
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 material: Find the value of $1+2+2^{2}+2^{3}+2^{4}+\ldots +2^{2017}$.  
Solution: Let $S=1+2+2^{2}+2^{3}+2^{4}+\ldots +2^{2017}$, multiply both sides of the equation by $2$ simultaneously:  
$2S=2+2^{2}+2^{3}+2^{4}+2^{5}+\ldots +2^{2018}$  
Subtract the above equation from the previous one: $2S-S=2^{2018}-1$, which means $S=2^{2018}-1$  
Therefore, $1+2+2^{2}+2^{3}+2^{4}+\ldots +2^{2017}=2^{2018}-1$  
Please calculate the following using the same method:  
$(1)1+2+2^{2}+2^{3}+2^{4}+\ldots +2^{10}$  
$(2)1+3+3^{2}+3^{3}+3^{4}+\ldots +3^{2016}$.

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