
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
Determine all pairs  $\{a, b\}$  of positive integers with the property that, in whatever manner you color the positive integers with two colors  $A$  and  $B$ , there always exist two positive integers of color  $A$  having their difference equal to  $a$  **or** of color  $B$  having their difference equal to  $b$ .

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