
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 sequences of positive integers  $(a_n)_{n\ge 1}$  which satisfy  $$ a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1) $$  for all  $n\in \mathbb{Z}_{\ge 1}$ . 
*(Bogdan Blaga)*

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