
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 positive integer pairs  $(a, b)$  such that  $\frac{2^{a+b}+1}{2^a+2^b+1}$  is an integer.**General**
Find all positive integer triples  $(t, a, b)$  such that  $\frac{t^{a+b}+1}{t^a+t^b+1}$  is an integer.


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