
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
Observe the following pattern to answer the questions: $\sqrt[3]{{-0.001}}=-0.1$, $\sqrt[3]{{-1}}=-1$, $\sqrt[3]{{-1000}}=-10$, $\sqrt[3]{{0.001}}=0.1$, $\sqrt[3]{1}=1$, $\sqrt[3]{{1000}}=10\ldots $<br/>$(1)$ Find $\sqrt[3]{{0.000001}}=$______; $\sqrt[3]{{{{10}^6}}}=$______; According to the above pattern, what is the relationship between the decimal point movement of a number $a$ and its cube root $\sqrt[3]{a}$?<br/>$(2)$ Given $\sqrt[3]{x}=1.587$, if $\sqrt[3]{y}=-0.1587$, express $y$ in terms of $x$: $y=$______;<br/>$(3)$ Based on the pattern, determine the relationship between $\sqrt[3]{a}$ and $a$.

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