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

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

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.

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 $$$a_j$$$ ($$$1 \leq j \lt i$$$) that $$$a_j \gt k$$$. Therefore, $$$b_1 = n + 1 = 5$$$.
When $$$i = 2$$$, $$$x = a_i = 3 \gt k$$$, 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 $$$a_j \gt k$$$ is $$$a_2$$$. Therefore, $$$b_2 = a_2 = 3$$$.
When $$$i = 4$$$, $$$x = a_i = 4 \gt k$$$, the last element $$$a_j$$$ that $$$a_j \leq k$$$ is $$$a_3$$$. Therefore, $$$b_4 = a_3 = 2$$$.

Finally, we obtain sequence $$$b = [5,3,1,2]$$$.

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 $$$a_j$$$ ($$$1 \leq j \lt i$$$) that $$$a_j \gt k$$$. Therefore, $$$b_1 = b_2 = b_3 = n + 1 = 7$$$.
When $$$i = 4, 5, 6$$$, $$$a_i \gt k$$$, the last element $$$a_j$$$ that $$$a_j \leq k$$$ is $$$a_3$$$. Therefore, $$$b_4 = b_5 = b_6 = a_3 = 3$$$.

Finally, we obtain sequence $$$b = [7,7,7,3,3,3]$$$.

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

When $$$i = 1, 2, 3$$$, $$$a_i \gt k$$$, there is no $$$a_j$$$ ($$$1 \leq j \lt i$$$) 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 $$$a_j \gt k$$$ is $$$a_3$$$. Therefore, $$$b_1 = b_2 = b_3 = a_3 = 4$$$.

Finally, we obtain sequence $$$b = [4,4,4,0,0,0]$$$.

Permutation Addicts

Examples