int i = 0; i < v.size(); i++)
//	{
//		num += v[i];
//		if (num < 0)
//		{
//			count += -1 * num;
//			num = 0;
//
//		}
//	}
//	return count;
//}
//int  Black_Square(string x, vector<int>v)
//{
//	int count = 0;
//	for (int i = 0; i < x.size(); i++)
//	{
//		if (x[i] == '1')
//			count += v[0];
//		else if (x[i] == '2')
//			count += v[1];
//		else if (x[i] == '3')
//			count += v[2];
//		else count += v[3];
//	}
//	return count;
//}
//int Night_at_the_Museum(string x)
//{
//	int count = 0;
//	int last = 'a';
//	for (int i = 0; i < x.size(); i++)
//	{
//		if (x[i] > last)
//		{
//			if (x[i] - last < 26 - (x[i] - last))
//				count += x[i] - last;
//			else count += 26 - (x[i] - last);
//			last = x[i];
//		}
//		else
//		{
//			if (last - x[i] < 26 - (last - x[i]))
//				count += last - x[i];
//			else count += 26 - (last - x[i]);
//			last = x[i];
//		}
//	}
//	return count;
//}
//int Games(vector<pair<int, int>>v)
//{
//	int count = 0;
//	for (int i = 0; i < v.size(); i++)
//	{
//		for (int j = 0; j < v.size(); j++)
//		{
//			if (v[i].first == v[j].second)
//				count++;
//		}
//	}
//	return count;
//}
//int Buy_a_Shovel(int price, int charge)
//{
//	int count = 1;
//	while (true)
//	{
//		if (price * count % 10 == charge|| price * count % 10 == 0)
//			break;
//		else count++;
//	}
//	return count;
//}
//int Colorful_Stones_Simplified_Edition(string x, string y)
//{
//	int index = 0;
//	for (int i = 0; i < y.size(); i++)
//	{
//		if (x[index] == y[i])
//			index++;
//
//	}
//	return index + 1;
//}
//int Roadside_Trees_Simplified_Edition(vector<int>x)
//{
//	int count = x[0];
//	count++;
//	for (int i = 1; i < x.size(); i++)
////	{
////		count++;
////		if (x[i - 1] > x[i])
////			count += x[i - 1] - x[i];
////		else count += x[i] - x[i - 1];
////		count++;
////	}
////	return count;
////}
////vector<int>Escape_from_Stones(string x)
////{
////	vector<long double>v(x.size(),long double(0.0));
////	v[0] =long double(0.5);
////	long double first=long double(0.0);
////	long double last=long double(1.0);
////	for (int i = 0; i < x.size()-1; i++)
////	{
////		if (x[i] == 'l')
////			last = (last + first) / long double(2.);
////		else first = (last + first) /long double(2.);
////		v[i + 1] = (last + first) / long double(2.);
////	}
////	vector<int> indices(v.size());
////	for (int i = 0; i < v.size(); ++i) {
////		indices[i] = i;
////	}
////
////
////	sort(indices.begin(), indices.end(), [&v](int a, int b) {
////		return v[a] < v[b];
////		});
////
////	return indices;
////}
//void Die_Roll(int x, int y)
//{
//	int count = max(x, y);
//	int remain = 6 - count + 1;
//	if (remain / 6. == .5)
//		cout << "1/2";
//	else if (remain / 6. == 0)
//		cout << "0";
//	else if (remain / 6. == 1 / 6.)
//		cout << "1/6";
//	else if (remain / 6. == 2 / 6.)
//		cout << "1/3";
//	else if (remain / 6. == 4 / 6.)
//		cout << "2/3";
//	else if (remain / 6. == 5 / 6.)
//		cout << "5/6";
//	else cout << "1/1";
//}
//void Running_Student(double speed_bus, double speed_student, vector<int>stations, int x, int y)
//{
//	double min = LONG_MAX;
//	int distance = 0;
//	int index = 0;
//	for (int i = 1; i < stations.size(); i++)
//	{
//		distance = stations[i];
//		if (distance / speed_bus + sqrt(pow((x - stations[i]), 2) + pow((y), 2)) / speed_student <= min)
//		{
//			min = distance / speed_bus + sqrt(pow((x - stations[i]), 2) + pow((y), 2)) / speed_student;
//			index = i;
//		}
//
//	}
//
//	cout << index + 1;
//}
//void Hexadecimal_Numbers(int y, int n, string x, int& num, int state)
//{
//	if (x.size() > n || x.size() > 0 && stoi(x) > y)
//	{
//		num--;
//		return;
//	}
//
//	if (state != 0)
//	{
//		num++;
//		Hexadecimal_Numbers(y, n, x + '0', num, state);
//	}
//
//	num++;
//	Hexadecimal_Numbers(y, n, x + '1', num, 1);
//
//}
//void  Shaass_and_Oskols(vector<int>v, vector<pair<int, int>>shot)
//{
//
//	for (int i = 0; i < shot.size(); i++)
//	{
//		if (shot[i].first > 0)
//		{
//			v[shot[i].first - 1] += shot[i].second;
//
//		}
//		if (shot[i].first < v.size() - 1)
//		{
//			v[shot[i].first + 1] += v[shot[i].first] - (shot[i].second + 1);
//		}
//		v[shot[i].first] = 0;
//	}
//	for (int i = 0; i < v.size(); i++)
//	{
//		cout << v[i] << endl;
//	}
//}
//void Juicer(vector<int>v, int b, int d)
//{
//	int waste = 0;
//	int result = 0;
//	for (int i = 0; i < v.size(); i++)
//	{
//		if (waste > d)
//		{
//			result++;
//			waste = 0;
//		}
//		if (v[i] > b)
//			continue;
//		waste += v[i];
//
//	}
//	if (waste > d)
//		result++;
//	cout << result;
//}
//void Checkpoints(vector<int>v, int a)
//{
//
//	sort(v.begin(), v.end());
//	int min = LONG_MAX;
//	int result = 0;
//	int first = 0;
//	std::sort(v.begin(), v.end(), [a](int x, int y) {
//		return std::abs(x - a) < std::abs(y - a);
//		});
//	int erased = v[0];
//	v.erase(v.begin());
//
//	if (!v.empty())
//	{
//		result = abs(v[0] - a);
//		first = v[0];
//	}
//	for (int i = 1; i < v.size(); i++)
//	{
//		result += abs(v[i] - first);
//		first = v[i];
//	}
//	if (result < min)
//		min = result;
//	v.insert(v.begin(), erased);
//	for (int j = 1; j < v.size(); j++)
//	{
//		int erased = v[j];
//
//		if (a < v[j - 1] && a > v[j] && j != v.size() - 1)
//		{
//			result += abs(v[j - 1] - a) + abs(v[j - 1] - a);
//		}
//		else if (a > v[j - 1] && a < v[j] && j != v.size() - 1)
//		{
//			result += abs(v[j - 1] - a) + abs(v[j - 1] - a);
//		}
//		else if (j == v.size() - 1)
//		{
//			result -= abs(v[j] - v[j - 1]);
//		}
//
//
//		if (result < min)
//			min = abs(result);
//
//
//	}
//	cout << min;
//}
//void Carrot_Cakes(int n, int t, int k, int d)
//{
//	int count = 0;
//	while (n > 0)
//	{
//		if (count > d)
//		{
//			cout << "YES";
//			return;
//		}
//
//		n -= k;
//		count += t;
//
//	}
//	cout << "NO";
//}
//void T_shirt_buying(vector<vector<long long>>v, vector<long long>buyer)
//{
//	vector<long long>sold;
//	vector<vector<long long>>v2 = v;
//	sort(v.begin(), v.end(), [](const vector<long long>& a, const vector<long long>& b) {
//		return a[1] < b[1];
//		});
//
//	sort(v2.begin(), v2.end(), [](const vector<long long>& a, const vector<long long>& b) {
//		return a[2] < b[2];
//		});
//
//	for (int i = 0; i < buyer.size(); i++)
//	{
//		long long maxVal = LLONG_MAX;
//		int maxIndex = -1;
//		auto lower = lower_bound(v.begin(), v.end(), buyer[i], [](const vector<long long>& a, long long value) {
//			return a[1] < value;
//			});
//		auto upper = upper_bound(v.begin(), v.end(), buyer[i], [](long long value, const vector<long long>& a) {
//			return value < a[1];
//			});
//		/////////////////
//		long long maxVal2 = LLONG_MAX;
//		int maxIndex2 = -1;
//		auto lower2 = lower_bound(v2.begin(), v2.end(), buyer[i], [](const vector<long long>& a, long long value) {
//			return a[2] < value;
//			});
//		auto upper2 = upper_bound(v2.begin(), v2.end(), buyer[i], [](long long value, const vector<long long>& a) {
//			return value < a[2];
//			});
//		// Loop through the range and find the maximum value of index 0
//		for (auto it = lower; it != upper; ++it) {
//			if ((*it)[0] < maxVal) {
//				maxVal = (*it)[0];
//				maxIndex = distance(v.begin(), it);  // Get the index of the maximum value
//			}
//		}
//		for (auto it2 = lower2; it2 != upper2; ++it2) {
//			if ((*it2)[0] < maxVal2) {
//				maxVal2 = (*it2)[0];
//				maxIndex2 = distance(v2.begin(), it2);  // Get the index of the maximum value
//			}
//		}
//		// Print the maximum value and its index
//		if (maxIndex != -1 || maxIndex2 != -1) {
//			sold.push_back(min(maxVal, maxVal2));
//			if (maxVal < maxVal2)
//			{
//				for (int k = 0; k < v2.size(); k++)
//				{
//					if (v2[k][0] == maxVal)
//					{
//						v2.erase(v2.begin() + k);
//						break;
//					}
//				}
//				v.erase(v.begin() + maxIndex);
//
//				//v2.erase(v2.begin() + maxIndex);
//			}
//			else
//			{
//				for (int k = 0; k < v.size(); k++)
//				{
//					if (v[k][0] == maxVal2)
//					{
//						v.erase(v.begin() + k);
//						break;
//					}
//				}
//
//				v2.erase(v2.begin() + maxIndex2);
//			}
//		}
//		else {
//			sold.push_back(-1);
//		}
//
//
//	}
//	for (int i = 0; i < sold.size(); i++)
//	{
//		cout << sold[i] << " ";
//	}
//}
//void Anton_and_Letters(string x)
//{
//	set<char>f;
//
//	for (int i = 1; i < x.size() - 1; i += 3)
//	{
//		f.insert(x[i]);
//
//	}
//	cout << f.size();
//}
//void Way_Too_Long_Words(vector<string>v)
//{
//	for (int i = 0; i < v.size(); i++)
//	{
//		if (v[i].size() > 10)
//		{
//			string x;
//			x.push_back(v[i][0]);
//			x += to_string(v[i].size() - 2);
//			x += v[i][v[i].size() - 1];
//			v[i] = x;
//		}
//
//	}
//	for (int i = 0; i < v.size(); i++)
//	{
//		cout << v[i] << endl;
//	}
//}
//void Free_Ice_Cream(vector<long long>v, vector<char>v2, long long n)
//{
//	long long count = n;
//	long long child = 0;
//	for (long long i = 0; i < v.size(); i++)
//	{
//		if (v2[i] == '+')
//			count += v[i];
//		else {
//			if (count >= v[i])
//				count -= v[i];
//			else child++;
//		}
//	}
//	cout << count << " " << child;
//}
//void Helpful_Maths(string x)
//{
//	vector<int>v;
//	for (int i = 0; i < x.size(); i += 2)
//	{
//		v.push_back(stoi(string(1, x[i])));
//
//	}
//	x = "";
//	sort(v.begin(), v.end());
//	for (int i = 0; i < v.size() - 1; i++)
//	{
//		x += char(v[i] + '0');
//		x += '+';
//	}
//	x += char(v[v.size() - 1] + '0');
//	cout << x;
//}
//void Team_Olympiad(vector<pair<int, int>>v)
//{
//	sort(v.begin(), v.end(), [](const pair<int, int>& a, const pair<int, int>& b) {
//		return a.first < b.first;
//		});
//	int count1 = 0;
//	int count2 = 0;
//	int count3 = 0;
//
//	for (int i = 0; i < v.size(); i++)
//	{
//		if (v[i].first == 1)
//			count1++;
//		else if (v[i].first == 2)
//			count2++;
//		else count3++;
//	}
//	int result = min(min(count1, count2), count3);
//
//	int start = 0;
//
//	int start2 = 0;
//
//	int start3 = 0;
//
//	bool flag1 = true;
//	bool flag2 = false;
//
//	if (result == 0)
//		cout << 0;
//	else
//	{
//		cout << result << endl;
//
//		for (int i = 0; i < v.size(); i++)
//		{
//			if (v[i].first == 2 && flag1 == true)
//			{
//				flag1 = false;
//				flag2 = true;
//				start2 = i;
//			}
//			else if (v[i].first == 3 && flag2 == true)
//			{
//
//				start3 = i;
//				break;
//			}
//		}
////
////	}
////	for (int i = 0; i < result; i++)
////	{
////		cout << v[start + i].second + 1 << " " << v[start2 + i].second + 1 << " " << v[start3 + i].second + 1 << endl;
////	}
////}
////void New_Password(int n, int k)
////{
////	char a = 'a';
////	string x = "";
////	int count = 0;
////	for (int i = 0; i < n; i++)
////	{
////		x += a + count;
////		count++;
////		if (count >= k)
////			count = 0;
////	}
////	cout << x;
////}
////void  Presents(vector<pair<int, int>>v)
////{
////	sort(v.begin(), v.end(), [](pair<int, int>a, pair<int, int>b)
////		{
////			return a.first < b.first;
////		});
////	for (int i = 0; i < v.size(); i++)
////	{
////		cout << v[i].second + 1 << " ";
////	}
////}
////void Lineland_Mail(vector<int>v)
////{
////	vector<pair<int, int>>result(v.size(), { 0,0 });
////	result[0].first = v[1] - v[0];
////	result[0].second = v[v.size() - 1] - v[0];
////	for (int i = 1; i < v.size() - 1; i++)
////	{
////		if (abs(v[i - 1] - v[i]) < abs(v[i] - v[i + 1]))
////			result[i].first = abs(v[i - 1] - v[i]);
