step 1:

It's obvious that the answer is "NO" if $$$a_{i} = 0$$$ for all $$$1 \le i \le n$$$.

step 2:

Lets prove that the answer is "YES" if $$$a_{i} = 1$$$ for at least one $$$1 \le i \le n$$$.

step 3:

If size of $$$a$$$ is equal to $$$k$$$, just use second type operation once and we are done.

step 4:

Otherwise (if $$$|a| \gt k$$$), there will be three cases: (assume that $$$a_{j} = 1$$$)

• if $$$j \gt 2$$$, you can use first type operation on first and second element and decrease size of $$$a$$$ by 1.
• else if $$$j \lt |a|-1$$$, you can use first type operation on last and second to last element and decrease size of $$$a$$$ by 1.
• else, it can be shown easily that $$$|a| = 3$$$ and $$$k = 2$$$ so you can use second type operation twice and turn $$$a$$$ into a single $$$1$$$.

In first and second case, you can continue decreasing size of $$$a$$$ until $$$|a| = k$$$ (or you reach $$$3$$$-rd case) and you can use second type operation to reach your aim.

step 5:

So we proved that the answer is "YES" iff $$$a_{i} = 1$$$ for at least one $$$1 \le i \le n$$$ or in other words, iff $$$\sum_{i = 1}^n a_{i} \gt 0$$$.

// In the name of God
#include <iostream>
 
using namespace std;
 
int main()
{
    int t;
    cin >> t;
    while(t--)
    {
        int n, k;
        cin >> n >> k;
        int sum = 0;
        for(int i = 0 ; i < n ; i++){
            int a;
            cin >> a;
            sum += a;
        }
        if(sum > 0) cout << "YES" << endl;
        else cout << "NO" << endl;
    }
    return 0;
}
// Thank God

step 1:

step 2:

step 3:

step 4:

/// In the name of God
#include <iostream>
 
using namespace std;
 
int main()
{
	int t;
	cin >> t;
	while(t--)
	{
		int n;
		cin >> n;
		int A[n], cnt[2][n+1];
		cnt[0][0] = cnt[1][0] = 0;
		for(int i = 0; i < n; i++)
		{
			cin >> A[i];
			cnt[0][i+1] = cnt[0][i]+(A[i]==0?1:0);
			cnt[1][i+1] = cnt[1][i]+(A[i]==1?1:0);
		}
		int ans = n-1;
		for(int last_zero = 0; last_zero <= n; last_zero++)
			ans= min(ans, max(cnt[1][last_zero], cnt[0][n]-cnt[0][last_zero]));
		cout << ans << endl;
	}
}
 
/// Thank God . . .

step 1:

step 2:

step 3:

step 4:

step 5:

/// In the name of God
#include <bits/stdc++.h>

using namespace std;

inline void solve()
{
	int n;
	cin >> n;
	int permutation[n], location[n];
	for(int i = 0; i < n; i++)
	{
		cin >> permutation[i];
		permutation[i]--;
		location[permutation[i]] = i;
	}
	for(int i = 0; i < n; i++)
	{
		if(location[i] == n-1)
			cout << rand()%n+1 << (i == n-1?'\n':' ');
		else
			cout << location[i]+2 << (i == n-1?'\n':' ');
	}
}

int main()
{
	int t;
	cin >> t;
	while(t--)
		solve();
	return 0;
}
/// Thank God . . .

Define $$$f(u, cnt)$$$ represents the maximum score of $$$cnt$$$ balls passing through the subtree of node $$$u$$$.

Define $$$num$$$ as the number of the sons of node $$$u$$$. The transition is only related to $$$\lceil cnt / num \rceil$$$ and $$$\lfloor cnt / num \rfloor$$$ two states of the subtree.

For each node $$$u$$$, $$$cnt$$$ can only be two adjacent integer $$$(x, x + 1)$$$.

