
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: $$(1-x)(1+x)=1-x^{2},$$ $$(1-x)(1+x+x^{2})=1-x^{3},$$ $$(1-x)(1+x+x^{2}+x^{3})=1-x^{4},\ldots$$

$(1)$ Observe the above equations, and conjecture: $$(1-x)(1+x+x^{2}+\ldots+x^{n})= \_\_\_\_\_\_.$$

$(2)$ Based on your conjecture, calculate: $$1+3+3^{2}+3^{3}+\ldots+3^{n}= \_\_\_\_\_\_.$$ (where $n$ is a positive integer)

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