
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 arithmetic sequence $\{a\_n\}$ with first term $3$ and common difference $1$, and the geometric sequence $\{b\_n\}$ with first term $\frac{1}{2}$ and common ratio also $\frac{1}{2}$, where $n∈N^{*}$, find the sum $S_n=$____ of the sequence $\{a\_n b\_n\}$ for the first $n$ terms.

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