
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 a geometric sequence $\{a_n\}$ be $S_n$. It is known that $S_1$, $S_2$, and $S_3$ form an arithmetic sequence.

 $(1)$ Find the common ratio $q$ of the sequence $\{a_n\}$;

 $(2)$ Given $a_1-a_3=3$, find $S_n$.

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