A16236 | MEXanization

Let's define $f(S)$ . Let $S$ be a multiset (i.e., it can contain repeated elements) of non-negative integers. In one operation, you can choose any non-empty subset of $S$ (which can also contain repeated elements), remove this subset (all elements in it) from $S$ , and add the MEX of the removed subset to $S$ . You can perform any number of such operations. After all the operations, $S$ should contain exactly $1$ number. $f(S)$ is the largest number that could remain in $S$ after any sequence of operations.

You are given an array of non-negative integers $a$ of length $n$ . For each of its $n$ prefixes, calculate $f(S)$ if $S$ is the corresponding prefix (for the $i$ -th prefix, $S$ consists of the first $i$ elements of array $a$ ).

The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For instance:

- The MEX of $[2,2,1]$ is $0$ , because $0$ does not belong to the array.
- The MEX of $[3,1,0,1]$ is $2$ , because $0$ and $1$ belong to the array, but $2$ does not.
- The MEX of $[0,3,1,2]$ is $4$ , because $0$ , $1$ , $2$ and $3$ belong to the array, but $4$ does not.

The first line contains a single integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. Then follows the description of the test cases.

The first line of each test case contains an integer $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the size of array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0 \leq a_i \leq 2 \cdot 10^5$ ) — the array $a$ .

It is guaranteed that the sum of all $n$ values across all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output $n$ numbers: $f(S)$ for each of the $n$ prefixes of array $a$ .