
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
$2^{1}-2^{0}=1=2^{0}$  
$2^{2}-2^{1}=2=2^{1}$  
$2^{3}-2^{2}=4=2^{2}$  
$\ldots $  
$(1)$ Observe the pattern of the above equations and try to write out the $n$th equation;  
$(2)$ Calculate $2^{0}+2^{1}+2^{2}+\ldots +2^{2023}$.

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