
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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Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x < 0$, $f(x)=e^{x}(x+1)$. Consider the following statements:  
$(1)$ When $x > 0$, $f(x)=e^{-x}(x-1)$;  
$(2)$ The function $f(x)$ has two zeros;  
$(3)$ The solution set of $f(x) < 0$ is $(-\infty,-1)\cup(0,1)$;  
$(4)$ For any $x_{1}$, $x_{2}\in\mathbb{R}$, it holds that $|f(x_{1})-f(x_{2})| < 2$.  
Among these statements, the correct ones are __________ (Fill in all correct statement numbers).

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