
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 $y=f(x)$ with domain $\mathbb{R}$, consider the following statements:
① If $f(x-1)=f(1-x)$ always holds, then the graph of the function $y=f(x)$ is symmetric about the line $x=1$.
② If $f(x+1)+f(1-x)=0$ always holds, then the graph of the function $y=f(x)$ is symmetric about the point $(1,0)$.
③ The graph of the function $y=f(x-1)$ and the graph of the function $y=f(1-x)$ are symmetric about the y-axis.
④ If $f(1+x)+f(x-1)=0$ always holds, then the function $y=f(x)$ has a period of 4. Among these statements, the true ones are.

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