
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
Xiao Ming is a smart and imaginative child. After learning about "powers of rational numbers," he came up with the concept of "division powers" by using the mathematical knowledge of "powers" to define the concept of "division powers" with great creativity. He stipulated that several division operations of the same rational numbers (none of which can be $0$) are called division powers, such as $5\div 5\div 5$, $\left(-2\right)\div \left(-2\right)\div \left(-2\right)\div \left(-2\right)$, etc., analogous to powers of rational numbers. Xiao Ming represents $5\div 5\div 5$ as $f\left(3,5\right)$ and $\left(-2\right)\div \left(-2\right)\div \left(-2\right)\div \left(-2\right)$ as $f\left(4,-2\right)$.

$(1)$ Write down the results directly: $f(5,\frac{1}{2})=\_\_\_\_\_\_$, $f\left(6,3\right)=\_\_\_\_\_\_$;  
$(2)$ After deep thinking, Xiao Ming found that "division powers" operations can be transformed into power operations, and the results can be written in the form of powers. Therefore, the formula for the "division powers" operation is $f\left(n,a\right)=\_\_\_\_\_\_$ (where $n$ is a positive integer, express the result in terms of $a$ and $n$);  
$(3)$ Using the formula from $(2)$, calculate: $f(5,\frac{1}{3})\times f(4,3)\times f(5,\frac{1}{2})+f(5,-\frac{1}{4})÷f(6,-\frac{1}{2})$.

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