It can be proved that the four numbers $$$\lfloor x/num \rfloor$$$, $$$\lceil x/num \rceil$$$, $$$\lceil (x + 1)/num \rceil$$$ and $$$\lfloor (x + 1)/num \rfloor$$$ can only have two kinds of numbers at most, and they are adjacent natural numbers.

So the number of states can be proved to be $$$\mathcal{O}(n)$$$.

/// In the name of God
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;

const int N = 200000;
ll V[N], dp[N];
int dad[N];
vector<int> child[N];
vector<pair<int, ll>> answers[N];

inline ll DP(int v, ll k)
{
	for(auto [kp, ans]: answers[v])
		if(k == kp)
			return ans;
	ll cnt_child = (ll)child[v].size();
	ll ans = k*V[v];
	if(cnt_child == 0)
		return ans;
	if(k%cnt_child == 0)
		for(auto u: child[v])
			ans += DP(u, k/cnt_child);
	else
	{
		ll dp1[cnt_child], dp2[cnt_child], diff[cnt_child];
		for(int i = 0; i < cnt_child; i++)
			dp1[i] = DP(child[v][i], k/cnt_child), dp2[i] = DP(child[v][i], k/cnt_child+1);
		for(int i = 0; i < cnt_child; i++)
			diff[i] = dp2[i] - dp1[i];
		sort(diff, diff+cnt_child, greater<int>());
		for(int i = 0; i < cnt_child; i++)
			ans += dp1[i];
		for(int i = 0; i < k%cnt_child; i++)
			ans += diff[i];
	}
	answers[v].push_back({k, ans});
	return ans;
}

inline ll solve()
{
	ll n, k;
	cin >> n >> k;
	for(int i = 0; i < n; i++)
		child[i].clear(), answers[i].clear();
	dad[0] = 0;
	for(int i = 0; i < n-1; i++)
	{
		cin >> dad[i+1];
		dad[i+1]--;
		child[dad[i+1]].push_back(i+1);
	}
	for(int i = 0; i < n; i++)
		cin >> V[i];
	return DP(0, k);
}

int main()
{
	int t;
	cin >> t;
	while(t--)
		cout << solve() << '\n';
	return 0;
}
/// Thank God . . .

step 1:

step 2:

step 3:

step 4:

/// In the name of God
#pragma GCC optimize("Ofast","unroll-loops","O3")
#include <bits/stdc++.h>

using namespace std;

inline bool get_answer(){string s; cin >> s; return s == "YES";}

inline bool f(int i){return i&1;}
inline bool g(int i){return i&2;}

void solve(const vector<int> &Valid)
{
    if(Valid.size() < 3u)
	{
	    cout << "! " << Valid[0] << endl;
		string s;
		cin >> s;
		if(s == ":)")return;
		cout << "! " << Valid[1] << endl;
		return;
	}
	else if(Valid.size() == 3u)
	{
		bool is[4];
		cout << "? 1 " << Valid[0] << endl;is[0] = get_answer();
		cout << "? 1 " << Valid[1] << endl;is[1] = get_answer();
		cout << "? 1 " << Valid[1] << endl;is[2] = get_answer();
		cout << "? 1 " << Valid[0] << endl;is[3] = get_answer();
		if(is[1] and is[2])return solve({Valid[1]});
		else if(!is[1] and !is[2])return solve({Valid[0], Valid[2]});
		else if((is[0] and is[1]) or (is[2] and is[3]))return solve({Valid[0], Valid[1]});
		else if((is[0] and !is[1]) or (!is[2] and is[3]))return solve({Valid[0], Valid[2]});
		else return solve({Valid[1], Valid[2]});
	}
	else
	{
		vector<int> query[2];
		for(int i = 0; i < (int)Valid.size(); i++)
		{
			if(f(i))query[0].push_back(Valid[i]);
			if(g(i))query[1].push_back(Valid[i]);
		}
		bool is[2];
		for(int i = 0; i < 2; i++)
		{
			cout << "? " << query[i].size();
			for(auto u: query[i])cout << ' ' << u;
			cout << endl;
			is[i] = get_answer();
		}
		vector<int> NewValid;
		for(int i = 0; i < (int)Valid.size(); i++)
		{
			if((!f(i) ^ is[0]) or (!g(i) ^ is[1]))NewValid.push_back(Valid[i]);
		}
		return solve(NewValid);
	}
}
int main()
{
	int n = 0;
	cin >> n;
	vector<int> all(n);
	for(int i = 0; i < n; i++)all[i] = i+1;
	solve(all);
	return 0;
}
/// Thank God . . .

