
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
An arithmetic sequence $\{a_{n}\}$ and a geometric sequence $\{b_{n}\}$ with all positive terms satisfy: $a_{1}=b_{1}=2$, $a_{3}=b_{3}=8$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Sequence $\{c_{n}\}$ is a new sequence obtained by arranging different terms from sequences $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order. Let $S_{n}$ denote the sum of the first $n$ terms of sequence $\{c_{n}\}$. Find $S_{100}$.

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