
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For the positive integers  $a, b, x_1$  we construct the sequence of numbers  $(x_n)_{n=1}^{\infty}$  such that   $x_n = ax_{n-1} + b$  for each  $n \ge  2$ . Specify the conditions for the given numbers  $a, b$  and  $x_1$  which are necessary and sufficient for all indexes  $m, n$  to apply the implication  $m | n \Rightarrow x_m | x_n$ .

(Jaromír Šimša)

Present the answer in LaTex format: \boxed{Your answer}
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