
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, $\{b_{n}\}$ is a monotonically increasing geometric sequence, $a_{1}=b_{1}=1$, $a_{2}+a_{4}=6$, $a_{3}b_{3}=12$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Find the sum $S_{n}$ of the first $n$ terms of the sequence $\{a_{n}+b_{n}\}$.

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