
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
Find all pairs  $(b,c)$  of positive integers, such that the sequence defined by  $a_1=b$ ,  $a_2=c$  and  $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$  for  $n \geq 1$  has only finite number of composite terms.

*Proposed by Oleg Mushkarov and Nikolai Nikolov*

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