
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 $\\{a_{n}\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:

$(1)$ Find the general term formula for the sequence;
$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$.

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