Lost Lists

Younghee, an elementary school student, received several lists from her teacher. Each list is made of distinct positive integers. Her homework was, for every list, to write down the sum of each pair of numbers in it. To stop students from copying, the teacher gave a different list to each student, so Younghee had to finish the work by herself.

After she was done, Younghee went out to play, and while she was away the original lists were thrown out as trash and lost. Only the sheets with the pairwise sums are left. She has to hand the lists in at the next class, so she wants to rebuild each original list from its pairwise sums.

For each list of sums, restore the list of distinct positive integers that produces exactly those sums. Some lists of sums cannot come from any valid list; in that case, report that it cannot be restored.

The first line contains an integer $T$, the number of test cases. Each test case is given on two lines. The first line contains an integer $n$ ($2 < n < 50$), the number of integers in the original list. The second line contains the $n(n-1)/2$ pairwise sums of that list in non-decreasing order. Every sum is a positive integer less than $10000$.

For each test case, print one line. If the original list can be restored, print its $n$ distinct positive integers in increasing order, separated by single spaces. If no list of distinct positive integers produces the given sums, print $-1$. When more than one valid list produces the given sums, print the lexicographically smallest such list (comparing the increasing sequences element by element).