311292: CF1966F. Missing Subarray Sum

There is a hidden array $a$ of $n$ positive integers. You know that $a$ is a palindrome, or in other words, for all $1 \le i \le n$, $a_i = a_{n + 1 - i}$. You are given the sums of all but one of its distinct subarrays, in arbitrary order. The subarray whose sum is not given can be any of the $\frac{n(n+1)}{2}$ distinct subarrays of $a$.

Recover any possible palindrome $a$. The input is chosen such that there is always at least one array $a$ that satisfies the conditions.

An array $b$ is a subarray of $a$ if $b$ can be obtained from $a$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

The first line of the input contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($3 \le n \le 1000$) — the size of the array $a$.

The next line of each test case contains $\frac{n(n+1)}{2} - 1$ integers $s_i$ ($1\leq s_i \leq 10^9$) — all but one of the subarray sums of $a$.

It is guaranteed that the sum of $n$ over all test cases does not exceed $1000$.

Additional constraint on the input: There is always at least one valid solution.

Hacks are disabled for this problem.

For each test case, print one line containing $n$ positive integers $a_1, a_2, \cdots a_n$ — any valid array $a$. Note that $a$ must be a palindrome.

If there are multiple solutions, print any.

7
3
1 2 3 4 1
4
18 2 11 9 7 11 7 2 9
4
5 10 5 16 3 3 13 8 8
4
8 10 4 6 4 20 14 14 6
5
1 2 3 4 5 4 3 2 1 1 2 3 2 1
5
1 1 2 2 2 3 3 3 3 4 5 5 6 8
3
500000000 1000000000 500000000 500000000 1000000000

1 2 1 
7 2 2 7 
3 5 5 3 
6 4 4 6 
1 1 1 1 1 
2 1 2 1 2 
500000000 500000000 500000000 

For the first example case, the subarrays of $a = [1, 2, 1]$ are:

• $[1]$ with sum $1$,
• $[2]$ with sum $2$,
• $[1]$ with sum $1$,
• $[1, 2]$ with sum $3$,
• $[2, 1]$ with sum $3$,
• $[1, 2, 1]$ with sum $4$.

For the second example case, the missing subarray sum is $4$, for the subarray $[2, 2]$.

For the third example case, the missing subarray sum is $13$, because there are two subarrays with sum $13$ ($[3, 5, 5]$ and $[5, 5, 3]$) but $13$ only occurs once in the input.