
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 the increasing geometric sequence $\{a_n\}$, satisfying $a_1 = 1$, and $a_2a_4 - 2a_3a_5 + a_4a_6 = 36$.
(Ⅰ) Find the general formula for the sequence $\{a_n\}$;
(Ⅱ) Let $b_n = \log_3 a_n + \frac{1}{2}$, calculate the sum of the first $n$ terms $S_n$ for the sequence $\{a_n^2 \cdot b_n\}$.

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