
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
1. In the arithmetic sequence $\{a\_n\}$, $a\_8=6$, $a\_{10}=0$. Find the general term formula $a\_n$ and the sum of the first $n$ terms $S\_n$. Also, indicate the value of $n$ when $S\_n$ reaches its maximum.
2. In the geometric sequence $\{a\_n\}$, $a\_1=\frac{1}{2}$, $a\_4=4$. Find the general term formula $a\_n$ and the sum of the first $n$ terms $S\_n$.

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