
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 its first $n$ terms denoted as $S_n$, and a geometric sequence $\{b_n\}$ with the sum of its first $n$ terms denoted as $T_n$, where $a_1=-1$, $b_1=1$, and $a_2+b_2=2$.
$(1)$ If $a_3+b_3=5$, find the general formula for $\{b_n\}$.
$(2)$ If $T_3=21$, find $S_3$.

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