
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  $a\in \{1,2,3,4\}$  such that if  $b=2a$ , then there exist infinitely many positive integers  $n$  such that   $$ \underbrace{aa\dots aa}_\textrm{ $2n$ }-\underbrace{bb\dots bb}_\textrm{ $n$ } $$   is a perfect square.

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