
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 $2\div 2\div 2$, $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$, etc. By analogy with rational number exponentiation, we denote $2\div 2\div 2$ as $2_{3}$, read as "the third quotient of $2$", $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ as $\left(-3\right)_{4}$, read as "the fourth quotient of $-3$". Generally, we denote the division of $n$ $a$'s ($a\neq 0$) as $a_{n}$, read as "$a$ to the power of $n$ quotient".  
Preliminary exploration  
$(1)$ Write down the result directly: $2_{3}=$______;  
$(2)$ Regarding division powers, which of the following statements is incorrect?  
① The second quotient of any non-zero number is equal to $1$;  
② For any positive integer $n$, $\left(-1\right)_{n}=-1$;  
③ $3_{4}=4_{3}$;  
④ The result of a negative number raised to an odd power is negative, and the result of a negative number raised to an even power is positive.  
In-depth 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: ${2_4}=2\div 2\div 2\div 2=2\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=({\frac{1}{2}})^2$.  
$(3)$ Try it out: Following the above formula, write the following operations directly in the form of exponentiation: $\left(-3\right)_{4}=$______; ${({\frac{1}{7}})_5}=$______;  
$(4)$ Think about it: Writing the $n$th quotient of a non-zero rational number $a$ in the form of exponentiation is equal to______;  
$(5)$ Calculate: ${5_2}÷{({-\frac{1}{2}})_4}×{({-\frac{1}{3}})_5}+{({-\frac{1}{4}})_3}×\frac{1}{4}=\_\_\_\_\_\_$.

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