
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 function $f(x)=x^2+(a+8)x+a^2+a-12$, and $f(a^2-4)=f(2a-8)$. Let $\{a_n\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted as $S_n$, where $n \in \mathbb{N}^*$. If $S_n=f(n)$, find the minimum value of $\frac{S_n-4a}{a_n-1}$.

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