
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]


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The domain of function $B_{1}$ is $A$. If $x_{1}, x_{2} \in A$ and $f(x_{1}) = f(x_{2})$ always implies $x_{1} = x_{2}$, then $f(x)$ is called an injective function. For example, the function $f(x) = x + 1 (x \in \mathbb{R})$ is injective. Consider the following statements:  
$(1)$ The function $f(x) = x^{2} - 2x (x \in \mathbb{R})$ is injective;  
$(2)$ The function $f(x) = \begin{cases} \log_{2}x, & x \geqslant 2 \\ 2 - x, & x < 2 \end{cases}$ is injective;  
$(3)$ If $y = f(x)$ is an injective function, and $x_{1}, x_{2} \in A$ with $x_{1} \neq x_{2}$, then $f(x_{1}) \neq f(x_{2})$;  
$(4)$ If the function $f(x)$ is monotonic in some interval $D$ within its domain, then $f(x)$ is necessarily an injective function.  
Among these statements, the true ones are ______. (Write down the numbers of all true statements.)

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