
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
Fill in the corresponding sets for the following numbers: $({-2})^2$, $-\frac{4}{3}$, $0$, $-\pi$, $\frac{22}{9}$, $0.050050005\ldots$, $\frac{\pi}{3}$, $300\%$, $-(-\frac{2}{5})$, $-|-3.14|$, $-0.\overline{1}\overline{5}$.

$(1)$ Set of negative numbers: $\{ \_\_\_\_\_\_\ldots \}$;<br/>$(2)$ Set of non-negative integers: $\{ \_\_\_\_\_\_\ldots \}$;<br/>$(3)$ Set of fractions: $\{ \_\_\_\_\_\_\ldots \}$;<br/>$(4)$ Set of irrational numbers: $\{ \_\_\_\_\_\_\ldots \}$.

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