
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 the number of pairs of positive integers  $(a, p)$  such that:
[list]
[*] $p$  is a prime greater than  $2.$ [*] $1 \le a \le 2024.$ [*] $a< p^4.$ [*] $ap^4 + 2p^3 + 2p^2 + 1$  is a perfect square.


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