A12622. Cute Sequences

Given a positive integer $m$ , we say that a sequence $x_1, x_2, \dots, x_n$ of positive integers is $m$ -cute if for every index $i$ such that $2 \le i \le n$ it holds that $x_i = x_{i - 1} + x_{i - 2} + \dots + x_1 + r_i$ for some positive integer $r_i$ satisfying $1 \le r_i \le m$ .

You will be given $q$ queries consisting of three positive integers $a$ , $b$ and $m$ . For each query you must determine whether or not there exists an $m$ -cute sequence whose first term is $a$ and whose last term is $b$ . If such a sequence exists, you must additionally find an example of it.

The first line contains an integer number $q$ ( $1 \le q \le 10^3$ ) — the number of queries.

Each of the following $q$ lines contains three integers $a$ , $b$ , and $m$ ( $1 \le a, b, m \le 10^{14}$ , $a \leq b$ ), describing a single query.

For each query, if no $m$ -cute sequence whose first term is $a$ and whose last term is $b$ exists, print $-1$ .

Otherwise print an integer $k$ ( $1 \le k \leq 50$ ), followed by $k$ integers $x_1, x_2, \dots, x_k$ ( $1 \le x_i \le 10^{14}$ ). These integers must satisfy $x_1 = a$ , $x_k = b$ , and that the sequence $x_1, x_2, \dots, x_k$ is $m$ -cute.

It can be shown that under the problem constraints, for each query either no $m$ -cute sequence exists, or there exists one with at most $50$ terms.

If there are multiple possible sequences, you may print any of them.

Consider the sample. In the first query, the sequence $5, 6, 13, 26$ is valid since $6 = 5 + \bf{\color{blue} 1}$ , $13 = 6 + 5 + {\bf\color{blue} 2}$ and $26 = 13 + 6 + 5 + {\bf\color{blue} 2}$ have the bold values all between $1$ and $2$ , so the sequence is $2$ -cute. Other valid sequences, such as $5, 7, 13, 26$ are also accepted.

In the second query, the only possible $1$ -cute sequence starting at $3$ is $3, 4, 8, 16, \dots$ , which does not contain $9$ .