////		else result[i].first = abs(v[i] - v[i + 1]);
////
////		if (abs(v[v.size() - 1] - v[i]) > abs(v[0] - v[i]))
////			result[i].second = abs(v[v.size() - 1] - v[i]);
////		else result[i].second = abs(v[0] - v[i]);
////
////	}
////	result[result.size()-1].first = v[v.size()-1] - v[v.size()-2];
////	result[result.size()-1].second = v[v.size() - 1] - v[0];
////	for (int i = 0; i < result.size(); i++)
////	{
////		cout << result[i].first << " " << result[i].second << endl;
////	}
////}
////void Mahmoud_and_Longest_Uncommon_Subsequence(string a, string b)
////{
////	if (a != b)
////	{
////		cout << (a.length() > b.length() ? a.length() : b.length());
////		return;
////	}
////	cout<<-1;
////
////
////}
//unordered_map<string, bool> memo; // Memoization map
//bool isteam(vector<int>v, set<pair<int, int>>s)
//{
//	for (int i = 0; i < v.size(); i++)
//	{
//		for (int j = 0; j < v.size(); j++)
//		{
//			if (s.find({ v[i],v[j] }) != s.end() || s.find({ v[j],v[i] }) != s.end())
//				return false;
//		}
//	}
//	return true;
//}
//string vectorToString(const vector<int>& v1, const vector<int>& v2) {
//	string result;
//	for (int i : v1) result += to_string(i) + ",";
//	result += "|";
//	for (int i : v2) result += to_string(i) + ",";
//	return result;
//}
//void Forming_Teams(set < pair<int, int>>s, int n, vector<int>v1, vector<int>v2, int& max)
//{
//	if (n == 0)
//	{
//		string memoKey = vectorToString(v1, v2);
//		if (memo[memoKey] == true)
//			return;
//
//		memo[memoKey] = true;
//		if (isteam(v1, s) && isteam(v2, s))
//		{
//			int size = min(v1.size(), v2.size());
//			if (size > max)
//				max = size;
//		}
//		return;
//	}
//
//
//	Forming_Teams(s, n - 1, v1, v2, max);
//	v1.push_back(n);
//	Forming_Teams(s, n - 1, v1, v2, max);
//	v1.pop_back();
//	Forming_Teams(s, n - 1, v1, v2, max);
//	v2.push_back(n);
//	Forming_Teams(s, n - 1, v1, v2, max);
//	v2.pop_back();
//}
//
//// Function to perform DFS-like traversal to count occurrences and component size
//void DFS(int index, const vector<pair<int, int>>& v, vector<bool>& visited, int& componentSize) {
//	visited[index] = true;
//	componentSize++;
//
//	// Recur for the next unvisited element in the vector
//	for (int i = 0; i < v.size(); i++) {
//		if (!visited[i]) {
//			// Check if the next element shares any pair with the current element
//			if (v[index].first == v[i].first || v[index].first == v[i].second ||
//				v[index].second == v[i].first || v[index].second == v[i].second) {
//				DFS(i, v, visited, componentSize);
//			}
//		}
//	}
//}
//std::vector<long long> sort_by_min_difference(vector<vector<long long>>& values) {
//	// Create a vector to store the original values with their minimum differences
//	vector<long long> diff_vector;
//
//	// Calculate the minimum difference for each value and store it along with the value
//	for (long long i = 0; i < values.size(); i++)
//	{
//		for (long long j = 1; j < values[i].size() - 1; j++)
//		{
//			long long first = values[i][0] - values[i][j];
//			long long last = values[i][values[i].size() - 1] - values[i][j];
//			diff_vector.push_back(min(first, last));
//		}
//	}
//
//	// Sort the vector of pairs based on the minimum difference
//	std::sort(diff_vector.begin(), diff_vector.end());
//
//
//
//	return diff_vector;
//}
//void K_Sort(vector<long long>v)
//{
//	long long count = 0;
//	vector<vector<long long>>willincrease;
//	for (int i = 1; i < v.size(); i++)
//	{
//
//		long long first;
//		long long last;
//		vector<long long>v23;
//		if (v[i] < v[i - 1])
//		{
//			first = v[i - 1];
//			v23.push_back(first);
//			v23.push_back(v[i]);
//			i++;
//
//			while (i < v.size() && v[i] < first)
//			{
//				v23.push_back(v[i]);
//				i++;
//			}
//			if (i != v.size())
//			{
//				last = v[i];
//				v23.push_back(last);
//			}
//			else
//			{
//				last = LLONG_MAX;
//				v23.push_back(last);
//			}
//			willincrease.push_back(v23);
//			// Create a vector to store pairs of (difference, 'first' or 'last')
//
//
//			i--;
//		}
//
//	}
//	vector<long long>result;
//	result = sort_by_min_difference(willincrease);
//	long long c = 0;
//	for (int i = 0; i < result.size(); i++)
//	{
//		count += (result.size() - i) * result[i];
//		count += result[i];
//		c += result[i];
//		if (i != result.size() - 1)
//			result[i + 1] -= c;
//	}
//	cout << count << endl;
//}
//void Matrix_Stabilization(vector<vector<long long>>v)
//{
//	for (int i = 0; i < v.size(); i++)
//	{
//		for (int j = 0; j < v[i].size(); j++)
//		{
//			bool top = false;
//			bool bottom = false;
//			bool right = false;
//			bool left = false;
//			if (i == 0)
//				top = true;
//			else if (v[i - 1][j] < v[i][j])
//				top = true;
//			else top = false;
//
//			if (j == 0)
//				left = true;
//			else if (v[i][j - 1] < v[i][j])
//				left = true;
//			else left = false;
//
//			if (i == v.size() - 1)
//				bottom = true;
//			else if (v[i + 1][j] < v[i][j])
//				bottom = true;
//			else bottom = false;
//
//			if (j == v[i].size() - 1)
//				right = true;
//			else if (v[i][j + 1] < v[i][j])
//				right = true;
//			else right = false;
//
//
//			if (right && left && top && bottom)
//			{
//				long long first = LLONG_MIN;
//				long long second = LLONG_MIN;
//				long long third = LLONG_MIN;
//				long long forth = LLONG_MIN;
//				if (i - 1 >= 0)
//					first = v[i - 1][j];
//				if (j - 1 >= 0)
//					second = v[i][j - 1];
//				if (i + 1 < v.size())
//					third = v[i + 1][j];
//				if (j + 1 < v[i].size())
//					forth = v[i][j + 1];
//				v[i][j] = max(first, max(second, max(third, forth)));
//			}
//		}
//	}
//
//	for (int i = 0; i < v.size(); i++)
//	{
//		for (int j = 0; j < v[i].size(); j++)
//		{
//			cout << v[i][j] << " ";
//		}
//		cout << endl;
//	}
//}
//
//void Good_Prefixes(vector<long long>v, set<long long>s)
//{
//	long long count = 0;
//	int result = 0;
//	for (int i = 0; i < v.size(); i++)
//	{
//		count += v[i];
//		s.insert(v[i]);
//		if (count % 2 == 0 && s.find(count / 2.) != s.end())
//			result++;
//	}
//	cout << result << endl;
//}
//string longLongToBinaryString(long long int num) {
//	if (num == 0) return "0";
//
//	string binaryStr;
//	while (num > 0) {
//		// Append '0' or '1' based on the least significant bit
//		binaryStr = (num % 2 == 0 ? '0' : '1') + binaryStr;
//		num /= 2;
//	}
//	return binaryStr;
//}
//void XOR_Sequences(long long a, long long b)
//{
//	long long count = 0;
//	string x = longLongToBinaryString(a);
//	string y = longLongToBinaryString(b);
//
//	if (x.size() < y.size())
//	{
//
//		while (x.size() < y.size())
//		{
//			if (x[0] == y[y.size() - x.size()])
//				x = '0' + x;
//			else
//			{
//				while (x.size() < y.size())
//					x = '5' + x;
//			}
//		}
//	}
//	if (y.size() < x.size())
//	{
//
//
//		while (y.size() < x.size())
//		{
//			if (x[x.size() - y.size()] == y[0])
//				y = '0' + y;
//			else
//			{
//				while (y.size() < x.size())
//					y = '5' + y;
//			}
//
//		}
//	}
//	int num = 0;
//	for (int i = 0; i < x.size(); i++)
//	{
//		if (x[i] == y[i])
//			num++;
//		else {
//			if (num > count)
//				count = num;
//			num = 0;
//		}
//	}
//	if (num > count)
//		count = num;
//	double result = std::pow(2, count);
//
//	// Output the result in fixed-point notation without scientific notation
//	std::cout << std::fixed << std::setprecision(0) << result << std::endl;
//}
//void Diverse_Game(vector<vector<long long>>v)
//{
//	if (v.size() == 1)
//	{
//		if (v[0].size() % 2 == 0)
//		{
//			for (int i = 0; i < v[0].size() - 1; i += 2)
//			{
//				long long temp = v[0][i];
//				v[0][i] = v[0][i + 1];
//				v[0][i + 1] = temp;
//			}
//			for (int i = 0; i < v[0].size(); i++)
//			{
//				cout << v[0][i] << " ";
//
//			}
//			cout << endl;
//			return;
//		}
//		else
//		{
//			if (v[0].size() == 1)
//			{
//				cout << -1 << endl;
//				return;
//			}
//			else
//			{
//				for (int i = 0; i < v[0].size(); i += 2)
//				{
//					if (i == 0)
//					{
//						long long temp = v[0][0];
//						v[0][0] = v[0][2];
//						long long temp2 = v[0][1];
//						v[0][1] = temp;
//						v[0][2] = temp2;
//						i += 1;
//					}
//					else
//					{
//						long long temp = v[0][i];
//						v[0][i] = v[0][i + 1];
//						v[0][i + 1] = temp;
//					}
//
//
//				}
//				for (int i = 0; i < v[0].size(); i++)
//				{
//					cout << v[0][i] << " ";
//
//				}
//				cout << endl;
//				return;
//			}
//		}
//	}
//	else
//	{
//		if (v.size() % 2 == 0)
//		{
//			for (int i = 0; i < v.size() - 1; i += 2)
//			{
//				vector<long long> temp = v[i];
//				v[i] = v[i + 1];
//				v[i + 1] = temp;
//			}
//			for (int i = 0; i < v.size(); i++)
//			{
//				for (int j = 0; j < v[i].size(); j++)
//				{
//					cout << v[i][j] << " ";
//				}
//				cout << endl;
//			}
//			return;
//		}
//		else
//		{
//			if (v.size() == 1)
//			{
//				cout << -1 << endl;
//				return;
//			}
//			else
//			{
//				for (int i = 0; i < v.size(); i += 2)
//				{
//					if (i == 0)
//					{
//						vector<long long> temp = v[0];
//						v[0] = v[2];
//						vector<long long> temp2 = v[1];
//						v[1] = temp;
//						v[2] = temp2;
//						i += 1;
//					}
//					else
//					{
//						vector<long long> temp = v[i];
//						v[i] = v[i + 1];
//						v[i + 1] = temp;
//					}
//
//
//				}
//				for (int i = 0; i < v.size(); i++)
//				{
//					for (int j = 0; j < v[i].size(); j++)
//					{
//						cout << v[i][j] << " ";
//					}
//					cout << endl;
//
//				}
//				return;
//			}
//		}
//	}
//}
//void Fun_Game(string x, string y)
//{
//
//	if (x == y)
//	{
//		cout << "YES" << endl;
//		return;
//	}
//	if (x.size() == 1 )
//	{
//		if (x == "0")
//		{
//			cout << "NO" << endl;
//			return;
//		}
//		else
//		{
//			cout << "YES" << endl;
//			return;
//		}
//	}
//
//	int count = -1;
//	bool flag = false;
//	for (int i = 0; i < x.size(); i++)
//	{
//
//		if (x[i] == '1')
//		{
//
//			count=i;
//			break;
//		}
//
//
//	}
//	int c = 0;
//	if (count == -1)
//	{
//		{
//			cout << "NO"<<endl;
//			return;
//		}
//	}
//	for (int i = 0; i < count; i++)
//	{
//		if (x[i] != y[i])
//		{
//			cout << "NO"<<endl;
//			return;
//		}
//	}
//
//	cout << "YES"<<endl;
//	return;
//
//}
//void Mocha_s_Array(vector<long long>v)
//{
//	long long  min1 = LLONG_MAX;
//	long long  min2 = LLONG_MAX;
//	sort(v.begin(), v.end());
//	for (long long i = 0; i < v.size(); i++)
//	{
//		if (v[i] < min1)
//		{
//			//if (min2 % v[i])
//			//{
//			//	min2 = LLONG_MAX;
//			//	min1 = v[i];
//			//}
//			min2 = min1;
//			min1 = v[i];
//		}
//		else if (v[i] > min1 && v[i] < min2&&v[i]%min1!=0)
//		{
//			min2 = v[i];
//		}
//
//	}
//	for (long long i = 0; i < v.size(); i++)
//	{
//		if (v[i] % min1 != 0 && v[i] % min2 != 0)
//		{
//			cout << "NO"<<endl;
//			return;
//		}
//	}
//	cout << "YES" << endl;
//	;
//
//}
//bool hasDuplicate(vector<int>& nums) {
//	unordered_map<int, int>m;
//	for (int i = 0; i < nums.size(); i++)
//	{
//		if (m.find(nums[i]) == m.end())
//		{
//			m[nums[i]] = 1;
//		}
//		else
//			return true;
//
//	}
//	return false;
//}
void Balanced_Shuffle_Easy(string x)
{
	stack<char>s;
	vector<pair<pair<int, int>, char>>v;
	for (int i = 0; i < x.size(); i++)
	{
		if (x[i] == '(')
		{
			s.push(x[i]);
			v.push_back({ { s.size() - 1,i},x[i] });

