
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 a geometric sequence with all positive terms, $a_{1}=8$, and $a_{3}=3a_{2}+32$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ Let $b_{n}=\log _{4}a_{n}$. Find the sum of the first $n$ terms of $\{b_{n}\}$, denoted as $T_{n}$.

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