
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 a function $f(x)$ defined on $\mathbb{R}$ such that for any $x \in \mathbb{R}$, we have $f(x+4) = -\frac{1}{f(x)}$. Let $a_n = f(n)$ for $n \in \mathbb{N}^*$. Determine the maximum number of different values in the sequence $\{a_n\}$.

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