		}
		else
		{
			v.push_back({ {s.size(),i},x[i] });
			if (!s.empty())
				s.pop();
		}

	}
	sort(v.begin(), v.end(), [](pair<pair<int, int>, char> a, pair<pair<int, int>, char> b) {
		if (a.first.first < b.first.first)
			return true;
		else if (a.first.first > b.first.first)
			return false;
		else
		{
			return a.first.second > b.first.second;
		}
		});

	for (int i = 0; i < v.size(); i++) {
		cout << v[i].second;
	}
}
void Shifts_and_Sorting(string x)
{
	long long count = 0;
	long long result = 0;
	bool flag = false;
	for (long long i = 0; i < x.size(); i++)
	{

		if (x[i] == '1')
			flag = true;
		if (x[i] == '0' && flag == true)
		{
			result += (i + 1 - count);
		}
		if (x[i] == '0')
			count++;
	}
	cout << result << endl;
}
void Assembly_via_Remainders(vector<long long>v)
{
	long long max = LLONG_MIN;
	for (long long i = 0; i < v.size(); i++)
	{
		if (v[i] > max)
			max = v[i];
	}
	max++;
	vector<long long>result;
	result.push_back(max);
	for (long long i = 0; i < v.size(); i++)
	{

		result.push_back(max + v[i]);
		max = v[i] + max;
	}
	for (long long i = 0; i < result.size(); i++)
	{
		cout << result[i] << " ";
	}
	cout << endl;
}
bool isAnagrams(string x, string y)
{
	sort(x.begin(), x.end());
	sort(y.begin(), y.end());
	if (x == y)
		return true;
	else return false;
}
vector<vector<string>> groupAnagrams(vector<string>& strs) {
	vector<vector<string>>result;

	for (int i = 0; i < strs.size(); i++)
	{
		bool flag = false;
		for (int j = 0; j < result.size(); j++)
		{
			if (!result[j].size() == 0 && isAnagrams(result[j][0], strs[i]))
			{
				result[j].push_back(strs[i]);
				flag = true;
			}
		}
		if (flag == false)
		{
			vector<string>x;
			x.push_back(strs[i]);
			result.push_back(x);
		}

