
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 the sequence $\{a_{n}\}$ is a geometric sequence with the first term $8$ and common ratio $\frac{1}{2}$.  
$(1)$ Find the value of $a_{1}+a_{2}+a_{3}+a_{4}+a_{3}$;  
$(2)$ Let the sum of the first $n$ terms of the sequence $\{\log _{2}a_{n}\}$ be $S_{n}$. Find the maximum value of $S_{n}$ and determine the corresponding value of $n$.

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