
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 $\{a\_n\}$ with a common ratio $q > 1$, $a\_1 + a\_4 = 9$, and $a\_2 a\_3 = 8$.
(1) Find the general term formula for the sequence;
(2) If $b\_n = a\_{n+1} \log \_2 a\_{n+1}$, and $S\_n$ denotes the sum of the first $n$ terms of the sequence $\{b\_n\}$, find the smallest positive integer $n$ such that $2^{n+1} + S\_n > 60n + 2$.

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