
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 progression $\{a_{n}\}$ and a geometric progression $\{b_{n}\}$ satisfying $a_{1}=1$, $b_{3}=8$, $a_{n}=\log _{2}b_{n}$, $n\in N^{*}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let the terms in the sequence $\{a_{n}\}$ that are not in the sequence $\{b_{n}\}$ be arranged in ascending order to form a sequence $\{c_{n}\}$. Denote the sum of the first $n$ terms of the sequence $\{c_{n}\}$ as $S_{n}$. Find $S_{50}$.

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