
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
Given a geometric sequence $a$, $a(a-1)$, $a(a-1)^{2}$, $...$, where $S_{n}$ represents the sum of its first $n$ terms.
1) Find the range of real values for $a$ and the expression for $S_{n}$;
2) Does there exist a real value of $a$ such that $S_{1}$, $S_{3}$, $S_{2}$ form an arithmetic sequence? If it exists, find the value of $a$; if not, explain the reason.

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