
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
It is stipulated that the division operation of several identical rational numbers (all not equal to $0$) is called a division power, 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 the exponentiation of rational numbers, we denote $2\div 2\div 2$ as $2^{③}$, read as "$2$ to the circle $3$ power", $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ as $\left(-3\right)^{④}$, read as "$-3$ to the circle $4$ power". Generally, we denote $\underbrace{a÷a÷a…÷a}_{n \text{ times}}(a\neq 0)$ as $a^{ⓝ}$, read as "$a$ to the circle $n$ power". Please read the above material and complete the following questions: <br/>【Direct Application】(1) Calculate directly: $a^{②}=$______, ${(-\frac{1}{3})}^{④}=\_\_\_\_\_\_.$ <br/>【Inductive Summary】(2) Raise a question: We know that the subtraction operation of rational numbers can be converted into addition operation, and the division operation can be converted into multiplication operation. How can the division power operation of rational numbers be converted into exponentiation operation? <br/> Carefully think and write the following operations directly in the form of powers. <br/> $5^{⑦}=\_\_\_\_\_\_;\left(-2\right)^{⑩}=$______; ${(-\frac{1}{2})}^{⑨}=\_\_\_\_\_\_.$ <br/> Inductive summary: $a^{⑧}=\_\_\_\_\_\_.$ <br/>【Flexible Application】(3) Calculate: ${(-\frac{1}{3})}^{④}×(-2)^{⑤}-{(-\frac{1}{3})}^{③}+{3}^{2}$.

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