
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 geometric sequence $\{a_n\}$ is monotonically increasing, and $a_2+a_3+a_4=28$, also $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ If $b_n=a_n\log_2a_n$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, find $S_n$.

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