
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, and $a_1= \frac{1}{8}$, the common ratio $q=2$.  
$(1)$ The general term formula for the sequence $\{a_n\}$ is $a_n=$ _______ ;  
$(2)$ The sequence $\{b_n\}$ satisfies $b_n=\log_2a_n$ $(n\in\mathbb{N}^*)$, find the minimum value of the sum of the first $n$ terms $S_n$ of the sequence $\{b_n\}$.

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