
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 the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, and it satisfies $a_n = 2 - S_n$ ($n \in \mathbb{N}^*$).

(1) Find the values of $a_1$, $a_2$, $a_3$, $a_4$ and write the general formula for the sequence;  
(2) Based on the general formula written in (1), use a three-step proof to demonstrate that the sequence $\{a_n\}$ is a geometric sequence.

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