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

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