step 1:

First note that in any question we ask, a set will be introduced as the set that contains $$$x$$$. If the answer is "YES", then the set that we asked is introduced and otherwise, its complement.

step 2:

step 3:

/// In the name of God
#pragma GCC optimize("Ofast","unroll-loops","O3")
#include <bits/stdc++.h>
 
using namespace std;
 
const int SUM = 50;
int dp[SUM][SUM];
pair<int,int> updater[SUM][SUM];
map<pair<int,int>, int> Dp;
map<pair<int,int>, pair<int,int>> Updater;
 
inline void preprocess()
{
	for(int i = 0; i < SUM; i++)
	{
		for(int j = 0; j < SUM; j++)
		{
			dp[i][j] = SUM;
			updater[i][j] = {i, j};
		}
	}
	dp[0][0] = dp[0][1] = dp[1][0] = dp[2][0] = dp[1][1] = dp[0][2] = 0;
	for(int sum = 0; sum < SUM; sum ++)for(int last = sum; last >= 0; last--)
	{
		int now = sum - last;
		for(int SelectLast = 0; SelectLast <= last; SelectLast++)for(int SelectNow = 0; SelectNow <= now; SelectNow++)
		{
			if(dp[last][now] > 1 + max(dp[now-SelectNow][SelectNow+SelectLast], dp[SelectNow][sum-SelectNow-SelectLast]))
			{
				dp[last][now] = 1 + max(dp[now-SelectNow][SelectNow+SelectLast], dp[SelectNow][now+last-SelectNow-SelectLast]);
				updater[last][now] = {SelectLast, SelectNow};
			}
		}
	}
}
 
inline int DP(const int last, const int now)
{
	if(last < 0 || now < 0)return SUM;
	if(last + now < SUM)return dp[last][now];
	if(Dp.find({last, now}) != Dp.end())return Dp[{last, now}];
	if((last&1) && (now&1))
	{
		Dp[{last, now}] = 1 + DP((now+1)/2, (last+ now)/2);
		Updater[{last, now}] = {(last+1)/2, now/2};
	}
	else
	{
		Dp[{last, now}] = 1 + DP((now+1)/2, (last+1)/2+(now+1)/2);
		Updater[{last, now}] = {(last+1)/2, (now+1)/2};
	}
	return Dp[{last, now}];
}
 
inline bool IsIn(const int x, const vector<int> &Sorted)
{
	auto u = lower_bound(Sorted.begin(), Sorted.end(), x);
	if(u == Sorted.end() or *u != x)return false;
	return true;
}
 
vector<int> solve(const vector<int> &LastAns, const vector<int> &Valid)
{
	if((int)Valid.size() < 3)
	{
		return Valid;
	}
	pair<int,int> Select;
	if((int)Valid.size() < SUM)
	{
		Select = updater[LastAns.size()][Valid.size()-LastAns.size()];
	}
	else
	{
		DP((int)LastAns.size(), (int)(Valid.size()-LastAns.size()));
		Select = Updater[{LastAns.size(), Valid.size()-LastAns.size()}];
	}
	vector<int> query;
	int p = 0;
	while(Select.first --)query.push_back(LastAns[p++]);
	p = 0;
	vector<int> LastAnsSorted = LastAns;
	sort(LastAnsSorted.begin(), LastAnsSorted.end());
	while(Select.second --)
	{
		while(IsIn(Valid[p], LastAnsSorted)) p++;
		query.push_back(Valid[p++]);
	}
	cout << "? " << query.size();
	for(auto u: query)cout << ' ' << u;
	cout << endl;
	
