
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
In a geometric sequence $\{a_{n}\}$, $a_{7}=8a_{4}$, and $\frac{1}{2}a_{2}$, $a_{3}-4$, $a_{4}-12$ form an arithmetic sequence.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ Let ${b}_{n}=(-1)^{n}\log_{2}{a}_{n}$. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. Find the value of $k$ such that $|T_{k}|=20$.

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