
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 geometric sequence $\{a\_n\}$ with a common ratio $q \neq 1$, $a\_1 = \frac{1}{2}$, and the sum of the first $n$ terms is $S\_n$. It is also known that $a\_2 + S\_2$, $a\_3 + S\_3$, $a\_4 + S\_4$ form an arithmetic sequence. Find $a\_n + S\_n=\_\_\_\_\_\_\_.$

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