
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
(a) Determine a natural number  $n$  such that  $n(n+2022)+2$  is a perfect square.
<details><summary>Spoiler</summary>In case you didn't realize,  $n=1$  works lol</details>
(b) Determine all natural numbers  $a$  such that for every natural number  $n$ , the number  $n(n+a)+2$  is never a perfect square.

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