
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
This semester we have learned about the operation of "rational number exponentiation" and know that the result of exponentiation is called "power". Now, we introduce a new operation related to "power".  
Definition: $a^{m}$ and $a^{n}$ (where $a\neq 0$, and $m$ and $n$ are positive integers) are called powers with the same base. The division of powers with the same base is denoted as $a^{m}\div a^{n}$.  
The operation rules are as follows:  
$a^{m}\div a^{n}=\begin{cases} \text{When } m>n, \frac{a^m}{a^n}=a^{m-n} \\ \text{When } m=n, \frac{a^m}{a^n}=1 \\ \text{When } m<n, \frac{a^m}{a^n}=\frac{1}{a^{n-m}} \end{cases}$  
According to the operation rules of "division of powers with the same base", answer the following questions:  
$(1)$ Fill in the blanks: ${(\frac{1}{2})^5}\div{(\frac{1}{2})^2}=$______, $4^{3}\div 4^{5}=\_\_\_\_\_\_$.  
$(2)$ If $3^{x-1}\div{3^{3x-4}}=\frac{1}{27}$, find the value of $x$.  
$(3)$ If $(x-1)^{2x+2}\div (x-1)^{x+6}=1$, please directly write down the value of $x$.

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