Playlist
Contest: Технокубок 2021 - Финал
Difficulty rating: 1900
Tags: data structures, dsu, implementation, shortest paths

Arkady has a playlist that initially consists of $$$n$$$ songs, numerated from $$$1$$$ to $$$n$$$ in the order they appear in the playlist. Arkady starts listening to the songs in the playlist one by one, starting from song $$$1$$$. The playlist is cycled, i. e. after listening to the last song, Arkady will continue listening from the beginning.

Each song has a genre $$$a_i$$$, which is a positive integer. Let Arkady finish listening to a song with genre $$$y$$$, and the genre of the next-to-last listened song be $$$x$$$. If $$$\operatorname{gcd}(x, y) = 1$$$, he deletes the last listened song (with genre $$$y$$$) from the playlist. After that he continues listening normally, skipping the deleted songs, and forgetting about songs he listened to before. In other words, after he deletes a song, he can't delete the next song immediately.

Here $$$\operatorname{gcd}(x, y)$$$ denotes the greatest common divisor (GCD) of integers $$$x$$$ and $$$y$$$.

For example, if the initial songs' genres were $$$[5, 9, 2, 10, 15]$$$, then the playlist is converted as follows: [5, 9, 2, 10, 15] $$$\to$$$ [5, 9, 2, 10, 15] $$$\to$$$ [5, 2, 10, 15] (because $$$\operatorname{gcd}(5, 9) = 1$$$) $$$\to$$$ [5, 2, 10, 15] $$$\to$$$ [5, 2, 10, 15] $$$\to$$$ [5, 2, 10, 15] $$$\to$$$ [5, 2, 10, 15] $$$\to$$$ [5, 2, 10, 15] $$$\to$$$ [5, 10, 15] (because $$$\operatorname{gcd}(5, 2) = 1$$$) $$$\to$$$ [5, 10, 15] $$$\to$$$ [5, 10, 15] $$$\to$$$ ... The bold numbers represent the two last played songs. Note that after a song is deleted, Arkady forgets that he listened to that and the previous songs.

Given the initial playlist, please determine which songs are eventually deleted and the order these songs are deleted.

Input format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10\,000$$$). Description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the number of songs.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the genres of the songs.

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

Output format:
For each test case, print a single line. First, print a single integer $$$k$$$ — the number of deleted songs. After that print $$$k$$$ distinct integers: deleted songs in the order of their deletion.

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
Explanation of the first test case is given in the statement.

In the second test case, the playlist is converted as follows: [1, 2, 4, 2, 4, 2] $$$\to$$$ [1, 2, 4, 2, 4, 2] $$$\to$$$ [1, 4, 2, 4, 2] (because $$$\operatorname{gcd}(1, 2) = 1$$$) $$$\to$$$ [1, 4, 2, 4, 2] $$$\to$$$ [1, 4, 2, 4, 2] $$$\to$$$ [1, 4, 2, 4, 2] $$$\to$$$ [1, 4, 2, 4, 2] $$$\to$$$ [1, 4, 2, 4, 2] $$$\to$$$ [4, 2, 4, 2] (because $$$\operatorname{gcd}(2, 1) = 1$$$) $$$\to$$$ [4, 2, 4, 2] $$$\to$$$ ...

In the third test case, the playlist is converted as follows: [1, 2] $$$\to$$$ [1, 2] $$$\to$$$ [1] (because $$$\operatorname{gcd}(1, 2) = 1$$$) $$$\to$$$ [1] $$$\to$$$ [1] (Arkady listened to the same song twice in a row) $$$\to$$$ [] (because $$$\operatorname{gcd}(1, 1) = 1$$$).

The fourth test case is same as the third after deletion of the second song.

In the fifth test case, the same song is listened to over and over again, but since $$$\operatorname{gcd}(2, 2) \ne 1$$$, it is not deleted.

Example input:
5
5
5 9 2 10 15
6
1 2 4 2 4 2
2
1 2
1
1
1
2

Example output:
2 2 3 
2 2 1 
2 2 1 
1 1 
0

Editorial:
Let's call a pair of songs bad if GCD of their genres is $$$1$$$. It is easy to check if two songs are bad with Euclid's algorithm in $$$O(\log{C})$$$.
There are at most $$$n$$$ deletions, so simulation is fine in terms of time limit, but we should be able to find the next deleted song quickly. Let's maintain an ordered set of songs on the playlist, and also another ordered set that stores pairs of bad consecutive songs. It is easy to see that after we delete a song, only constant number of changes is needed to these sets. Let the song being deleted be $$$b$$$, while the previous song be $$$a$$$, and the next song in the playlist be $$$c$$$. It is easy to find $$$a$$$ and $$$c$$$ using the playlist set. Then we should:
• remove $$$b$$$ from the playlist set;
• remove $$$(a, b)$$$ from the bad pairs set;
• remove $$$(b, c)$$$ from the bad pairs set if this pair is bad;
• add $$$(a, c)$$$ to the bad pairs set if this pair is bad.
If we store the sets in some fast enough data structure (e. g. balanced binary tree, standard C++'s set is enough), we have fast running time ($$$O(n (\log{n} + \log{C}))$$$).
It is also possible to solve this problem with double-linked lists or DSU with $$$O(n \log{C})$$$ complexity.