
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 pairs of positive integers \((a, b)\) satisfying the following conditions:
(i) \(a\) divides \(b^{4}+1\),
(ii) \(b\) divides \(a^{4}+1\),
(iii) \(\lfloor\sqrt{a}\rfloor=\lfloor\sqrt{b}\rfloor\).

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