
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 the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as "division exponentiation," denoted as $a^{ⓝ}$, read as "$a$ circle $n$ times." For example, $2\div 2\div 2$ is denoted as $2^{③}$, read as "$2$ circle $3$ times"; $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ is denoted as $\left(-3\right)^{④}$, read as "$-3$ circle $4$ times".<br/>$(1)$ Write down the results directly: $2^{③}=$______, $(-\frac{1}{2})^{④}=$______; <br/>$(2)$ Division exponentiation can also be converted into the form of powers, such as $2^{④}=2\div 2\div 2\div 2=2\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=(\frac{1}{2})^{2}$. Try to directly write the following operation results in the form of powers: $\left(-3\right)^{④}=$______; ($\frac{1}{2})^{⑩}=$______; $a^{ⓝ}=$______; <br/>$(3)$ Calculate: $2^{2}\times (-\frac{1}{3})^{④}\div \left(-2\right)^{③}-\left(-3\right)^{②}$.

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