
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 an arithmetic sequence, and $a_1+a_2+a_3=15$, the sequence $\{b_n\}$ is a geometric sequence, and $b_1b_2b_3=27$, with $a_1=b_2$ and $a_4=b_3$.  
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ For the sequence $\{c_n\}$, which satisfies $c_n=2a_n+b_n$, find the sum of the first $n$ terms of $\{c_n\}$.

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