
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: $-\frac{1}{7}$, $2.5$, $(\sqrt{\frac{4}{5}})^{2}$, $\sqrt[3]{9}$, $0$, $\frac{π}{2}$, $-\sqrt{(-3)^{2}}$, $0.5757757775\ldots$ (the number of 7s between adjacent 5s increases by 1 each time).<br/>$(1)$ Rational number set: $\{ \_\_\_\_\_\_\ldots \}$;<br/>$(2)$ Irrational number set: $\{ \_\_\_\_\_\_\ldots \}$.

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