	string s;
	cin >> s;
	bool correct = (s == "YES");
	sort(query.begin(), query.end());
	vector<int> NewLast, NewValid;
	for(auto u: Valid)
	{
		if(!IsIn(u, LastAnsSorted) and (correct ^ IsIn(u, query)))
		{
			NewLast.push_back(u);
		}
		if(!IsIn(u, LastAnsSorted) or !(correct ^ IsIn(u, query)))
		{
			NewValid.push_back(u);
		}
	}
	vector<int> ans = solve(NewLast, NewValid);
	return ans;
}
 
int main()
{
	preprocess();
 
	int n;
	cin >> n;
	vector<int> all(n);
	for(int i = 0; i < n; i++)all[i] = i+1;
	vector<int> valid = solve({}, all);
	for(auto guess: valid)
	{
		cout << "! " << guess << endl;
		string s;
		cin >> s;
		if(s == ":)")return 0;
	}
	
	return 0;
}
/// Thank God . . .

step 1:

First of all, we can compress $$$a_{i}$$$'s and $$$x$$$'s (in second type query). so we can assume that all numbers are less than $$$n+q \le 6 \cdot 10^5$$$.

step 2:

step 3:

step 4:

#include <bits/stdc++.h>
 
using namespace std;
 
typedef long long int ll;
 
mt19937 rnd(time(0));
 
const int N = 300'000 + 5; 
const int Q = 300'000 + 5; 
const int T = 50;
bitset<N+Q> RandomSet[T];
unordered_map<int, int> id; int cnt_id = 0;
int n, q, A[N];
 
struct fenwick
{
	int PartialSum[N];
	fenwick()
	{
		for(int i = 0; i < N; i++)PartialSum[i] = 0;
	}
	inline void add(int index, bool increase)
	{
		while(index < N)
		{
			PartialSum[index] += (increase? 1 : -1);
			index += index&-index;
		}
	}
	inline int get(int index)
	{
		int sum = 0;
		while(index)
		{
			sum += PartialSum[index];
			index -= index&-index;
		}
		return sum;
	}
}Fen[T];
 
inline int GetId(const int x)
{
	auto id_iterator = id.find(x);
	if(id_iterator == id.end())
	{
		return id[x] = cnt_id++;
	}
	else return (*id_iterator).second;
}
 
inline void ChooseRandomSets()
{
	for(int i = 0; i < T; i++)
	{
		for(int j = 0; j < N+Q; j++)
		{
			if(rnd()&1)RandomSet[i].set(j);
		}
	}
}
 
inline void AddArrayToFenwick()
{
	for(int i = 0; i < n; i++)
	{
		int MyId = GetId(A[i]);
		for(int j = 0; j < T; j++)
		{
			if(RandomSet[j][MyId])Fen[j].add(i+1, true);
		}
	}
}
	
inline void Query()
{
	int index, l, r, k, x, type;
	for(int i = 0; i < q; i++)
	{
		cin >> type;
		if(type == 1)
		{
			cin >> index >> x;
			index --;
			int IdPre = GetId(A[index]);
			int IdNew = GetId(x);
			A[index] = x;
			for(int j = 0; j < T; j++)
			{
				if(RandomSet[j][IdPre])Fen[j].add(index+1, false);
				if(RandomSet[j][IdNew])Fen[j].add(index+1, true);
			}
		}
		if(type == 2)
		{
			cin >> l >> r >> k;
			l--;
			if(k == 1){cout << "YES\n"; continue;}
			else if((r-l)%k != 0){cout << "NO\n"; continue;}
			bool answer = true;
			for(int j = 0; j < T; j++)
			{
				if((Fen[j].get(r)-Fen[j].get(l))%k != 0){answer = false; break;}
			}
			cout << (answer?"YES":"NO") << '\n';
		}
	}
}
 
int main()
{
    ios::sync_with_stdio(false) , cin.tie(0);
    ChooseRandomSets();
    cin >> n >> q;
    for(int i = 0; i < n; i++) cin >> A[i];
    AddArrayToFenwick();
    Query();
    return 0;
}

