
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 that $\{a_n\}$ is an arithmetic sequence satisfying $a_1=1$ and $a_4=-5$, and the sequence $\{b_n\}$ satisfies $b_1=1$ and $b_4=21$, and $\{a_n+b_n\}$ forms a geometric sequence.  
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Find the sum of the first $n$ terms of sequence $\{b_n\}$, denoted as $S_n$.

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