
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
Let \( p \) and \( q \) be relatively prime positive integers. A subset \( S \) of \(\{0,1,2, \ldots\} \) is called ideal if \(0 \in S\) and for each element \(n \in S\), the integers \(n+p\) and \(n+q\) belong to \(S\). Determine the number of ideal subsets of \(\{0,1,2, \ldots\}\).

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