
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 $f(x) = \begin{cases} \frac{1}{2} - |x - \frac{3}{2}|, & x \leqslant 2 \\ e^{x-2}(-x^2 + 8x - 12), & x > 2 \end{cases}$, if there exist $n (n \geqslant 2)$ distinct numbers $(x_1, x_2, x_3, \cdots, x_n)$ in the interval $(1, \infty)$ such that $\frac{f(x_1)}{x_1} = \frac{f(x_2)}{x_2} = \cdots \frac{f(x_n)}{x_n}$ holds, then the set of possible values for $n$ is ( ).

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