
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
Write the following numbers into the corresponding sets: $-\frac{1}{2}$, $\sqrt[3]{-11}$, $0.1$, $\frac{π}{2}$, $\sqrt{36}$, $\sqrt[3]{-8}$, $0$, $0.1212212221\ldots$ (the number where the number of 2s between two adjacent 1s increases by 1 each time).<br/>$(1)$ Set of positive numbers $\{\_\_\_\_\_\_\ldots \}$;<br/>$(2)$ Set of rational numbers $\{\_\_\_\_\_\_\ldots \}$;<br/>$(3)$ Set of irrational numbers $\{\_\_\_\_\_\_\ldots \}$.

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