
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 increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$ such that $a_{3}+a_{5}=40$, $a_{4}=16$, and $b_{n}=\log _{2}a_{n}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let $c_{n}=a_{n}\cdot b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $S_{n}$.

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