
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]


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Given a sequence $\{a_n\}$ whose sum of the first $n$ terms $S_n$ satisfies $S_n=\frac{3}{2}a_n-\frac{1}{2}$, and a sequence $\{b_n\}$ satisfies: $b_n=2\log_3{a_n}+1$, where $n\in \mathbb{N}^*$. 

(I) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.

(II) Let $c_n=\frac{b_n}{a_n}$, and the sum of the first $n$ terms of sequence $\{c_n\}$ be $T_n$. If $T_n < c^2-2c$ holds for all $n\in \mathbb{N}^*$, find the range of the real number $c$.

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