B. Build the Permutation
constructive algorithms; greedy
You are given three integers \(n, a, b\). Determine if there exists a permutation \(p_1, p_2, \ldots, p_n\) of integers from \(1\) to \(n\), such that:There are exactly \(a\) integers \(i\) with \(2 \le i \le n-1\) such that \(p_{i-1} < p_i > p_{i+1}\) (in other words, there are exactly \(a\) local maximums).There are exactly \(b\) integers \(i\) with \(2 \le i \le n-1\) such that \(p_{i-1} > p_i < p_{i+1}\) (in other words, there are exactly \(b\) local minimums).If such permutations exist, find any such permutation.
The first line of the input contains a single integer \(t\) (\(1 \le t \le 10^4\)) — the number of test cases. The description of test cases follows.The only line of each test case contains three integers \(n\), \(a\) and \(b\) (\(2 \leq n \leq 10^5\), \(0 \leq a,b \leq n\)).The sum of \(n\) over all test cases doesn't exceed \(10^5\).
For each test case, if there is no permutation with the requested properties, output \(-1\).Otherwise, print the permutation that you are found. If there are several such permutations, you may print any of them.
In the first test case, one example of such permutations is \([1, 3, 2, 4]\). In it \(p_1 < p_2 > p_3\), and \(2\) is the only such index, and \(p_2> p_3 < p_4\), and \(3\) the only such index.One can show that there is no such permutation for the third test case.