
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
It is given the sequence defined by  $$ \{a_{n+2}=6a_{n+1}-a_n\}_{n \in \mathbb{Z}_{>0}},a_1=1, a_2=7 \text{.} $$  Find all  $n$  such that there exists an integer  $m$  for which  $a_n=2m^2-1$ .

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