
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
Analogous to exponentiation, we define the operation of dividing $n$ identical rational numbers (all non-zero) as "dividing operation." For example, $2\div 2\div 2\div 2$ is denoted as $2^{"4"}$, read as "the quotient of $2$ raised to the power of $4$." Generally, we denote $\underset{\underbrace{a÷a÷a÷a}}{n}$ ($a\neq 0$, $n\geqslant 2$, and both are integers) as $a^{"n"}$, read as "$a$ raised to the power of $n$."  
$(1)$ Calculate the following directly: $(\frac{1}{2})^{"4"}=\_\_\_\_\_\_$, $\left(-3\right)^{"3"}=\_\_\_\_\_\_$.  
$(2)$ Induction: The result of a negative number raised to a positive odd power is ______, and the result of a negative number raised to a positive even power is ______ (fill in "positive" or "negative").  
$(3)$ Calculate: $\left(-8\right)\div 2^{"3"}+11\times (-\frac{1}{4})^{"4"}$.

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