
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 $y=f(x)$, where $x\in \mathbb{R}$, consider the following statements:  
$(1)$ If for any $x_{1}$, $x_{2}$ and $x_{1} \neq x_{2}$, it holds that $\dfrac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} < 0$, then $f(x)$ is a decreasing function on $\mathbb{R}$;  
$(2)$ If $f(x)$ is an even function on $\mathbb{R}$, and it is a decreasing function within $(-\infty,0)$, with $f(-2)=0$, then the solution set of $f(x) > 0$ is $(-2,2)$;  
$(3)$ If $f(x)$ is an odd function on $\mathbb{R}$, then $y=f(x)-f(|x|)$ is also an odd function on $\mathbb{R}$;  
$(4)$ If $t$ is a constant, and for any $x$, it holds that $f(x-t)=f(x+t)$, then the graph of $f(x)$ is symmetric about $x=t$.  
Among these statements, the correct ones are \_\_\_\_\_\_.

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