
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
A non-empty number set $A$ is called a "reciprocal set" if it satisfies: $(1) 0 \notin A$; $(2)$ For every $x \in A$, $\frac{1}{x} \in A$. Consider the following number sets:

$(1) \{x \in \mathbb{R} | x^{2} + ax + 1 = 0\}$; $(2) \{x | x^{2} - 4x + 1 < 0\}$;

$(3) \{y | y = \frac{\ln x}{x}, x \in [\frac{1}{e}, 1) \cup (1, e]\}$;

$(4) \left\{ y \left| y = \begin{cases} \frac{2x + \frac{2}{5}}{x + \frac{1}{x}}, & x \in [0,1) \\ \frac{2x + \frac{2}{5}}{x + \frac{1}{x}}, & x \in [1,2] \end{cases} \right. \right\}$.

The number of "reciprocal sets" among them is $\boxed{\text{2}}$.

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