	}
	return result;
}

void Progressive_Square(vector<long long>v, long long c, long long d, long long n)
{
	long long min = LLONG_MAX;
	for (int i = 0; i < v.size(); i++)
	{
		if (v[i] < min)
			min = v[i];
	}
	vector<vector<long long>>result(n, vector<long long>(n, 0));
	result[0][0] = min;
	vector<long long >check;
	for (long long i = 0; i < n; i++)
	{
		for (long long j = 0; j < n; j++)
		{
			if (i + 1 < result.size())
			{
				result[i + 1][j] = result[i][j] + c;

			}
			if (j + 1 < result.size())
			{
				result[i][j + 1] = result[i][j] + d;

			}
		}
	}
	for (long long i = 0; i < n; i++)
	{
		for (long long j = 0; j < n; j++)
		{
			check.push_back(result[i][j]);
		}
	}
	sort(check.begin(), check.end());
	sort(v.begin(), v.end());
	if (check == v)
	{
		cout << "YES" << endl;
	}
	else
	{
		cout << "NO" << endl;
	}

}
void Submission_Bait(vector<long long>v)
{
	sort(v.begin(), v.end(), greater<long long>());
	unordered_map<long long, long long>m;
	int count = v[0];
	int c = 0;

	for (long long i = 0; i < v.size(); i++)
	{
		if (v[i] == count)
			c++;
		m[v[i]]++;
	}
	if (m[count] % 2 == 1)
	{
		cout << "YES" << endl;
		return;
	}
	else
	{
		for (int i = c; i < v.size(); i++)
		{
			if (m[v[i]] % 2 == 1)
			{
				cout << "YES" << endl;
				return;
			}
		}
		cout << "NO" << endl;
		return;
	}

}
void Rudolf_and_121(vector<long long>v)
{
	for (long long i = 1; i < v.size() - 1; i++)
	{
		if (v[i - 1] != 0 && float(v[i]) / v[i - 1] >= 2. && v[i - 1] <= v[i + 1])
		{
			v[i] -= v[i - 1] * 2;
			v[i + 1] -= v[i - 1];
			v[i - 1] = 0;
		}
		else if (v[i - 1] == 0)
			continue;
		else
		{
			cout << "NO" << endl;
			return;
		}

	}

	if (v[v.size() - 1] == 0 && v[v.size() - 2] == 0)
	{
		cout << "YES" << endl;
		return;
	}
	else {
		cout << "NO" << endl;
		return;
	}
}
void Make_Equal_Again(vector<long long>v)
{
	int countfirst = 1;
	int countlast = 1;
	set<long long>s;
	for (int i = 0; i < v.size(); i++)
	{
		s.insert(v[i]);
	}
	if (s.size() == 1)
	{
		cout << 0 << endl;
		return;
	}
	bool flagstart = false;
	bool flagend = false;
	for (int i = 1; i < v.size() - 1; i++)
	{
		if (v[i] == v[0] && flagstart == false)
			countfirst++;
		else flagstart = true;

		if (v[v.size() - 1 - i] == v[v.size() - 1] && flagend == false)
			countlast++;
		else flagend = true;
	}
	if (v[0] == v[v.size() - 1])
	{
		cout << v.size() - (countfirst + countlast) << endl;
	}
	else if (countfirst > countlast)
	{
		cout << v.size() - countfirst << endl;

	}
	else
	{
		cout << v.size() - countlast << endl;
	}
}
void  Permutation_Printing(vector<long long>v)
{
	vector<long long>result(v.size(), 0);
	int count = 0;
	if (v.size() % 2 == 0)
	{
		for (int i = v.size() / 2; i < v.size(); i++)
		{
			result[count] = v[i];
			count += 2;
		}
		count = 1;
		for (int i = 0; i < v.size() / 2; i++)
		{
			result[count] = v[i];
			count += 2;
		}
	}
	else
	{
		count = 1;
		for (int i = v.size() / 2 + 1; i < v.size(); i++)
		{
			result[count] = v[i];
			count += 2;
		}
		count = 0;
		for (int i = 0; i < v.size() / 2 + 1; i++)
		{
			result[count] = v[i];
			count += 2;
		}
	}
	for (int i = 0; i < v.size(); i++)
	{
		cout << result[i] << " ";
	}
	cout << endl;

}
void Choose_the_Different_Ones(vector<long long>v, vector<long long>v2, long long n)
{
	set<long long>s1;
	set<long long >s2;
	for (int i = 0; i < v.size(); i++)
	{
		if (v[i] <= n)
			s1.insert(v[i]);
	}
	for (int i = 0; i < v2.size(); i++)
	{
		if (v2[i] <= n)
			s2.insert(v2[i]);
	}
	if (s1.size() < n / 2 || s2.size() < n / 2)
	{
		cout << "NO" << endl;
		return;
	}
	if (s1.size() == n / 2 && s2.size() == n / 2)
	{
		set<long long>target;
		for (auto it = s1.begin(); it != s1.end(); ++it) {
			target.insert(*it);
		}
		for (auto it = s2.begin(); it != s2.end(); ++it) {
			target.insert(*it);
		}

		if (target.size() == n) {
			cout << "YES" << endl;
		}
		else {
			cout << "NO" << endl;
		}
		return;
	}
	int count = 0;
	set<long long>result;
	int size1 = s1.size();
	int size2 = s2.size();
	while (result.size() != n)
	{
		if (s1.empty() && s2.empty())
		{
			cout << "NO" << endl;
			return;
		}
		if (s2.empty())
		{
			result.insert(*s1.begin());
			s1.erase(s1.begin());
			continue;
		}
		if (s1.empty())
		{
			result.insert(*s2.begin());
			s2.erase(s2.begin());
			continue;

		}

		if (*s1.begin() < *s2.begin())
		{
			result.insert(*s1.begin());
			s1.erase(s1.begin());
		}
		else if (*s1.begin() > *s2.begin())
		{
			result.insert(*s2.begin());
			s2.erase(s2.begin());
		}
		else
		{
			result.insert(*s1.begin());
			s1.erase(s1.begin());
			s2.erase(s2.begin());
			count++;
		}
	}
	if (size1 - s1.size() == n / 2 && size2 - s2.size() == n / 2)
	{
		cout << "YES" << endl;
		return;
	}
	if (size1 - s1.size() + count >= n / 2 && size2 - s2.size() + count >= n / 2)
	{
		cout << "YES" << endl;
		return;
	}
	else

	{
		cout << "NO" << endl;
	}

}
vector<int> topKFrequent(vector<int>& nums, int k) {
	unordered_map<int, int>v;
	for (int i = 0; i < nums.size(); i++)
	{
		v[nums[i]]++;
	}
	vector<pair<int, int>> vec(v.begin(), v.end());
	sort(vec.begin(), vec.end(), [](pair<int, int> a, pair<int, int> b) {
		return a.second > b.second;
		});
	vector<int>result;
	for (int i = 0; i < k; i++)
	{
		result.push_back(vec[i].first);
	}
	return result;
}
vector<int> productExceptSelf(vector<int>& nums) {
	int n = nums.size();
	vector<int> result(n, 1);

	int prefix = 1;
	for (int i = 0; i < n; i++) {
		result[i] = prefix;
		prefix = prefix * nums[i];
	}

	int postfix = 1;
	for (int i = n - 1; i >= 0; i--) {
		result[i] = result[i] * postfix;
		postfix = postfix * nums[i];
	}

	return result;
}
void Rudolfand_the_Ticket(vector<long long>v, vector<long long >v2, long long k)
{
	vector<pair<long long, long long>>result;
	for (int i = 0; i < v.size(); i++)
	{
		for (int j = 0; j < v2.size(); j++)
		{
			if (v[i] + v2[j] <= k)
				result.push_back({ i + 1,j + 1 });
		}
	}
	cout << result.size() << endl;
}
long long MAD(vector<long long>v, long long end)
{
	unordered_map<long long, long long>m;
	for (int i = 0; i <= end; i++)
	{
		m[v[i]]++;
	}
	vector<pair<int, int>> vec(m.begin(), m.end());
	sort(vec.begin(), vec.end(), [](pair<int, int> a, pair<int, int> b) {
		if (a.second > b.second)
			return true;
		else if (a.second < b.second)
			return false;
		else
		{
			return a.first > b.first;
		}
		});
	if (vec[0].second == 1)
		return 0;
	return vec[0].first;
}
void Mad_MAD_Sum(vector<long long>v)
{
	long long sum = 0;
	vector<long long>v2 = v;
	while (*max_element(v.begin(), v.end()) != 0)
	{

		int sum1 = 0;
		for (long long i = 0; i < v.size(); i++)
		{
			sum1 += v[i];
		}
		sum += sum1;

		for (long long i = 0; i < v.size(); i++)
		{
			long long val = MAD(v, i);

			v2[i] = val;

		}
		v = v2;
	}
	cout << sum << endl;

