A14746. Build the Permutation

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.