
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 the sequence $\{a\_n\}$ is an increasing geometric sequence, and $a\_1+a\_4=9$, $a\_2a\_3=8$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If the sequence $\{b\_n\}$ satisfies $\frac{a\_1}{b\_1} + \frac{a\_2}{b\_2} + ... + \frac{a\_n}{b\_n} = (n^2+n+2) \cdot 2^n (n \in \mathbb{N}^*)$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$.

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