
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
If for any $x_{1}$, $x_{2}$ in the domain of function $f(x)$, $f(x_{1})=f(x_{2})$ always implies $x_{1}=x_{2}$, then the function $f(x)$ is called an injective function. For example, the function $f(x)=x$ is injective, but the function $f(x)=x^{2}$ is not injective. Consider the following statements:  
- (1) The function $f(x)= \begin{cases} \log_{2}x & \text{for } x\geqslant 2\\ x-1 & \text{for } x < 2 \end{cases}$ is injective;  
- (2) When $a > -2$, the function $f(x)= \frac{x^{2}+ax+1}{x}$ is injective on $(0,+\infty)$;  
- (3) If the function $f(x)$ is injective within its domain and $x_{1}\neq x_{2}$, then $f(x_{1})\neq f(x_{2})$;  
- (4) If the function $f(x)$ is injective and differentiable within its domain, then there must exist an $x_{0}$ in its domain such that its derivative $f'(x_{0})=0$.  
Among these statements, the correct ones are __________. (Fill in all correct statement numbers)

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