CF1736D.Equal Binary Subsequences

Everool has a binary string $s$ of length $2n$ . Note that a binary string is a string consisting of only characters $0$ and $1$ . He wants to partition $s$ into two disjoint equal subsequences. He needs your help to do it.

You are allowed to do the following operation exactly once.

• You can choose any subsequence (possibly empty) of $s$ and rotate it right by one position.

In other words, you can select a sequence of indices $b_1, b_2, \ldots, b_m$ , where $1 \le b_1 < b_2 < \ldots < b_m \le 2n$ . After that you simultaneously set $$$$s_{b_1} := s_{b_m}, $$ $$ s_{b_2} := s_{b_1}, $$ $$ \ldots, $$ $$ s_{b_m} := s_{b_{m-1}}. $$

Can you partition $s$ into two disjoint equal subsequences after performing the allowed operation exactly once?

A partition of $s$ into two disjoint equal subsequences $s^p$ and $s^q$ is two increasing arrays of indices $p\_1, p\_2, \\ldots, p\_n$ and $q\_1, q\_2, \\ldots, q\_n$ , such that each integer from $1$ to $2n$ is encountered in either $p$ or $q$ exactly once, $s^p = s\_{p\_1} s\_{p\_2} \\ldots s\_{p\_n}$ , $s^q = s\_{q\_1} s\_{q\_2} \\ldots s\_{q\_n}$ , and $s^p = s^q$ .

If it is not possible to partition after performing any kind of operation, report $-1$ .

If it is possible to do the operation and partition $s$ into two disjoint subsequences $s^p$ and $s^q$ , such that $s^p = s^q$ , print elements of $b$ and indices of $s^p$ , i. e. the values $p\_1, p\_2, \\ldots, p\_n$$$.

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

The first line of each test case contains a single integer $n$ ( $1 \le n \le 10^5$ ), where $2n$ is the length of the binary string.

The second line of each test case contains the binary string $s$ of length $2n$ .

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

For each test case, follow the following output format.

If there is no solution, print $-1$ .

Otherwise,

• In the first line, print an integer $m$ ( $0 \leq m \leq 2n$ ), followed by $m$ distinct indices $b_1$ , $b_2$ , ..., $b_m$ (in increasing order).
• In the second line, print $n$ distinct indices $p_1$ , $p_2$ , ..., $p_n$ (in increasing order).

If there are multiple solutions, print any.

In the first test case, $b$ is empty. So string $s$ is not changed. Now $s^p = s_1 s_2 = \mathtt{10}$ , and $s^q = s_3s_4 = \mathtt{10}$ .

In the second test case, $b=[3,5]$ . Initially $s_3=\mathtt{0}$ , and $s_5=\mathtt{1}$ . On performing the operation, we simultaneously set $s_3=\mathtt{1}$ , and $s_5=\mathtt{0}$ .

So $s$ is updated to 101000 on performing the operation.

Now if we take characters at indices $[1,2,5]$ in $s^p$ , we get $s_1=\mathtt{100}$ . Also characters at indices $[3,4,6]$ are in $s^q$ . Thus $s^q=100$ . We are done as $s^p=s^q$ .

In fourth test case, it can be proved that it is not possible to partition the string after performing any operation.