
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-geometric sequence $\{a\_n\}$ with a sum of the first $n$ terms denoted as $S\_n$ and a common ratio of $\frac{3}{2}$.
(1) If $S\_4 = \frac{65}{24}$, find $a\_1$;
(2) If $a\_1=2$, $c\_n = \frac{1}{2}a\_n + bn$, and $c\_2$, $c\_4$, $c\_5$ form an arithmetic sequence, find $b$.

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