
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 progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let the sum of the first $n$ terms of $\{c_{n}\}$ be denoted as $S_{n}$. If $c_{k}=101$, find the value of $k$ and determine $S_{k}$.

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