
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 that $S_{n}$ is the sum of the first $n$ terms of an infinite geometric sequence ${a_{n}}$, with common ratio $q \neq 1$. It is known that $1$ is the arithmetic mean of $\frac{1}{2}S_{2}$ and $\frac{1}{3}S_{3}$, and $6$ is the geometric mean of $2S_{2}$ and $3S_{3}$.

(1) Find $S_{2}$ and $S_{3}$;
(2) Find the sum of the first $n$ terms of the sequence ${a_{n}}$;
(3) Find the sum of the first $n$ terms of the sequence ${S_{n}}$.

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