1701D · Permutation Restoration

Problem: Monocarp had a permutation $a$ of $n$ integers $1$, $2$, ..., $n$ (a permutation is an array where each element from $1$ to $n$ occurs exactly once).

Then Monocarp calculated an array of integers $b$ of size $n$, where $b_i = \left\lfloor \frac{i}{a_i} \right\rfloor$. For example, if the permutation $a$ is $[2, 1, 4, 3]$, then the array $b$ is equal to $\left[ \left\lfloor \frac{1}{2} \right\rfloor, \left\lfloor \frac{2}{1} \right\rfloor, \left\lfloor \frac{3}{4} \right\rfloor, \left\lfloor \frac{4}{3} \right\rfloor \right] = [0, 2, 0, 1]$.

Unfortunately, the Monocarp has lost his permutation, so he wants to restore it. Your task is to find a permutation $a$ that corresponds to the given array $b$. If there are multiple possible permutations, then print any of them. The tests are constructed in such a way that least one suitable permutation exists.

Input Format: The first line contains a single integer $t$ ($1 \le t \le 10^5$) — number of test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 5 \cdot 10^5$).

The second line contains $n$ integers $b_1, b_2, \dots, b_n$ ($0 \le b_i \le n$).

Additional constrains on the input:

• the sum of $n$ over test cases does not exceed $5 \cdot 10^5$;
• there exists at least one permutation $a$ that would yield this array $b$.

Output Format: For each test case, print $n$ integers — a permutation $a$ that corresponds to the given array $b$. If there are multiple possible permutations, then print any of them.