
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 the pairs of positive integers $(a, b)$ for which the following statement is true: The set of positive integers can be partitioned into two disjoint sets, $H_{1}$ and $H_{2}$, such that neither $a$ nor $b$ can be expressed as the difference of two numbers both belonging to $H_{1}$ or both belonging to $H_{2}$.

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