}
void Diagonals(long long n, long long k)
{
	long long numofdiagonals = 1;
	if (k == 0)
	{
		cout << 0 << endl;
		return;
	}
	k -= n;
	if (k <= 0)
	{
		cout << 1 << endl;
	}
	else
	{
		for (long long i = n - 1; i > 0; i--)
		{
			k -= i;
			numofdiagonals++;
			if (k <= 0)
			{
				cout << numofdiagonals << endl;
				return;
			}
			k -= i;
			numofdiagonals++;
			if (k <= 0)
			{
				cout << numofdiagonals << endl;
				return;
			}

		}
	}
}
struct pair_hash {
	template <class T1, class T2>
	std::size_t operator() (const std::pair<T1, T2>& pair) const {
		return std::hash<T1>()(pair.first) ^ std::hash<T2>()(pair.second);
	}
};

void Bouquet_Easy_Version(vector<long long> v, long long m) {
	unordered_map<long long, long long> m1;
	for (long long i = 0; i < v.size(); i++) {
		m1[v[i]]++;
	}

	vector<pair<long long, long long>> vec(m1.begin(), m1.end());
	long long sum = 0;

	sort(vec.begin(), vec.end(), [](pair<long long, long long> a, pair<long long, long long> b) {
		return a.first < b.first;
		});

	unordered_map<pair<long long, long long>, long long, pair_hash> result;
	for (long long i = 0; i < vec.size() - 1; i++) {
		if (abs(vec[i].first - vec[i + 1].first) == 1) {
			result[{vec[i].first, vec[i + 1].first}] = vec[i].second * vec[i].first + vec[i + 1].first * vec[i + 1].second;
		}
		else if (i != vec.size() - 2) {
			result[{vec[i].first, vec[i].first}] = vec[i].second * vec[i].first;
		}
		else {
			result[{vec[i].first, vec[i].first}] = vec[i].second * vec[i].first;
			result[{vec[i + 1].first, vec[i + 1].first}] = vec[i + 1].second * vec[i + 1].first;
		}
	}

	long long max1 = LLONG_MIN;
	for (const auto& pair : result) {
		if (pair.second <= m && pair.second > max1) {
			max1 = pair.second;
		}
		else if (pair.second > m) {
			int diff = pair.second - m;
			long long num1 = m1[min(pair.first.first, pair.first.second)];

			// Subtract the largest possible value less than or equal to diff
			while (diff >= min(pair.first.first, pair.first.second) && num1 != 0) {
				if (diff % max(pair.first.first, pair.first.second) == 0 && diff / max(pair.first.first, pair.first.second) <= m1[max(pair.first.first, pair.first.second)])
					diff = 0;
				diff -= min(pair.first.first, pair.first.second);
				num1--;

			}
			m1[min(pair.first.first, pair.first.second)] = num1;  // Update the count in the map

			if (max(pair.first.first, pair.first.second) != min(pair.first.first, pair.first.second)) {
				num1 = m1[max(pair.first.first, pair.first.second)];
				while (diff >= max(pair.first.first, pair.first.second) && num1 != 0) {
					diff -= max(pair.first.first, pair.first.second);
					num1--;
				}
				m1[max(pair.first.first, pair.first.second)] = num1;  // Update the count in the map
			}
			long long val = diff + m;
			if (val > max1 && val <= m)
				max1 = val;
		}
	}
	cout << max1 << endl;
}
void generate(int& n, vector<string >& x, string v, int open, int closed, stack <char>s)
{
	if (v.size() == n * 2)
	{
		x.push_back(v);
		return;
	}
	if (v.empty())
	{
		s.push('(');
		v.push_back('(');
		generate(n, x, v, open + 1, closed, s);
		s.pop();
	}
	else
	{
		if (open >= closed && open < n)
		{
			v.push_back('(');
			s.push('(');
			generate(n, x, v, open + 1, closed, s);
			v.pop_back();
			s.pop();
			if (!s.empty())
			{
				s.pop();
				v.push_back(')');
				generate(n, x, v, open, closed + 1, s);
				v.pop_back();
				s.push('(');
			}
		}
		else if (closed < n)
		{
			s.pop();
			v.push_back(')');
			generate(n, x, v, open, closed + 1, s);
			v.pop_back();
			s.push('(');
		}

	}

}
vector<string> generateParenthesis(int& n) {
	vector<string>x;
	string v;
	stack<char>s;
	generate(n, x, v, 0, 0, s);
	return x;

}
int lengthOfLongestSubstring(string s) {
	unordered_set<char> charSet;
	int l = 0, maxLength = 0;

	for (int r = 0; r < s.size(); r++) {
		// If duplicate character found, move left pointer until it's removed
		while (charSet.find(s[r]) != charSet.end()) {
			charSet.erase(s[l]);
			l++;
		}
		charSet.insert(s[r]);
		maxLength = max(maxLength, r - l + 1);
	}

	return maxLength;
}
void Legs(long long n)
{
	long long result;
	if (n % 4 == 0)
		result = n / 4;
	else
	{
		result = n / 4;
		result += (n % 4) / 2;
	}
	cout << result << endl;
}
void Scale(vector<vector<long long>>v, long long k)
{
	vector<vector<long long>>result(v.size() / k, vector<long long>(v.size() / k));
	for (long long i = 0; i < v.size(); i += k)
	{
		for (long long j = 0; j < v.size(); j += k)
		{
			result[i / k][j / k] = v[i][j];
		}
	}
	for (long long i = 0; i < result.size(); i++)
	{
		for (long long j = 0; j < result.size(); j++)
		{
			cout << result[i][j];
		}
		cout << endl;
	}
}
void Sort()
{

	long long n, k;

	cin >> n >> k;
	string a, b; cin >> a >> b;

	map<char, vector<int>>ma;
	map<char, vector<int>>mb;

	for (int i = 0; i < a.size(); ++i) {
		for (char j = 'a'; j <= 'z'; ++j) {
			if (ma[j].empty())ma[j].push_back(0);
			else {
				ma[j].push_back(ma[j].back());
			}
			if (j == a[i])ma[j][ma[j].size() - 1]++;
		}
	}for (int i = 0; i < b.size(); ++i) {
		for (char j = 'a'; j <= 'z'; ++j) {
			if (mb[j].empty())mb[j].push_back(0);
			else {
				mb[j].push_back(mb[j].back());
			}
			if (j == b[i])mb[j][mb[j].size() - 1]++;
		}
	}
	while (k--) {
		int l, r; cin >> l >> r;
		l--, r--;
		long long ans{ 0 };
		long long A{ 0 }, B{ 0 };
		for (char j = 'a'; j <= 'z'; ++j) {
			A = ma[j][r] - (l > 0 ? ma[j][l - 1] : 0);
			B = mb[j][r] - (l > 0 ? mb[j][l - 1] : 0);
			ans += abs(A - B);
		}
		cout << ans / 2 << endl;
	}

}
void Fun(long long n, long long x)
{
	long long count = 0;
	for (long long i = 1; i < x; i++)
	{
		for (long long j = 1; j < x; j++)
		{
			for (long long k = 1; k < x; k++)
			{
				if (i + j + k <= x && (i * j + i * k + j * k) <= n)
					count++;

			}
		}
	}
	cout << count << endl;
}
void merge(vector<int>& nums1, int m, vector<int>& nums2, int n) {
	int i = 0;
	int j = 0;
	int count = 0;
	if (m == 0)
	{
		nums1 = nums2;
		return;
	}
	while (i != n)
	{
		if (j == m)
		{

			for (int k = m; k < n + m - count; k++)
			{
				nums1[k] = nums2[i];
				i++;
			}
		}

		else if (nums2[i] < nums1[j])
		{
			nums1.insert(nums1.begin() + j, nums2[i]);
			j++;
			m++;
			count++;
			i++;
			nums1.erase(nums1.end() - 1);
		}
		else j++;
	}
}
int removeElement(vector<int>& nums, int val) {

	int count = 0;
	vector<int>v;
	for (int i = 0; i < nums.size(); i++)
	{
		if (nums[i] == val)
		{
			count++;

		}
		else v.push_back(nums[i]);

	}
	nums = v;
	return count;
}
void Maximize_the_Last_Element(vector<long long>v)
{
	long long max = LLONG_MIN;
	for (long long i = 0; i < v.size(); i += 2)
	{
		if (v[i] > max)
			max = v[i];
	}
	cout << max << endl;
}
#include <bitset>

// Function to convert long long to binary string
std::string longLongToBinary(long long num) {
	std::string binary = std::bitset<64>(num).to_string();

	// Removing leading zeros
	const auto pos = binary.find('1');
	if (pos != std::string::npos) {
		return binary.substr(pos);
	}

	// If the number is zero
	return "0";
}

// Function to convert binary string to long long
long long binaryToLongLong(const std::string& binary) {
	return std::bitset<64>(binary).to_ullong();
}
void AND_Reconstruction(vector<long long>v)
{
	vector<long long>result;
	result.push_back(v[0]);
	long long max = LLONG_MIN;
	for (long long i = 0; i < v.size(); i++)
	{
		if (v[i] > max)
			max = v[i];
	}
	string maximum = longLongToBinary(max);
	for (long long i = 0; i < maximum.size(); i++)
	{
		maximum[i] = '1';
	}
	//result.push_back(binaryToLongLong(maximum));
	bool flag = true;
	string last = longLongToBinary(v[0]);
	string stemp = "";
	if (last.size() < maximum.size())
	{
		for (long long i = last.size(); i < maximum.size(); i++)
			stemp += "0";
		last = stemp + last;
	}
	bool flag2 = false;
	for (long long i = 1; i < v.size(); i++)
	{
		string current = longLongToBinary(v[i]);

		string temp = "";

		//last = current;
		if (current.size() < maximum.size())
		{
			for (long long i = current.size(); i < maximum.size(); i++)
				temp += "0";
			current = temp + current;
		}
		for (long long j = 0; j < current.size(); j++)
		{
			if (last[j] == '0' && current[j] == '0')
				current[j] = '1';
			else if (current[j] == '1' && last[j] == '0' && flag == false)
			{
				cout << -1 << endl;
				return;
			}
			else if (current[j] == '1' && last[j] == '0' && flag == true)
			{
				result.push_back(binaryToLongLong(maximum));
				flag = false;
				i--;
				last = maximum;
				flag2 = true;
				break;
			}
		}
		if (flag2 == false)
		{
			result.push_back(binaryToLongLong(current));
			last = current;
		}
		else flag2 = false;

