MEX and Increments
Description
Dmitry has an array of $n$ non-negative integers $a_1, a_2, \dots, a_n$.
In one operation, Dmitry can choose any index $j$ ($1 \le j \le n$) and increase the value of the element $a_j$ by $1$. He can choose the same index $j$ multiple times.
For each $i$ from $0$ to $n$, determine whether Dmitry can make the $\mathrm{MEX}$ of the array equal to exactly $i$. If it is possible, then determine the minimum number of operations to do it.
The $\mathrm{MEX}$ of the array is equal to the minimum non-negative integer that is not in the array. For example, the $\mathrm{MEX}$ of the array $[3, 1, 0]$ is equal to $2$, and the array $[3, 3, 1, 4]$ is equal to $0$.
The first line of input data contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases in the input.
The descriptions of the test cases follow.
The first line of the description of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the length of the array $a$.
The second line of the description of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le n$) — elements of the array $a$.
It is guaranteed that the sum of the values $n$ over all test cases in the test does not exceed $2\cdot10^5$.
For each test case, output $n + 1$ integer — $i$-th number is equal to the minimum number of operations for which you can make the array $\mathrm{MEX}$ equal to $i$ ($0 \le i \le n$), or -1 if this cannot be done.
Input
The first line of input data contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases in the input.
The descriptions of the test cases follow.
The first line of the description of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the length of the array $a$.
The second line of the description of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le n$) — elements of the array $a$.
It is guaranteed that the sum of the values $n$ over all test cases in the test does not exceed $2\cdot10^5$.
Output
For each test case, output $n + 1$ integer — $i$-th number is equal to the minimum number of operations for which you can make the array $\mathrm{MEX}$ equal to $i$ ($0 \le i \le n$), or -1 if this cannot be done.
Samples
5
3
0 1 3
7
0 1 2 3 4 3 2
4
3 0 0 0
7
4 6 2 3 5 0 5
5
4 0 1 0 4
1 1 0 -1
1 1 2 2 1 0 2 6
3 0 1 4 3
1 0 -1 -1 -1 -1 -1 -1
2 1 0 2 -1 -1
Note
In the first set of example inputs, $n=3$:
- to get $\mathrm{MEX}=0$, it is enough to perform one increment: $a_1$++;
- to get $\mathrm{MEX}=1$, it is enough to perform one increment: $a_2$++;
- $\mathrm{MEX}=2$ for a given array, so there is no need to perform increments;
- it is impossible to get $\mathrm{MEX}=3$ by performing increments.