
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
Read the following equations: $(a\cdot b)^{2}=a^{2}\cdot b^{2}$, $(a\cdot b)^{3}=a^{3}\cdot b^{3}$, $(a\cdot b)^{4}=a^{4}\cdot b^{4}$, $(a\cdot b)^{5}=a^{5}\cdot b^{5}\cdots$.  
Answer the following questions:  
$(1)$ Conjecture: $(a\cdot b)^{n}=\_\_\_\_\_\_$.  
$(2)$ Calculate: $2^5\times(-\frac{1}{2})^5$;  
$(3)$ Calculate: $\left(-0.125\right)^{2022}\times 2^{2021}\times 4^{2020}$.

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