
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
Observe the following expressions:<br/>$(x^{2}-1)\div \left(x-1\right)=x+1$<br/>$(x^{3}-1)\div \left(x-1\right)=x^{2}+x+1$<br/>$(x^{4}-1)\div \left(x-1\right)=x^{3}+x^{2}+x+1$<br/>$\ldots $<br/>Based on the pattern you discovered, answer the following questions:<br/>$(1)$ Write down the result directly: $(x^{5}-1)\div \left(x-1\right)=$______；<br/>$(2)$ If $n$ is a positive integer and $n\geqslant 2$, then $(x^{n}-1)\div \left(x-1\right)=$______；<br/>$(3)$ According to the pattern you discovered, calculate the value of $1+2+2^{2}+2^{3}+\ldots +2^{2020}+2^{2021}+2^{2022}$.

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