
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and a non-constant geometric sequence $\{b_n\}$ with a common ratio $q$, satisfying: $a_1=2$, $b_1=1$, $S_2=3b_2$, $a_2=b_3$.  
(I) Find $a_n$ and $b_n$;  
(II) Let $c_n=2b_n-\lambda\cdot3^{\frac{a_n}{2}}$, if the sequence $\{c_n\}$ is decreasing, find the range of the real number $\lambda$.

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