
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
The sequence \(a_n\) is defined by \(a_1 = 20\), \(a_2 = 30\), and \(a_{n+1} = 3a_n - a_{n-1}\). Find all \(n\) for which \(5a_{n+1} \cdot a_n + 1\) is a perfect square.

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