
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) Let \( n \) be a positive integer that is not a perfect square. Find all pairs \((a, b)\) of positive integers for which there exists a positive real number \( r \), such that
\[ r^{a} + \sqrt{n} \]
and
\[ r^{b} + \sqrt{n} \]
are both rational numbers.

(b) Let \( n \) be a positive integer that is not a perfect square. Find all pairs \((a, b)\) of positive integers for which there exists a real number \( r \), such that
\[ r^{a} + \sqrt{n} \]
and
\[ r^{b} + \sqrt{n} \]
are both rational numbers.

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