Consider all subsets of the original set in which tasks can be performed in some order to satisfy the constraints on their completion, and also that the size of each of them does not exceed $$$a+b+c$$$. It is easy to show that this set is a weighted matroid. This means that the optimal set of tasks can be found using a greedy algorithm.

So, we can find a set of tasks with maximum total usefulness, which:

• satisfies all deadline constraints
• has no more than $$$a+b+c$$$ elements

For the given parameters $$$(x, y, z)$$$, we introduce the following transformation:

• Increase the usefulness of all tasks of the first type by $$$x$$$, the second type~--- by $$$y$$$, and the third type~--- by $$$z$$$.

Let's assume that the optimal solution after the transformation is described by the triple $$$[a',b',c']$$$~--- the number of problems of each type, respectively. We need to get a solution described by the triple $$$[a,b,c]$$$. For $$$[a,b,c] \neq [a',b',c']$$$, $$$a+b+c=a'+b'+c'$$$ holds. Then (without limitation of generality, since there are at least two inequalities among $$$a\vee a',b\vee b',c\vee c'$$$) we assume that $$$a' \gt a,b' \lt b$$$.

Consider the following coordinate system on a two-dimensional plane (modulo the addition of $$$\overrightarrow{(1;1;1)})$$$:

For the coordinates of the point, we take the value $$$\vec{x}\times x+\vec{y}\times y + \vec{z}\times z$$$$$$(\vec{x}+\vec{y}+\vec{z} =\vec{0})$$$. It is easy to understand that the coordinates of the point $$$(x; y)$$$ on the plane are uniquely set by the triple $$$(x; y; z)$$$ up to the addition of $$$\overrightarrow{(1;1;1)} \times k, k \in\mathbb{Z}$$$

Let's assume that there are parameters $$$(x,y,z)$$$ for which the optimal solution is unique and equal to $$$[a,b,c]$$$ (see "But what if..?" to see what needs to be done to the input to make this hold).

Consider the solution $$$[a',b',c']$$$ for the "center" of the current polygon (initially it is a hexagon with infinitely distant coordinates). The "center" is the center of the triangle formed by the middle lines of our polygon.

On this plane, decreasing $$$z$$$ or $$$y$$$ does not increase $$$b'+c'$$$, therefore, it does not decrease $$$a'$$$. Thus, the red area in the picture below does not suit us.

Increasing $$$x$$$ or $$$z$$$ does not decrease $$$a'+c'$$$, therefore, it does not increase $$$b'$$$. Because of this, the blue area in the picture below does not suit us either.

So, we need to cut off one half-plane along the $$$z$$$ axis. It can be shown that the area of the polygon under consideration decreases by a constant time for every 2-3 iterations.

But what if the number of tasks of the desired color jumps from $$$a-1$$$ to $$$a+1$$$ at once? This is possible if and only if with the addition of some $$$x$$$ we get several pairs of elements of equal weight at the same time. To get rid of this, we can simply add a random number from $$$0$$$ to $$$\frac{1}{3}$$$ to each weight. The fact that such a change will fix the problem described above and will not change the optimal answer to the original problem remains to the reader as an exercise (the authors have proof).

