
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
For the function $y=f(x)$ ($x \in \mathbb{R}$), the following statements are given:  
$(1)$ In the same Cartesian coordinate system, the graphs of $y=f(-1-x)$ and $y=f(x-1)$ are symmetric about the line $x=0$;  
$(2)$ If $f(1-x)=f(x-1)$, then the graph of $y=f(x)$ is symmetric about the line $x=1$;  
$(3)$ If $f(1+x)=f(x-1)$, then the function $y=f(x)$ is a periodic function;  
$(4)$ If $f(1-x)=-f(x-1)$, then the graph of $y=f(x)$ is symmetric about the point $(0,0)$.  
Among these, the correct statement numbers are $\boxed{1, 3, 4}$.

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