
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 \( 2 \leq k \leq n-1 \).

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