
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 $\{a_{n}\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$, $a_{3}=6$, $a_{7}=14$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$ and $S_{n}$;  
$(2)$ If ___, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$.  
Choose one of the following conditions to supplement in question (2):  
① $b_{n}=2{}^{{a}_{n}}\cdot a_{n}$;  
② $b_{n}=\frac{{a}_{n}^{2}+{{a}_{n+1}}^{2}}{{S}_{n}}$;  
③ $b_{n}=\left(-1\right)^{n}\cdot a_{n}$, and solve it.

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