
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
Consider a geometric sequence with terms $a$, $a(a-1)$, $a(a-1)^2$, ..., and let the sum of the first $n$ terms be denoted as $S_n$.  
(1) Determine the range of the real number $a$ and the expression for $S_n$;  
(2) Does there exist a real number $a$ such that $S_1$, $S_3$, $S_2$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain why.

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