You are given a permutation $$$p$$$$$$^{\text{∗}}$$$ of integers from $$$1$$$ to $$$n$$$.

For an arbitrary permutation $$$q$$$ of length $$$n$$$, we say that index $$$j$$$ dominates index $$$i$$$ if and only if all of the following conditions hold:

$$$j \gt i$$$,
$$$p_j \gt p_i$$$, and
$$$q_j \gt q_i$$$.

You are also given an array $$$d$$$ of length $$$n$$$, where $$$d_i$$$ denotes the number of indices that dominate index $$$i$$$.

Your task is to construct a permutation $$$q$$$ of integers from $$$1$$$ to $$$n$$$ such that for every index $$$i$$$ ($$$1 \le i \le n$$$), the number of indices $$$j$$$ that dominate $$$i$$$ is exactly $$$d_i$$$.

If such $$$q$$$ exists, output any valid one. Otherwise, report that it does not exist.

$$$^{\text{∗}}$$$A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

Input

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 1000$$$). The description of the test cases follows.

The first line of each test case contains an integer $$$n$$$ ($$$1 \le n \le 5000$$$).

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$. It is guaranteed that $$$p$$$ forms a permutation of integers from $$$1$$$ to $$$n$$$.

The third line of each test case contains $$$n$$$ integers $$$d_1, d_2, \ldots, d_n$$$ ($$$0 \le d_i \le n$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$5000$$$.

Output

For each test case:

If it is possible to construct $$$q$$$, output a line containing $$$n$$$ integers $$$q_1, q_2, \ldots, q_n$$$ — a valid permutation. If multiple valid answers exist, you may output any of them.
Otherwise, output $$$-1$$$.

Note

In the first test case, a valid output is $$$q = [1, 2, 3]$$$:

For $$$i=1$$$, we have $$$p_1=2$$$ and $$$q_1=1$$$. The index $$$j=2$$$ dominates $$$i=1$$$ because $$$2 \gt 1$$$, $$$p_2=3 \gt p_1$$$, and $$$q_2=2 \gt q_1$$$. Index $$$j=3$$$ does not dominate $$$i=1$$$ because $$$p_3=1 \lt p_1$$$. Thus, exactly $$$1$$$ index dominates $$$i=1$$$, which matches $$$d_1=1$$$.
For $$$i=2$$$, we have $$$p_2=3$$$ and $$$q_2=2$$$. The only larger index is $$$j=3$$$, but $$$p_3=1 \lt p_2$$$, so it does not dominate. Thus, $$$0$$$ indices dominate $$$i=2$$$, matching $$$d_2=0$$$.
For $$$i=3$$$, there are no larger indices $$$j \gt 3$$$, so $$$0$$$ indices dominate, matching $$$d_3=0$$$.

In the second test case, $$$p = [3, 4, 1, 2]$$$ and $$$d = [2, 1, 1, 0]$$$. Let's look at $$$i=2$$$, where $$$p_2=4$$$. Since there is no index $$$j \gt 2$$$ with $$$p_j \gt p_2$$$, no index can dominate $$$i=2$$$. However, the array $$$d$$$ requires $$$d_2=1$$$, which makes it impossible to construct a valid permutation $$$q$$$. Hence, the answer is -1.

Definitely Larger

Examples