
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
Let $\left\{A_{1}, A_{2}, A_{3}, \cdots, A_{n}\right\}$ be the set of $n$ vertices of a polygon inscribed in a circle. 

(1) Which type of subset is more numerous: the subsets containing $A_{1}$ or the subsets not containing $A_{1}$?

(2) Among the polygons formed using the vertices $\left\{A_{1}, A_{2}, \cdots, A_{n}\right\}$, which type of polygon is more numerous: the polygons containing vertex $A_{1}$ or the polygons not containing vertex $A_{1}$?

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