
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 an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\{b_{n}\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$. <br/>$(1)$ If $b_{2}=-2a_{4}$, find the general formula for the sequence $\{b_{n}\}$; <br/>$(2)$ If $T_{3}=13$, find $S_{3}$.

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