You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 512.0 MB

# Problem
Given a permutation $$$a_1, a_2, \dots, a_n$$$ of integers from $$$1$$$ to $$$n$$$, and a threshold $$$k$$$ with $$$0 \leq k \leq n$$$, you compute a sequence $$$b_1, b_2, \dots, b_n$$$ as follows.

For every $$$1 \leq i \leq n$$$ in increasing order, let $$$x = a_i$$$.

- If $$$x \leq k$$$, set $$$b_{x}$$$ to the last element $$$a_j$$$ ($$$1 \leq j < i$$$) that $$$a_j > k$$$. If no such element $$$a_j$$$ exists, set $$$b_{x} = n+1$$$.
- If $$$x > k$$$, set $$$b_{x}$$$ to the last element $$$a_j$$$ ($$$1 \leq j < i$$$) that $$$a_j \leq k$$$. If no such element $$$a_j$$$ exists, set $$$b_{x} = 0$$$.

Unfortunately, after the sequence $$$b_1, b_2, \dots, b_n$$$ has been completely computed, the permutation $$$a_1, a_2, \dots, a_n$$$ and the threshold $$$k$$$ are discarded.

Now you only have the sequence $$$b_1, b_2, \dots, b_n$$$. Your task is to find any possible permutation $$$a_1, a_2, \dots, a_n$$$ and threshold $$$k$$$ that produce the sequence $$$b_1, b_2, \dots, b_n$$$. It is guaranteed that there exists at least one pair of permutation $$$a_1, a_2, \dots, a_n$$$ and threshold $$$k$$$ that produce the sequence $$$b_1, b_2, \dots, b_n$$$.

A permutation of integers from $$$1$$$ to $$$n$$$ is a sequence of length $$$n$$$ which contains all integers from $$$1$$$ to $$$n$$$ exactly once.

## Input Format
Each test contains multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases. The following lines contain the description of each test case.

The first line of each test case contains an integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$), indicating the length of the permutation $$$a$$$.

The second line of each test case contains $$$n$$$ integers $$$b_1, b_2, \dots, b_n$$$ ($$$0 \leq b_i \leq n+1$$$), indicating the elements of the sequence $$$b$$$.

It is guaranteed that there exists at least one pair of permutation $$$a_1, a_2, \dots, a_n$$$ and threshold $$$k$$$ that produce the sequence $$$b_1, b_2, \dots, b_n$$$.

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

## Output Format
For each test case, output the threshold $$$k$$$ ($$$0 \leq k \leq n$$$) in the first line, and then output the permutation $$$a_1, a_2, \dots, a_n$$$ ($$$1 \leq a_i \leq n$$$) in the second line such that the permutation $$$a_1, a_2, \dots, a_n$$$ and threshold $$$k$$$ produce the sequence $$$b_1, b_2, \dots, b_n$$$. If there are multiple solutions, you can output any of them.

## Examples
```input
3
4
5 3 1 2
6
7 7 7 3 3 3
6
4 4 4 0 0 0
```
```output
2
1 3 2 4
3
1 2 3 4 5 6
3
6 5 4 3 2 1
```

## Note
For the first test case, permutation $$$a = [1,3,2,4]$$$ and threshold $$$k = 2$$$ will produce sequence $$$b$$$ as follows.

- When $$$i = 1$$$, $$$x = a_i = 1 \leq k$$$, there is no $$$a_j$$$ ($$$1 \leq j < i$$$) that $$$a_j > k$$$. Therefore, $$$b_1 = n + 1 = 5$$$.
- When $$$i = 2$$$, $$$x = a_i = 3 > k$$$, the last element $$$a_j$$$ that $$$a_j \leq k$$$ is $$$a_1$$$. Therefore, $$$b_3 = a_1 = 1$$$.
- When $$$i = 3$$$, $$$x = a_i = 2 \leq k$$$, the last element $$$a_j$$$ that $$$a_j > k$$$ is $$$a_2$$$. Therefore, $$$b_2 = a_2 = 3$$$.
- When $$$i = 4$$$, $$$x = a_i = 4 > k$$$, the last element $$$a_j$$$ that $$$a_j \leq k$$$ is $$$a_3$$$. Therefore, $$$b_4 = a_3 = 2$$$.

For the second test case, permutation $$$a = [1,2,3,4,5,6]$$$ and threshold $$$k = 3$$$ will produce sequence $$$b$$$ as follows.

- When $$$i = 1, 2, 3$$$, $$$a_i \leq k$$$, there is no $$$a_j$$$ ($$$1 \leq j < i$$$) that $$$a_j > k$$$. Therefore, $$$b_1 = b_2 = b_3 = n + 1 = 7$$$.
- When $$$i = 4, 5, 6$$$, $$$a_i > k$$$, the last element $$$a_j$$$ that $$$a_j \leq k$$$ is $$$a_3$$$. Therefore, $$$b_4 = b_5 = b_6 = a_3 = 3$$$.

For the third test case, permutation $$$a = [6,5,4,3,2,1]$$$ and threshold $$$k = 3$$$ will produce sequence $$$b$$$ as follows.

- When $$$i = 1, 2, 3$$$, $$$a_i > k$$$, there is no $$$a_j$$$ ($$$1 \leq j < i$$$) that $$$a_j \leq k$$$. Therefore, $$$b_4 = b_5 = b_6 = 0$$$.
- When $$$i = 4, 5, 6$$$, $$$a_i \leq k$$$, the last element $$$a_j$$$ that $$$a_j > k$$$ is $$$a_3$$$. Therefore, $$$b_1 = b_2 = b_3 = a_3 = 4$$$.

Now solve the problem and return the code.