
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
Given the sequence \(\{a_n\}\) such that \(a_1 = a_2 = a_3 = 1\), and for any positive integer \(n\), 

$$
a_{n+3} = \frac{a_{n+1} a_{n+2} + 1}{a_n},
$$

find the general term formula for the sequence \(\{a_n\}\).

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