Suppose you had an array $A$ of $n$ elements, each of which is $0$ or $1$.

Let us define a function $f(k,A)$ which returns another array $B$, the result of sorting the first $k$ elements of $A$ in non-decreasing order. For example, $f(4,[0,1,1,0,0,1,0]) = [0,0,1,1,0,1,0]$. Note that the first $4$ elements were sorted.

Now consider the arrays $B_1, B_2,\ldots, B_n$ generated by $f(1,A), f(2,A),\ldots,f(n,A)$. Let $C$ be the array obtained by taking the element-wise sum of $B_1, B_2,\ldots, B_n$.

For example, let $A=[0,1,0,1]$. Then we have $B_1=[0,1,0,1]$, $B_2=[0,1,0,1]$, $B_3=[0,0,1,1]$, $B_4=[0,0,1,1]$. Then $C=B_1+B_2+B_3+B_4=[0,1,0,1]+[0,1,0,1]+[0,0,1,1]+[0,0,1,1]=[0,2,2,4]$.

You are given $C$. Determine a binary array $A$ that would give $C$ when processed as above. It is guaranteed that an array $A$ exists for given $C$ in the input.


-----Input-----

The first line contains a single integer $t$ ($1 \leq t \leq 1000$)  — the number of test cases.

Each test case has two lines. The first line contains a single integer $n$ ($1 \leq n \leq 2 \cdot 10^5$).

The second line contains $n$ integers $c_1, c_2, \ldots, c_n$ ($0 \leq c_i \leq n$). It is guaranteed that a valid array $A$ exists for the given $C$.

The sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.


-----Output-----

For each test case, output a single line containing $n$ integers $a_1, a_2, \ldots, a_n$ ($a_i$ is $0$ or $1$). If there are multiple answers, you may output any of them.


-----Examples-----

Input
5
4
2 4 2 4
7
0 3 4 2 3 2 7
3
0 0 0
4
0 0 0 4
3
1 2 3
Output
1 1 0 1 
0 1 1 0 0 0 1 
0 0 0 
0 0 0 1 
1 0 1


-----Note-----

Here's the explanation for the first test case. Given that $A=[1,1,0,1]$, we can construct each $B_i$:

$B_1=[\color{blue}{1},1,0,1]$;

$B_2=[\color{blue}{1},\color{blue}{1},0,1]$;

$B_3=[\color{blue}{0},\color{blue}{1},\color{blue}{1},1]$;

$B_4=[\color{blue}{0},\color{blue}{1},\color{blue}{1},\color{blue}{1}]$

And then, we can sum up each column above to get $C=[1+1+0+0,1+1+1+1,0+0+1+1,1+1+1+1]=[2,4,2,4]$.

Difficulty: HARD

Solutions:
from builtins import map

def main():
	for t in range(int(input())):
		n = int(input())
		ans = [0 for i in range(n)]
		a = list(map(lambda ai: int(ai), input().split(' ')))
		k = sum(a) // n
		lb = n - k
		for i in range(1, k + 1):
			a[n - i] -= i
		for i in range(n - 1, -1, -1):
			if a[i] == i and k > 0:
				k -= 1
				ans[i] = 1
			else:
				ans[i] = 0
				lb -= 1
				if lb >= 0:
					a[lb] -= k
		for i in range(n):
			print(ans[i], end=' ')
		print(end='\n')
main()


def solve():
	n = int(input())
	c = list(map(int, input().split()))
	sum = 0
	for i in range(n):
		sum += c[i]
	ones = sum // n
	ans = [1 for _ in range(n + 1)]
	first_0 = c[0]
	ans[first_0] = 0
	for i in range(n):
		if ans[i] == 1:
			if c[i] == 0:
				ans[i] = 0
		if ans[i] == 1:
			next_0 = c[i]
			ans[next_0] = 0
		else:
			next_0 = i + c[i]
			ans[next_0] = 0
		print(ans[i], end=' ')
	print()
t = int(input())
for _ in range(t):
	solve()