
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: $\left(a\times b\right)^{2}=a^{2}\times b^{2}$, $\left(a\times b\right)^{3}=a^{3}\times b^{3}$, $\left(a\times b\right)^{4}=a^{4}\times b^{4}$, $\left(a\times b\right)^{5}=a^{5}\times b^{5}\ldots $ Answer the following questions:<br/>$(1)$ Conjecture: $\left(a\times b\right)^{n}=\_\_\_\_\_\_.$<br/>$(2)$ Calculate: $\left(-0.125\right)^{2022}\times 2^{2021}\times 4^{2020}$.

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