
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 a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and $S_n=2a_n-2$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;  
$(2)$ Let $b_n=\log_{2}a_1+\log_{2}a_2+\ldots+\log_{2}a_n$, find the range of real numbers $k$ for which $(n-8)b_n \geqslant nk$ holds true for any $n\in\mathbb{N}^*$.

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