
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
Calculate the following expressions:
$$(x-1)(x+1) = \_\_\_\_\_\_;$$
$$(x-1)(x^2+x+1) = \_\_\_\_\_\_;$$
$$(x-1)(x^3+x^2+x+1) = \_\_\_\_\_\_;$$
...
1. Based on the pattern above, write down the result directly for: $$(x-1)(x^6+x^5+x^4+x^3+x^2+x+1) = \_\_\_\_\_\_;$$
2. Can you deduce a general conclusion for: $$(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+\ldots+x+1) = \_\_\_\_\_\_;$$ where $n$ is a positive integer?
3. Based on the conclusion from (2), find the result of: $$1+2+2^2+2^3+2^4+\ldots+2^{35}.$$

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