
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
Given $\sqrt{0.01}=0.1$, $\sqrt{1}=1$, $\sqrt{100}=10$, $\sqrt{10000}=100$, $\ldots \ldots $  
$(1)$ Fill in the blanks: $\sqrt{0.0001}=$______, $\sqrt{1{0}^{6}}=$______;  
$(2)$ According to the above rule, what is the relationship between the movement of the decimal point of a number $a$ and its square root $\sqrt{a}$?  
$(3)$ Solve the following problems according to the rule in (2):  
① Given $\sqrt{0.1}≈0.316$, then $\sqrt{0.001}≈$______;  
② Given $\sqrt{x}=1.414$, $\sqrt{y}=141.4$, express $y$ in terms of $x$, then $y=$______;

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