
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
[Concept Learning]  
Definition: Several divisions of the same rational number (all not equal to $0$) are called division powers, such as $2\div 2\div 2$, etc. Analogous to rational number exponentiation, we denote $2\div 2\div 2$ as $2_{3}$, read as "the $3$rd power of $2$", generally, $n$ divisions of $a$ ($a\neq 0$) are denoted as $a_{n}$, read as "$a$ to the power of $n$".  
[Preliminary Exploration]  
$(1)$ Write down the result directly: $2_{3}=\_\_\_\_\_\_$.  
$(2)$ Regarding division powers, the correct statements are:  
① The $2$nd power of any non-zero number is equal to $1$;  
② For any positive integer $n$, $1_{n}=1$;  
③ $3_{4}=4_{3}$;  
④ The result of a negative number to an odd power is negative, and to an even power is positive.  
[Further Thinking]  
We know that subtraction of rational numbers can be converted into addition, and division can be converted into multiplication. How can division powers of rational numbers be converted into exponentiation?  
For example: $2_{4}=2\div 2\div 2\div 2=2\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}={(\frac{1}{2})}^{2}$ (in exponent form);  
$(1)$ Try to write the following division powers directly in exponent form. $5_{6}=$______; ${(-\frac{1}{2})}_{10}=$______;  
$(2)$ Calculate: ${(-\frac{1}{4})}_{4}\div 2^{3}+(-8)\times 2_{3}$.

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