A16147. Rivalries

Ntarsis has an array $a$ of length $n$ .

The power of a subarray $a_l \dots a_r$ ( $1 \leq l \leq r \leq n$ ) is defined as:

- The largest value $x$ such that $a_l \dots a_r$ contains $x$ and neither $a_1 \dots a_{l-1}$ nor $a_{r+1} \dots a_n$ contains $x$ .
- If no such $x$ exists, the power is $0$ .

Call an array $b$ a rival to $a$ if the following holds:

- The length of both $a$ and $b$ are equal to some $n$ .
- Over all $l, r$ where $1 \leq l \leq r \leq n$ , the power of $a_l \dots a_r$ equals the power of $b_l \dots b_r$ .
- The elements of $b$ are positive.

Ntarsis wants you to find a rival $b$ to $a$ such that the sum of $b_i$ over $1 \leq i \leq n$ is maximized. Help him with this task!

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^5$ ). The description of the test cases follows.

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

The next line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \leq a_i \leq 10^9$ ).

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

For each test case, output $n$ integers $b_1, b_2, \ldots, b_n$ — a valid rival to $a$ such that $b_1 + b_2 + \cdots + b_n$ is maximal.

If there exist multiple rivals with the maximum sum, output any of them.

For the first test case, one rival with the maximal sum is $[2, 4, 2, 3, 3]$ .

$[2, 4, 2, 3, 3]$ can be shown to be a rival to $[1, 4, 1, 3, 3]$ .

All possible subarrays of $a$ and $b$ and their corresponding powers are listed below:

- The power of $a[1:1] = [1] = 0$ , the power of $b[1:1] = [2] = 0$ .
- The power of $a[1:2] = [1, 4] = 4$ , the power of $b[1:2] = [2, 4] = 4$ .
- The power of $a[1:3] = [1, 4, 1] = 4$ , the power of $b[1:3] = [2, 4, 2] = 4$ .
- The power of $a[1:4] = [1, 4, 1, 3] = 4$ , the power of $b[1:4] = [2, 4, 2, 3] = 4$ .
- The power of $a[1:5] = [1, 4, 1, 3, 3] = 4$ , the power of $b[1:5] = [2, 4, 2, 3, 3] = 4$ .
- The power of $a[2:2] = [4] = 4$ , the power of $b[2:2] = [4] = 4$ .
- The power of $a[2:3] = [4, 1] = 4$ , the power of $b[2:3] = [4, 2] = 4$ .
- The power of $a[2:4] = [4, 1, 3] = 4$ , the power of $b[2:4] = [4, 2, 3] = 4$ .
- The power of $a[2:5] = [4, 1, 3, 3] = 4$ , the power of $b[2:5] = [4, 2, 3, 3] = 4$ .
- The power of $a[3:3] = [1] = 0$ , the power of $b[3:3] = [2] = 0$ .
- The power of $a[3:4] = [1, 3] = 0$ , the power of $b[3:4] = [2, 3] = 0$ .
- The power of $a[3:5] = [1, 3, 3] = 3$ , the power of $b[3:5] = [2, 3, 3] = 3$ .
- The power of $a[4:4] = [3] = 0$ , the power of $b[4:4] = [3] = 0$ .
- The power of $a[4:5] = [3, 3] = 3$ , the power of $b[4:5] = [3, 3] = 3$ .
- The power of $a[5:5] = [3] = 0$ , the power of $b[5:5] = [3] = 0$ .

It can be shown there exists no rival with a greater sum than $2 + 4 + 2 + 3 + 3 = 14$ .