After $$$O(\log C)$$$ iterations, our algorithm will converge to a point (a triple of real numbers). It remains only to check that the solution found is suitable for us.

$$$O(n\log n\log C)$$$

#include <bits/stdc++.h>
using namespace std;
 
#ifdef SG
	#include <debug.h>
#else
	template<typename T> struct outputer;
	struct outputable {};
	#define PRINT(...)
	#define OUTPUT(...)
	#define show(...)
	#define debug(...)
	#define deepen(...)
	#define timer(...)
	#define fbegin(...)
	#define fend
	#define pbegin(...)
	#define pend
#endif
 
#define ARG4(_1,_2,_3,_4,...) _4
 
#define forn3(i,l,r) for (int i = int(l); i < int(r); ++i)
#define forn2(i,n) forn3 (i, 0, n)
#define forn(...) ARG4(__VA_ARGS__, forn3, forn2) (__VA_ARGS__)
 
#define ford3(i,l,r) for (int i = int(r) - 1; i >= int(l); --i)
#define ford2(i,n) ford3 (i, 0, n)
#define ford(...) ARG4(__VA_ARGS__, ford3, ford2) (__VA_ARGS__)
 
#define ve vector
#define pa pair
#define tu tuple
#define mp make_pair
#define mt make_tuple
#define pb emplace_back
#define fs first
#define sc second
#define all(a) (a).begin(), (a).end()
#define sz(a) ((int)(a).size())
 
typedef long double ld;
typedef signed __int128 lll;
typedef unsigned __int128 ulll;
typedef int64_t ll;
typedef uint64_t ull;
typedef uint32_t ui;
typedef uint16_t us;
typedef uint8_t uc;
typedef pa<int, int> pii;
typedef pa<int, ll> pil;
typedef pa<ll, int> pli;
typedef pa<ll, ll> pll;
typedef ve<int> vi;
 
template<typename T> inline auto sqr (T x) -> decltype(x * x) {return x * x;}
template<typename T1, typename T2> inline bool umx (T1& a, T2 b) {if (a < b) {a = b; return 1;} return 0;}
template<typename T1, typename T2> inline bool umn (T1& a, T2 b) {if (b < a) {a = b; return 1;} return 0;}
 
const int N = 100000;
const int X = 1000000000;
const int T = 3;
 
struct Input {
	int n;
	std::array<int, T> k;
	int a[N], t[N], d[N];
	
	bool read() {
		if (!(cin >> n)) {
			return 0;
		}
		forn (i, T) {
			cin >> k[i];
		}
		forn (i, n) {
			cin >> a[i] >> t[i] >> d[i];
			--t[i];
			--d[i];
		}
		return 1;
	}
 
	void init(const Input& input) {
		*this = input;
	}
};
 
struct Data: Input {
	ll ans;
	
	void write() {
		cout << ans << endl;
	}
};
 
 
namespace Main {
	const lll P = 2 * N + 1;
	const lll Q = 2 * N + 2;
	const lll R = 2 * N + 3;
	const lll A[3] = {3 * P * Q, 3 * P * R, 3 * Q * R};
	const lll M = 3 * P * Q * R * N;
	const lll inf = (X + 1) * M;
 
	struct SNM {
		int rnd = 0;
		int n;
		int pr[N];
 
		void init(int cnt) {
			n = cnt;
			forn (i, n) {
				pr[i] = i;
			}
		}
 
		int get_p(int v) {
			if (v < 0 || pr[v] == v) {
				return v;
			}
			return pr[v] = get_p(pr[v]);
		}
 
		bool add(int v) {
			v = get_p(v);
			if (v == -1) {
				return 0;
			}
			pr[v] = v - 1;
			return 1;
		}
	};
	
	struct Solution: Data {
		SNM snm;
 
		int m;
		lll b[N], c[N];
		vi ord[T];
 
		array<int, T> check(const array<lll, T>& adds) {
			forn (i, n) {
				c[i] = b[i] + adds[t[i]];
			}
			auto cmp = [&](int lhs, int rhs) {
				return c[lhs] > c[rhs];
			};
			static vi q;
			q.clear();
			forn (i, T) {
				static vi w;
				w.resize(sz(q) + sz(ord[i]));
				merge(all(q), all(ord[i]), w.begin(), cmp);
				q.swap(w);
			}
 
			snm.init(n);
			array<int, T> cnt{};
			int tot = 0;
			ll val = 0;
			for (int i : q) {
				if (tot >= m) {
					break;
				}
				if (snm.add(d[i])) {
					cnt[t[i]]++;
					tot++;
					val += a[i];
				}
			}
			if (cnt == k) {
				ans = val;
			}
			return cnt;
		}
 
