
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
(1) Fill in the blank: $(a-1)(a+1)=$ $(a-1)(a^2+a+1)=$ $(a-1)(a^3+a^2+a+1)=$  
(2) Have you noticed the pattern? Please use the pattern you found to fill in the blank: $(a-1)(a^n+a^{n-1}+\ldots+a^2+a+1)=$  
(3) Based on the pattern above, please calculate the value of $4^{2012}+4^{2011}+4^{2010}+\ldots+4+1$.

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