
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 positive integers  $n$  such that there exists a sequence of positive integers  $a_1$ ,  $a_2$ , $\ldots$ ,  $a_n$  satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every  $k$  with  $2\leq k\leq n-1$ .

*Proposed by North Korea*

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