
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 $\{a_n\}$ is a sequence of positive numbers, and the sum of its first $n$ terms $2S_n = a_n^2 + a_n$ ($n \in \mathbb{N}^*$), the sequence $\{b_n\}$ satisfies $b_1 = \frac{3}{2}$, $b_{n+1} = b_n + 3^{a_n}$ ($n \in \mathbb{N}^*$).

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

(II) If $c_n = a_n b_n$ ($n \in \mathbb{N}^*$), and the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$, find $\lim_{n \to \infty} \frac{T_n}{c_n}$.

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