	}
	for (long long i = 0; i < result.size(); i++)
	{
		cout << result[i] << " ";
	}
	if (result.size() == v.size())
		cout << binaryToLongLong(maximum);
	cout << endl;
}
void Absolute_Zero(vector<long long>v)
{
	vector<long long>result;
	long long sum = 0;
	for (long long i = 0; i < v.size(); i++)
	{
		sum += v[i];
	}
	int count = sum / v.size();
	if (sum == 0)
	{
		cout << 0 << endl;
		return;
	}

	for (int i = 0; i < 40; i++)
	{
		sum = 0;
		result.push_back(count);
		for (long long i = 0; i < v.size(); i++)
		{
			v[i] = abs(v[i] - count);
			sum += v[i];
		}

		count = ceil(float(sum) / v.size());
		if (count == 0 && sum != 0)
		{
			cout << -1 << endl;
			return;
		}
		if (sum == 0)
		{
			cout << result.size() << endl;
			for (long long i = 0; i < result.size(); i++)
			{
				cout << result[i] << " ";
			}
			cout << endl;
			return;
		}

	}
	cout << -1 << endl;
	return;
}

void Update_Queries(string x, string y, vector<long long>v)
{
	string temp = y;
	sort(y.begin(), y.end());
	sort(v.begin(), v.end());
	char min = y[0];
	int j = 0;
	int count = 0;

	// Count the number of consecutive 1's from the start of vector v
	while (j < v.size() && v[j] == 1) {
		count++;
		j++;
	}
	count--;
	j--;
	// Check if count is within the bounds of y
	if (count < y.size()) {
		// Swap the character at index 'count' with the first character

		y[0] = y[count];
		y[count] = min;
	}
	bool flag = false;
	for (long long i = 0; i < v.size(); i++)
	{
		x[v[i] - 1] = y[i];

	}

	cout << x << endl;
}
std::vector<int> splitDigits(long long num) {
	std::vector<int> digits;
	while (num > 0) {
		digits.push_back(num % 10);
		num /= 10;
	}
	// Reverse the vector to have the most significant digit at the beginning
	std::reverse(digits.begin(), digits.end());
	return digits;
}

//void Large_Addition(long long x)
//{
//	bool flag = false;
//	// 5 6 7 8 9
//
//	// 5 6 7 8 9
//	while (x >=10)
//	{
//		long long result = x % 10;
//		result += 10;
//		result -= flag;
//		if (result < 10||result==19)
//		{
//			cout << "NO" << endl;
//			return;
//		}
//		flag = true;
//		x/= 10;
//	}
//	if (x == 1)
//		cout << "YES" << endl;
//	else cout << "NO" << endl;
//}
void  Binary_Colouring(long long n)
{

	vector<long long>result;
	int size = 0;
	while (n != 0)
	{
		if (n % 2 == 1)
		{
			result.push_back(pow(2, size));
		}
		else result.push_back(0);
		size++;
		n /= 2;
	}
	//reverse(result.begin(), result.end());
	for (int i = 0; i < result.size() - 1; i++)
	{
		if (result[i] != 0 && result[i + 1] != 0)
		{
			result[i] = -1;
			i++;
			while (i < result.size() && result[i] != 0)
			{
				result[i] = 0;
				i++;
			}

			if (i == result.size())
				result.push_back(1);
			else
			{
				result[i] = pow(2, i);
				i--;
			}
		}
		else if (result[i] != 0)
			result[i] = (1);
	}
	if (result[result.size() - 1] > 1)
		result[result.size() - 1] = 1;
	cout << result.size() << endl;
	for (int i = 0; i < result.size(); i++)
	{
		cout << result[i] << " ";
	}
	cout << endl;
}
int removeDuplicates(vector<int>& nums) {
	unordered_map<int, int>m;
	for (int i = 0; i < nums.size(); i++)
	{
		if (m[nums[i]] < 2)
			m[nums[i]]++;
	}
	nums.clear();
	for (auto it = m.begin(); it != m.end(); ++it)
	{

		if (it->second == 2)
		{
			nums.push_back(it->first);
			nums.push_back(it->first);
		}
		else
		{
			nums.push_back(it->first);
		}
	}
	return nums.size();
}
void letterCombinations2(vector<int>v, vector<string>& result, string x, int index = 0)
{
	if (index == v.size())
	{
		result.push_back(x);
		x.clear();

		return;
	}

}
void combinationSum22(int target, set<multiset<int>>& result, int index, vector<int>candidates, multiset<int>v, set<int>indexes)
{
	if (target == 0)
	{
		result.insert(v);
		return;
	}
	if (index == candidates.size())
		return;

	for (index; index < candidates.size(); index++)
	{
		if (target == 0)
		{
			result.insert(v);
			return;
		}
		if (candidates[index] <= target)
		{
			//int sizebefore = indexes.size();
			//indexes.insert(i);
			//if (indexes.size() > sizebefore)
			{
				v.insert(candidates[index]);

				combinationSum22(target - candidates[index], result, index + 1, candidates, v, indexes);

				auto it = v.find(candidates[index]); // Find the first occurrence of candidates[i]
				if (it != v.end()) {
					v.erase(it); // Erase only the first occurrence
				}

				//if (!indexes.empty()) {
				//	// Erase the specific element i
				//	indexes.erase(i);
				//}
			}
		}
		else return;
	}

}
void Klee_s_SUPER_DUPER_LARGE_Array(long long n, long long k)
{
	long long first = n;
	long long last = n+k-1;
	long long mid = (first + last) / 2;
	long long min = LLONG_MAX;
	while(first<last)
	{

		long long val = ((n + k - 1 + 1) * (n + k - 1) / 2 - (mid + 1) * mid / 2) - ((mid + 1) * mid / 2 - n * (n - 1) / 2);
		if (val < min&&val>0 )
		{
			min = val;
			first = mid;
			mid = (first + last) / 2;
		}
		else
		{
			last = mid;
			mid = (first + last) / 2;
		}

	}
	 first = n;
	 last = n + k - 1;
	 mid = (first + last) / 2;
	 long long min2 = LLONG_MAX;
	while (first < last)
	{

		long long val = abs(((mid + 1) * mid / 2 - n * (n - 1) / 2) - ((n + k - 1 + 1) * (n + k - 1) / 2 - (mid + 1) * mid / 2));
		if (val < min2 )
		{
			min2 = val;
			first = mid;
			mid = (first + last) / 2;
		}
		else
		{
			last = mid;
			mid = (first + last) / 2;
		}

	}
	if (min < min2)
		cout << min<<endl;
	else cout << min2<<endl;
}
int main() {
	long long num;
	cin >> num;
	for(long long i=0;i<num;i++)
	{
		long long k, n;
		cin >> k >> n;
		Klee_s_SUPER_DUPER_LARGE_Array(n, k);
	}
	return 0;
}

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

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

Think step by step.

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


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Klee's SUPER DUPER LARGE Array!!!
time_limit_ms: 2000
memory_limit_mb: 256
Description: Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.

Output the minimum possible value of $$$x$$$.

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

Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.

Output: For each test case, output the minimum value of $$$x$$$ on a new line.

Examples:
input:
4
2 2
7 2
5 3
1000000000 1000000000
output:
1
5
1
347369930

Note: In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

Buggy Code:
#include "bits/stdc++.h"
#define fast ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
#define ll long long
#define pb push_back
#define in insert
#define F first
#define S second
#define vll vector<ll>
#define all(v) v.begin(),v.end()
#define endl '\n'
#define pii pair<ll,ll>
using namespace std;
const ll INF =1e18, mod = 1e9 + 7, N = 2e5 + 5;
ll gcd(ll a, ll b){
 if(b == 0){
 return a;
 }
 return gcd(b, a%b);
}
ll lcm(ll a, ll b){
    return (a / gcd(a,b)) * b;
}
ll modpow(ll a, ll b){
    ll ans = 1;
    while(b){
        if(b % 2 == 1){
            ans *= a;
            ans %= mod;
        }
        b /= 2;
        a *= a;
        a %= mod;
    }
    return ans;
}
ll n,k;
ll f(ll l, ll r){
    return (((r * (r + 1)) / 2) - (l * (l - 1) / 2));
}
ll dif(ll a){
    return abs(f(k,a) - f(a + 1, n + k - 1));
}
void solve(){
    cin >> n >> k;
    //cout << f(3,6) << endl;
    ll l = k, r = n + k - 1;
    ll ans = INF;
    while(l < r){
        ll mid = (l + r) / 2;
        if(dif(mid) < dif(mid + 1)){
            r = mid;
        }
        else if(dif(mid) > dif(mid + 1)){
            l = mid + 1;
        }
        else{
            break;
        }
    }
   // cout << l << endl;
    cout << dif(l) << endl;
}
int main(){
    fast;
    ll 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:
To generate a challenging test input that exposes the bug in the given buggy code, we need to consider edge cases and scenarios that maximize the failure probability. Here are the steps to think through this:

1. **Understand the Bug**: The bug in the code seems to be related to the binary search mechanism used to find the optimal index `i`. The `dif` function calculates the absolute difference between the sum of the first `i` elements and the sum of the remaining elements. The binary search is supposed to find the minimum value of this difference, but it might not always converge correctly, especially for large values of `n` and `k`.

2. **Edge Cases**:
   - **Small Values**: Test with small values of `n` and `k` to ensure the basic logic works.
   - **Large Values**: Test with large values of `n` and `k` to see if the binary search converges correctly.
   - **Boundary Values**: Test with boundary values like `n = 2` and `k = 10^9` or `n = 10^9` and `k = 2`.

