E. MEX and Incrementstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputDmitry 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.InputThe 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.OutputFor 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.ExampleInput
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
Output
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 
NoteIn 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. 
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  
['constructive algorithms', 'data structures', 'dp', 'greedy', 'implementation', 'math', 'sortings', '*1700']