
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 $S_{n}$, and let $\{b_{n}\}$ be a geometric sequence with the sum of the first $n$ terms denoted as $T_{n}$. Given $a_{1}=1$, $b_{1}=2$, and $a_{3}+b_{3}=13$. <br/>$(Ⅰ)$ If $a_{2}+b_{2}=7$, find the general formula for $\{b_{n}\}$; <br/>$(Ⅱ)$ If $T_{3}=14$, find $S_{n}$.

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