		// x[i] == adds[(i + 1) % T] - adds[i]
 
		array<lll, T> get_middle_point(const array<lll, T>& lb, const array<lll, T>& rb) {
			array<lll, T> x;
			forn (i, T) {
				x[i] = lb[i] + rb[i];
			}
			lll sum = accumulate(all(x), lll(0));
			forn (i, T) {
				x[i] = T * x[i] - sum;
				if (x[i] >= 0) {
					x[i] /= (2 * T);
				} else {
					x[i] = (x[i] - 2 * T + 1) / (2 * T);
				}
			}
			sum = accumulate(all(x), lll(0));
			forn (i, T) {
				if (sum < 0 && x[i] < rb[i]) {
					x[i]++;
					sum++;
				}
				assert(x[i] >= lb[i]);
				assert(x[i] <= rb[i]);
			}
			assert(sum == 0);
			return x;
		}
 
		bool search() {
			array<lll, T> lb, rb;
			forn (i, T) {
				lb[i] = -2 * inf;
				rb[i] = 2 * inf;
			}
 
			while (true) {
				{
					lll sum_l = accumulate(all(lb), lll(0));
					lll sum_r = accumulate(all(rb), lll(0));
					forn (i, T) {
						lll sol = sum_l - lb[i];
						lll sor = sum_r - rb[i];
						umx(lb[i], -sor);
						umn(rb[i], -sol);
						if (lb[i] > rb[i]) {
							return 0;
						}
					}
				}
				array<lll, T> x = get_middle_point(lb, rb);
				array<lll, T> adds{};
				forn (i, T - 1) {
					adds[i + 1] = adds[i] + x[i];
				}
				array<int, T> cnt = check(adds);
				assert(accumulate(all(cnt), 0) == m);
				if (cnt == k) {
					return 1;
				}
				forn (i, T) {
					lll d1 = cnt[i] - k[i];
					lll d2 = cnt[(i + 1) % T] - k[(i + 1) % T];
					if (d1 > 0 && d2 < 0) {
						lb[i] = x[i] + 1;
					}
					if (d1 < 0 && d2 > 0) {
						rb[i] = x[i] - 1;
					}
				}
			}
		}
		
		void solve() {
			forn (i, n) {
				b[i] = a[i] * M + i * A[t[i]];
			}
			forn (i, n) {
				ord[t[i]].pb(i);
			}
			forn (i, T) {
				sort(all(ord[i]), [&](int lhs, int rhs) {
					return b[lhs] > b[rhs];
				});
			}
			m = accumulate(all(k), 0);
 
			{
				array<int, T> r = check({});
				if (accumulate(all(r), 0) != m) {
					ans = -1;
					return;
				}
			}
			forn (i, T) {
				array<lll, T> adds{};
				adds[i] = inf;
				if (check(adds)[i] < k[i]) {
					ans = -1;
					return;
				}
				adds[i] = -inf;
				if (check(adds)[i] > k[i]) {
					ans = -1;
					return;
				}
			}
			assert(search());
		}
		
		void clear() {
			forn (i, T) {
				ord[i].clear();
			}
		}
	};
}
 
 
Main::Solution sol;
 
int main() {
	#ifdef SG
		freopen((problemname + ".in").c_str(), "r", stdin);
//		freopen((problemname + ".out").c_str(), "w", stdout);
	#endif
	
	int t;
	cin >> t;
	forn (i, t) {
		sol.read();
		sol.solve();
		sol.write();
		sol.clear();
	}
	
	return 0;
}