
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
There are six numbers \(a(b+8), b(a+8), ab, a(a+8), b(b+8), (a+8)(b+8)\) for any pair of natural numbers \(a\) and \(b\), where \(a \neq b\). Among them, for some \(a\) and \(b\), one can find perfect squares of integers. For which \(a\) and \(b\) will the number of squares be the maximum possible?

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