Chiaki has just learned hash in today’s lesson. A hash function is any function that can be used to map data of arbitrary size to data of fixed size. As a beginner, Chiaki simply chooses a hash table of size $n$ with hash function $h(x) = x \mod n$.
Unfortunately, the hash function may map two distinct values to the same hash value. For example, when $n = 9$ we have $h(7) = h(16) = 7$. It will cause a failure in the procession of insertion. In this case, Chiaki will check whether the next position is available or not. This task will not be finished until an available position is found. If we insert $\lbrace 7, 8, 16 \rbrace$ into a hash table of size $9$, we will finally get $ \lbrace 16, -1, -1, -1, -1, -1, -1, 7, 8 \rbrace$. Available positions are marked as $-1$.
After done all the exercises, Chiaki became curious to the inverse problem. Can we rebuild the insertion sequence from a hash table? If there are multiple available insertion sequences, Chiaki would like to find the smallest one under lexicographical order.
Sequence $a_1, a_2, …, a_n$ is lexicographically smaller than sequence $b_1, b_2, …, b_n$ if and only if there exists $i$ ($1 ≤ i ≤ n$) satisfy that $a_i < b_i$ and $a_j = b_j$ for all $1 ≤ j < i$.
There are multiple test cases. The first line of input contains an integer $T$, indicating the number of test cases. For each test case:
The first line of each case contains a positive integer $n$ ($1 ≤ n ≤ 2 \times 10^5$) – the length of the hash table.
The second line contains exactly $n$ integers $a_1,a_2,…,a_n$ ($-1 ≤ a_i ≤ 10^9$).
It is guaranteed that the sum of all $n$ does not exceed $2 \times 10^6$.
For each case, please output smallest available insertion sequence in a single line. Print an empty line when the available insertion sequence is empty. If there’s no such available insertion sequence, just output $-1$ in a single line.
7 8 16
- 当$x$插入到$j$而应该插入到$i = x \pmod n$时，如果$[i, j)$中间有$-1$，则序列不合法。