3. **Challenging Scenarios**:
   - **Equal Sums**: Test cases where the sum of the first `i` elements and the sum of the remaining elements are close to each other.
   - **Symmetric Arrays**: Test cases where the array is symmetric around the middle.

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

```plaintext
4
2 2
7 2
5 3
1000000000 1000000000
```

This test input includes:
- A small case (`2 2`) to check basic functionality.
- A medium case (`7 2`) to check intermediate values.
- Another small case (`5 3`) to check different starting points.
- A large case (`1000000000 1000000000`) to stress-test the binary search and ensure it handles large values correctly.

This should help expose any issues in the binary search logic and the `dif` function.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Klee's SUPER DUPER LARGE Array!!!
time_limit_ms: 2000
memory_limit_mb: 256
Description: Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.

Output the minimum possible value of $$$x$$$.

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

Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.

Output: For each test case, output the minimum value of $$$x$$$ on a new line.

Examples:
input:
4
2 2
7 2
5 3
1000000000 1000000000
output:
1
5
1
347369930

Note: In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

Buggy Code:
import sys
input = sys.stdin.readline
def fnc (p,n,k):
    ans = p*(2*k + p - 1) - (n*(2*k + n -1))//2
    return ans
def solve():
    n,k = map(int,input().split())
    low = 1
    high = n
    mid = 0
    ans1 = []
    while low <= high:
        mid = (low + high) // 2
        fnc_mid = fnc(mid, n, k)
        ans1.append(abs(fnc(mid-1, n, k)))
        ans1.append(abs(fnc_mid))
        ans1.append(abs(fnc(mid+1, n, k)))
        if fnc_mid == 0:
            return

        if fnc_mid < 0:
            low = mid+1
        else:
            high = mid -1
    ans1.append(abs(fnc(n//2, n, k)))
    print(min(ans1))

for _ in range(int(input())):
    solve()

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

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

Think step by step.

Response:
To generate a challenging test input that exposes the bug in the provided buggy code, we need to consider edge cases and scenarios that can maximize the failure probability. Let's break down the requirements and analyze the code:

1. **Edge Cases:**
   - Small values of `n` and `k` to test basic functionality.
   - Large values of `n` and `k` to test performance and correctness with large inputs.
   - Values of `n` and `k` that can cause integer overflow or precision issues.
   - Values of `n` and `k` that can lead to incorrect partitioning in the binary search.

2. **Critical Points:**
   - The formula used in the `fnc` function: `p*(2*k + p - 1) - (n*(2*k + n -1))//2`.
   - The binary search logic in the `solve` function, particularly the conditions for updating `low` and `high`.

3. **Challenging Test Input:**
   - We need to create a test case where the binary search logic fails to find the correct minimum value of `x`.

Let's consider the following test case:
- `n = 3` and `k = 1000000000` (large `k` and small `n`).
- This will test the precision and correctness of the formula and the binary search logic.

Here is the test input:
```plaintext
1
3 1000000000
```

### Explanation:
- **Test Case:**
  - `n = 3`: The array will be `[1000000000, 1000000001, 1000000002]`.
  - `k = 1000000000`: Large starting value to test precision and overflow issues.

- **Expected Output:**
  - The minimum value of `x` should be `1` when `i = 2` (i.e., `x = |1000000000 + 1000000001 - 1000000002| = 1`).

This test case should expose any issues in the binary search logic or the formula used in the `fnc` function, particularly with large values of `k`.


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Klee's SUPER DUPER LARGE Array!!!
time_limit_ms: 2000
memory_limit_mb: 256
Description: Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.

Output the minimum possible value of $$$x$$$.

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

Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.

Output: For each test case, output the minimum value of $$$x$$$ on a new line.

Examples:
input:
4
2 2
7 2
5 3
1000000000 1000000000
output:
1
5
1
347369930

Note: In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

Buggy Code:
def fun(mid):
    v = (k+mid+k-1)*mid //2
    v1 = (k+n+k-1)*n//2-v
    return v,v1

for _ in range(int(input())):
    n,k=map(int,input().split())
    l,r = 1,n
    curr = 1
    while l < r:
        mid = (l+r)//2
        a,b=fun(mid)
        if b > a:
            curr = mid
            l = mid+1
        else:
            r = mid-1
    a1,b1 = fun(curr)
    a2,b2 = fun(curr+1)
    print(min(abs(a1-b1) , abs(a2-b2)))

