
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 $\{a_{n}\}$ that satisfies $a_{n+1}-a_n=3(n∈N^*)$, and the sum of its first $5$ terms is $45$; sequence $\{b_{n}\}$ is a geometric sequence with $b_{1}=2$, $4b_{2}$, $2b_{3}$, $b_{4}$ forming an arithmetic sequence.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Find the sum of the first $n$ terms of the sequence $\{\frac{a_n⋅b_n}{3}\}$ denoted as $S_{n}$.

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