
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
In the arithmetic sequence $\left\{ a_n \right\}$, $a_7=4$, $a_{19}=2a_{9}$,

(1) Find the general formula for $\left\{ a_n \right\}$;

(2) Let $\left\{b_n\right\}$ be a geometric sequence with positive terms, where $b_1=a_1$, $b_3=a_7$. Find the sum of the first $n$ terms of $\left\{b_n\right\}$, $S_n$.

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