**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
1000000000 1
```


**Task:**
Generate a challenging test input that exposes the bug in the buggy code of the algorithm problem:
Algorithm Problem:
Title: Klee's SUPER DUPER LARGE Array!!!
time_limit_ms: 2000
memory_limit_mb: 256
Description: Klee has an array $$$a$$$ of length $$$n$$$ containing integers $$$[k, k+1, ..., k+n-1]$$$ in that order. Klee wants to choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) such that $$$x = |a_1 + a_2 + \dots + a_i - a_{i+1} - \dots - a_n|$$$ is minimized. Note that for an arbitrary integer $$$z$$$, $$$|z|$$$ represents the absolute value of $$$z$$$.

Output the minimum possible value of $$$x$$$.

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

Each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \leq n, k \leq 10^9$$$) — the length of the array and the starting element of the array.

Output: For each test case, output the minimum value of $$$x$$$ on a new line.

Examples:
input:
4
2 2
7 2
5 3
1000000000 1000000000
output:
1
5
1
347369930

Note: In the first sample, $$$a = [2, 3]$$$. When $$$i = 1$$$ is chosen, $$$x = |2-3| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

In the third sample, $$$a = [3, 4, 5, 6, 7]$$$. When $$$i = 3$$$ is chosen, $$$x = |3 + 4 + 5 - 6 - 7| = 1$$$. It can be shown this is the minimum possible value of $$$x$$$.

Buggy Code:
import java.io.BufferedInputStream;import java.io.File;import java.io.FileInputStream;import java.io.InputStream;import java.util.*;import static java.lang.System.out;import java.io.*;import java.util.Arrays;

public class E {
    static String INPUT = """
      4
      2 2
      7 2
      5 3
      1000000000 1000000000
            """;

    static boolean oj = System.getProperty("ONLINE_JUDGE") != null;    static InputStream is = oj ? System.in : new ByteArrayInputStream(INPUT.getBytes());    static FastScanner sc = new FastScanner(is);

        public static void main(String[] args) {
            int t = sc.nextInt();
            outer:
            while(t-- > 0){
                long n = sc.nextInt();
    //            int[] arr = readIntArray(sc, n);
                long k = sc.nextInt();

    //            int n = ;'
                long nn = n;
                long ans = Long.MAX_VALUE;

                n = k + n - 1;

                long start = 0, end = nn - 1;
                while (start <= end) {
                    long mid = start + (end - start) / 2;
                  long center= Math.abs(getSum(k, n - mid - 1) - getSum(n - mid, n));
                  long left = mid > 1 ? Math.abs(getSum(k, n - mid) - getSum(n - mid + 1, n)) :
                      Long.MAX_VALUE;
                    long right = mid < nn - 1 ?
                      Math.abs(getSum(k, n - mid - 2) - getSum(n - mid - 1,
                      n)) : Long.MAX_VALUE;
                    ans = Math.min(ans, center);
                    if (left < center) {
                        end = mid - 1;
                    } else if (right < center) {
                        start = mid + 1;
                    } else {
                        break;
                    }
                }
                out.println(ans);
            }
        }

        private static long getSum(long num1, long num2) {
            long sum = num2 * (num2 + 1) / 2 - num1 * (num1 + 1) / 2 + num1;
            return sum;
        }
//
//        if (347369930 == ans) {
//        out.println(i);
//        break;
//    }

    private static int[] readIntArray(FastScanner sc, int n) {        int[] arr = new int[n];        for (int i = 0; i < n; i++) arr[i] = sc.nextInt();        return arr;    }    private static long[] readLongArray(FastScanner sc, int n) {        long[] arr = new long[n];        for (int i = 0; i < n; i++) arr[i] = sc.nextLong();        return arr;    } private static String[] readStringArray(FastScanner sc, int n) {        String[] arr = new String[n];        for (int i = 0; i < n; i++) arr[i] = sc.next();        return arr;    } private static int[][] read2dIntArray(FastScanner sc, int n, int m) { int[][] arr = new int[n][m]; for (int i = 0; i < n; i++) { arr[i] = readIntArray(sc, m); } return arr; } private static long[][] read2dLongArray(FastScanner sc, int n, int m) { long[][] arr = new long[n][m]; for (int i = 0; i < n; i++) { arr[i] = readLongArray(sc, m); } return arr; } private static String[][] read2dStringArray(FastScanner sc, int n, int m) { String[][] arr = new String[n][m]; for (int i = 0; i < n; i++) { arr[i] = readStringArray(sc, m); } return arr; }
    private static void yes() {out.println("YES");}private static void no() {out.println("NO");}
    public static void print(int[] arr) {        for (int x : arr)            out.print(x + " ");        out.println();    }    public static void print(long[] arr) {        for (long x : arr)            out.print(x + " ");        out.println();    }    public static void print(String[] arr) {        for (String x : arr)            out.print(x + " ");        out.println();    }
    public static long totient(long n) {        long result = n;        for (int p = 2; (long) p * p <= n; ++p)            if (n % p == 0) {                while (n % p == 0)                    n /= p;                result -= result / p;            }        if (n > 1)            result -= result / n;        return result;        /*        find phi(i) from 1 to N fast        O(N*loglogN)        long[] arr = new long[N+1];        for(int i=1; i <= N; i++)            arr[i] = i;        for(int v=2; v <= N; v++)            if(arr[v] == v)                for(int a=v; a <= N; a+=v)                    arr[a] -= arr[a]/v;         */    }
    public static ArrayList<Integer> findDiv(int N) {                ArrayList<Integer> ls1 = new ArrayList<Integer>();        ArrayList<Integer> ls2 = new ArrayList<Integer>();        for (int i = 1; i <= (int) (Math.sqrt(N) + 0.00000001); i++)            if (N % i == 0) {                ls1.add(i);                ls2.add(N / i);            }        Collections.reverse(ls2);        for (int b : ls2)            if (b != ls1.get(ls1.size() - 1))                ls1.add(b);        return ls1;    } public static ArrayList<Long> findDiv(long N) {        ArrayList<Long> ls1 = new ArrayList<Long>();        ArrayList<Long> ls2 = new ArrayList<Long>();        for (long i = 1; i <= (long) (Math.sqrt(N) + 0.00000001); i++)            if (N % i == 0) {                ls1.add(i);                ls2.add(N / i);            }        Collections.reverse(ls2);        for (long b : ls2)            if (b != ls1.get(ls1.size() - 1))                ls1.add(b);        return ls1;    }
    public static ArrayList<Integer> findPrimeFactors(int N) {    ArrayList<Integer> ls1 = new ArrayList<Integer>();    int n = N;    if (n % 2 == 0) {      ls1.add(2);    }    while (n%2==0)    {      n /= 2;    }    for (int i = 3; i <= Math.sqrt(n); i+= 2)    {      if (n % i == 0) {        ls1.add(i);      }      while (n%i == 0)      {        n /= i;      }    }    if (n > 2)      ls1.add(n);    return ls1;  }  public static ArrayList<Long> findPrimeFactors(long N) {    ArrayList<Long> ls1 = new ArrayList<>();    long n = N;    if (n % 2 == 0) {      ls1.add(2L);    }    while (n%2==0)    {      n /= 2;    }    for (long i = 3; i <= Math.sqrt(n); i+= 2)    {      if (n % i == 0) {        ls1.add(i);      }      while (n%i == 0)      {        n /= i;      }    }    if (n > 2)      ls1.add(n);    return ls1;  }
    public static void sort(int[] arr) {         ArrayList<Integer> ls = new ArrayList<Integer>();        for (int x : arr)            ls.add(x);        Collections.sort(ls);        for (int i = 0; i < arr.length; i++)            arr[i] = ls.get(i);    }        public static void sort(long[] arr) {          ArrayList<Long> ls = new ArrayList<Long>();        for (long x : arr)            ls.add(x);        Collections.sort(ls);        for (int i = 0; i < arr.length; i++)            arr[i] = ls.get(i);    }    public static void sort(String[] arr) {        ArrayList<String> ls = new ArrayList<String>();        for (String x : arr)            ls.add(x);        Collections.sort(ls);        for (int i = 0; i < arr.length; i++)            arr[i] = ls.get(i);    } public static void reverseSort(int[] arr) { ArrayList<Integer> ls = new ArrayList<Integer>(); for (int x : arr) ls.add(x); Collections.sort(ls, Collections.reverseOrder()); for (int i = 0; i < arr.length; i++) arr[i] = ls.get(i); } public static void reverseSort(long[] arr) { ArrayList<Long> ls = new ArrayList<Long>(); for (long x : arr) ls.add(x); Collections.sort(ls, Collections.reverseOrder()); for (int i = 0; i < arr.length; i++) arr[i] = ls.get(i); } public static void reverseSort(String[] arr) { ArrayList<String> ls = new ArrayList<String>(); for (String x : arr) ls.add(x); Collections.sort(ls, Collections.reverseOrder()); for (int i = 0; i < arr.length; i++) arr[i] = ls.get(i); }
    public static long power(long x, long y, long p) {        long res = 1L;        x = x % p;        while (y > 0) {            if ((y & 1) == 1)                res = (res * x) % p;            y >>= 1;            x = (x * x) % p;        }        return res;    }
    public static void push(HashMap<Integer, Integer> map, int k, int v) {        if (!map.containsKey(k))            map.put(k, v);        else            map.put(k, map.get(k) + v);    }    public static void pull(HashMap<Integer, Integer> map, int k, int v) {        int lol = map.get(k);        if (lol == v)            map.remove(k);        else            map.put(k, lol - v);    }    public static void push(HashMap<String, Integer> map, String k, int v) {        if (!map.containsKey(k))            map.put(k, v);        else            map.put(k, map.get(k) + v);    }    public static void pull(HashMap<String, Integer> map, String k, int v) {        int lol = map.get(k);        if (lol == v)            map.remove(k);        else            map.put(k, lol - v);    }    public static void push(HashMap<Long, Integer> map, Long k, int v) {        if (!map.containsKey(k))            map.put(k, v);        else            map.put(k, map.get(k) + v);    }    public static void pull(HashMap<Long, Integer> map, Long k, int v) {        int lol = map.get(k);        if (lol == v)            map.remove(k);        else            map.put(k, lol - v);    }
    public static int[] compress(int[] arr) {        ArrayList<Integer> ls = new ArrayList<Integer>();        for (int x : arr)            ls.add(x);        Collections.sort(ls);        HashMap<Integer, Integer> map = new HashMap<Integer, Integer>();        int boof = 1; /* min value */        for (int x : ls)            if (!map.containsKey(x))                map.put(x, boof++);        int[] brr = new int[arr.length];        for (int i = 0; i < arr.length; i++)            brr[i] = map.get(arr[i]);        return brr;    }
    public static long[][] multiply(long[][] left, long[][] right) {        long MOD = 1000000007L;        int N = left.length;        int M = right[0].length;        long[][] res = new long[N][M];        for (int a = 0; a < N; a++)            for (int b = 0; b < M; b++)                for (int c = 0; c < left[0].length; c++) {                    res[a][b] += (left[a][c] * right[c][b]) % MOD;                    if (res[a][b] >= MOD)                        res[a][b] -= MOD;                }        return res;    }    public static long[][] power(long[][] grid, long pow) {        long[][] res = new long[grid.length][grid[0].length];        for (int i = 0; i < res.length; i++)            res[i][i] = 1L;        long[][] curr = grid.clone();        while (pow > 0) {            if ((pow & 1L) == 1L)                res = multiply(curr, res);            pow >>= 1;            curr = multiply(curr, curr);        }        return res;    }
    public static long sqrt(long x) {long start = 0, end = (long) 3e9, ans = 1; while (start <= end) {long mid = (start + end) / 2; if (mid * mid <= x) {ans = mid;start = mid + 1;} else end = mid - 1;} return ans;}
    private static class FastScanner {              private final int BS = 1 << 16;        private final char NC = (char) 0;        private final byte[] buf = new byte[BS];        private int bId = 0, size = 0;        private char c = NC;        private double cnt = 1;        private BufferedInputStream in;        public FastScanner() {            in = new BufferedInputStream(System.in, BS);        }  public FastScanner(InputStream ff) {            in = new BufferedInputStream(ff);        }        public FastScanner(String s) {            try {                in = new BufferedInputStream(new FileInputStream(new File(s)), BS);            } catch (Exception e) {                in = new BufferedInputStream(System.in, BS);            }        }        private char getChar() {            while (bId == size) {                try {                    size = in.read(buf);                } catch (Exception e) {                    return NC;                }                if (size == -1) return NC;                bId = 0;            }            return (char) buf[bId++];        }        public int nextInt() {            return (int) nextLong();        }        public int[] nextInts(int N) {            int[] res = new int[N];            for (int i = 0; i < N; i++) {                res[i] = (int) nextLong();            }            return res;        }        public long[] nextLongs(int N) {            long[] res = new long[N];            for (int i = 0; i < N; i++) {                res[i] = nextLong();            }            return res;        }        public long nextLong() {            cnt = 1;            boolean neg = false;            if (c == NC) c = getChar();            for (; (c < '0' || c > '9'); c = getChar()) {                if (c == '-') neg = true;            }            long res = 0;            for (; c >= '0' && c <= '9'; c = getChar()) {                res = (res << 3) + (res << 1) + c - '0';                cnt *= 10;            }            return neg ? -res : res;        }        public double nextDouble() {            double cur = nextLong();            return c != '.' ? cur : cur + nextLong() / cnt;        }        public double[] nextDoubles(int N) {            double[] res = new double[N];            for (int i = 0; i < N; i++) {                res[i] = nextDouble();            }            return res;        }        public String next() {            StringBuilder res = new StringBuilder();            while (c <= 32) c = getChar();            while (c > 32) {                res.append(c);                c = getChar();            }            return res.toString();        }        public String nextLine() {            StringBuilder res = new StringBuilder();            while (c <= 32) c = getChar();            while (c != '\n') {                res.append(c);                c = getChar();            }            return res.toString();        }        public boolean hasNext() {            if (c > 32) return true;            while (true) {                c = getChar();                if (c == NC) return false;                else if (c > 32) return true;            }        }    }private void tr(Object... o) { if(!oj)System.out.println(Arrays.deepToString(o));
    }
  }

**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 edge cases and scenarios that might cause the algorithm to fail. The key aspects to consider are:

1. **Large Values of \( n \) and \( k \):** The code should handle large values of \( n \) and \( k \) efficiently without running into overflow or performance issues.
2. **Symmetric and Asymmetric Arrays:** The code should handle both symmetric and asymmetric arrays correctly.
3. **Edge Cases:** Small values of \( n \) and \( k \) can also reveal bugs.

Let's construct a test input that covers these scenarios:

1. **Large Values:**
   - \( n = 10^9 \) and \( k = 10^9 \)
2. **Small Values:**
   - \( n = 2 \) and \( k = 2 \)
3. **Symmetric Array:**
   - \( n = 5 \) and \( k = 3 \)
4. **Asymmetric Array:**
   - \( n = 7 \) and \( k = 2 \)

Here is the test input:

```plaintext
4
1000000000 1000000000
2 2
5 3
7 2
```

This input includes a mix of large and small values, as well as symmetric and asymmetric arrays, which should help in identifying any potential bugs in the code.