
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
Find the pattern through calculation:  
(1) Calculate: $(x+1)(x-1)=$ . $(x-1)(x^2+x+1)=$ . $(x-1)(x^3+x^2+x+1)=$ .  
(2) Conjecture: $(x-1)(x^n+x^{n-1}+\ldots+x+1)=$ .  
(3) Based on the conjecture, write out the following results: $2^5+2^4+\ldots+2+1=$ . $2^{n-1}+2^{n-2}+\ldots+2+1=$ .

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