
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 a function $f(x)$ defined on $\mathbb{R}$, the following propositions are given:  
① If $f(x)$ is an odd function, then the graph of $f(x-1)$ is symmetric about point A(1, 0);  
② If the graph of the function $f(x-1)$ is symmetric about the line $x=1$, then $f(x)$ is an even function;  
③ If for all $x \in \mathbb{R}$, $f(x-1) = -f(x)$, then the period of $f(x)$ is 2;  
④ The graphs of the functions $y=f(x-1)$ and $y=f(1-x)$ are symmetric about the line $x=0$.  
Among these, the correct propositions are numbered as follows.

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