
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 an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, and a geometric sequence $\{b_n\}$ whose terms are all positive numbers with a common ratio $q$, and it is known that: $a_1=3$, $b_1=1$, $b_2+S_2=12$, $S_2=b_2q$.  
$(1)$ Find $a_n$ and $b_n$;  
$(2)$ Let $c_n=3b_n-2\lambda \cdot \frac{a_n}{3} (\lambda \in \mathbb{R})$. If the sequence $\{c_n\}$ is increasing, find the range of $\lambda$.

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