
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
In the previous month, we learned about the operation of "powers of rational numbers" and understood that the result of exponentiation is called "power." Now, we introduce a new operation related to "power," defined as: $a^{m}$ 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: 

$a^{m}\div a^{n}=\left\{\begin{array}{l}{\text{When } m>n, \ a^{m}\div a^{n}=a^{m-n}}\\{\text{When } m=n, \ a^{m}\div a^{n}=1}\\{\text{When } m<n, \ a^{m}\div a^{n}=\frac{1}{a^{n-m}}}\end{array}\right.$

**Solve the problem:**

According to the operation rules of "division of powers with the same base," answer the following questions:

$(1)$ Fill in the blanks: $(\frac{1}{3})^4\div(\frac{1}{3})^2=$______, $2^{3}\div 2^{7}=$______;

$(2)$ If $3\div3^{3x+4}=\frac{1}{81}$, find the value of $x$;

$(3)$ If $\left(5-2x\right)^{3x-1}\div \left(5-2x\right)^{x+7}=1$, please directly write out the value of $x$.

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