
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
Let $S\subseteq N^{*}$, $S\neq \varnothing$ and satisfies the following conditions: ① $1\notin S$; ② if $x\in S$, then $1+\frac{12}{x-1} \in S$.  
$(1)$ Can $S$ be a singleton set? Why or why not?  
$(2)$ Find the set $S$ that contains only two elements.  
$(3)$ How many sets $S$ satisfy the given conditions? Can you list them?

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