An element $b_i$ ($1\le i\le m$) in an array $b_1, b_2, \ldots, b_m$ is a local minimum if at least one of the following holds:

• $2\le i\le m - 1$ and $b_i \lt b_{i - 1}$ and $b_i \lt b_{i + 1}$, or
• $i = 1$ and $b_1 \lt b_2$, or
• $i = m$ and $b_m \lt b_{m - 1}$.

Similarly, an element $b_i$ ($1\le i\le m$) in an array $b_1, b_2, \ldots, b_m$ is a local maximum if at least one of the following holds:

• $2\le i\le m - 1$ and $b_i \gt b_{i - 1}$ and $b_i \gt b_{i + 1}$, or
• $i = 1$ and $b_1 \gt b_2$, or
• $i = m$ and $b_m \gt b_{m - 1}$.

Note that local minima and maxima are not defined for arrays with only one element.

There is a hidden permutation$^{\text{∗}}$ $p$ of length $n$. The following two operations are applied to permutation $p$ alternately, starting from operation 1, until there is only one element left in $p$:

• Operation 1 — remove all elements of $p$ which are not local minima.
• Operation 2 — remove all elements of $p$ which are not local maxima.

More specifically, operation 1 is applied during every odd iteration, and operation 2 is applied during every even iteration, until there is only one element left in $p$.

For each index $i$ ($1\le i\le n$), let $a_i$ be the iteration number that element $p_i$ is removed, or $-1$ if it was never removed.

It can be proven that there will be only one element left in $p$ after at most $\lceil \log_2 n\rceil$ iterations (in other words, $a_i \le \lceil \log_2 n\rceil$).

You are given the array $a_1, a_2, \ldots, a_n$. Your task is to construct any permutation $p$ of $n$ elements that satisfies array $a$.

$^{\text{∗}}$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 $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$) — the number of elements in permutation $p$.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le \lceil\log_2 n\rceil$ or $a_i = -1$) — the iteration number that element $p_i$ is removed.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

It is guaranteed that there exists at least one permutation $p$ that satisfies array $a$.

Output

For each test case, output $n$ integers representing the elements of the permutation satisfying array $a$.

If there are multiple solutions, you may output any of them.

Note

In the first test case, operations will be applied to permutation $[3, 2, 1]$ as follows:

1. The only local minimum in $[3, 2, 1]$ is $1$. Hence, elements $3$ and $2$ are removed. There is only one remaining element; hence the process terminates.

This satisfies array $a = [1, 1, -1]$ as both $p_1$ and $p_2$ were removed on iteration number $1$, while $p_3$ was not removed.

In the second test case, operations will be applied to permutation $p = [4, 3, 5, 1, 2]$ as follows:

1. The local minima in $[4, 3, 5, 1, 2]$ are $3$ and $1$. Hence, elements $4$, $5$, and $2$ are removed.
2. The only local maximum in $[3, 1]$ is $3$. Hence, element $1$ is removed. There is only one remaining element; hence the process terminates.

This satisfies array $a = [1, -1, 1, 2, 1]$ as elements $p_1 = 4$, $p_3 = 5$, and $p_5 = 2$ were removed on iteration $1$, element $p_4 = 1$ was removed on iteration $2$, and element $p_2 = 3$ was not removed.

In the third test case, operations will be applied on permutation $[6, 7, 2, 4, 3, 8, 5, 1]$ as follows:

1. The local minima in $[6, 7, 2, 4, 3, 8, 5, 1]$ are $6$, $2$, $3$, and $1$. Hence, elements $7$, $4$, $8$, and $5$ are removed.
2. The local maxima in $[6, 2, 3, 1]$ are $6$ and $3$. Hence, elements $2$ and $1$ are removed.
3. The only local minimum in $[6, 3]$ is $3$. Hence, element $6$ is removed. There is only one remaining element; hence the process terminates.

In the fourth test case, one permutation satisfying the constraints is [$6$, $5$, $2$, $1$, $3$, $4$, $7$]. $1$ is the only local minimum, so only it will stay after the first iteration. Note that there are other valid permutations; for example, [$6$, $4$, $3$, $1$, $2$, $5$, $7$] would also be considered correct.