
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
Let the sequence  $ \{a_n\}_{n\geq 1}$  be defined by  $ a_1 \equal{} 20$ ,  $ a_2 \equal{} 30$  and  $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$  for all  $ n\geq 1$ . Find all positive integers  $ n$  such that  $ 1 \plus{} 5a_n a_{n \plus{} 1}$  is a perfect square.

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