
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
A geometric sequence $\{a_n\}$ satisfies $a_3a_4a_5=512$ and $a_3+a_4+a_5=28$, with a common ratio greater than $1$.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ Let $b_n=2n-1$, find the sum of the first $n$ terms $S_n$ of the sequence $\{a_n+b_n\}$.

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