
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 $\left\{ a_n \right\}$ with $a_1=2$ and $a_3=18$, and an arithmetic sequence $\left\{ b_n \right\}$ with $b_1=2$, and $a_1+a_2+a_3=b_1+b_2+b_3+b_4 > 20$.

(1) Find the general formula for the $n$-th term $a_n$ of the sequence $\left\{ a_n \right\}$;

(2) Find the sum of the first $n$ terms $S_n$ of the sequence $\left\{ b_n \right\}$.

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