
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 that the domain of the function $f(x)$ is $\mathbb{R}$, among the following propositions:  
① If $y=f(x)$ is an even function, then the graph of $y=f(x+2)$ is symmetric about the y-axis;  
② If $y=f(x+2)$ is an even function, then the graph of $y=f(x)$ is symmetric about the line $x=2$;  
③ If $f(x-2)=f(2-x)$, then the graph of $y=f(x)$ is symmetric about the line $x=2$;  
④ The graphs of $y=f(x-2)$ and $y=f(2-x)$ are symmetric about the line $x=2$.  
Indicate the correct propositions. (Fill in all correct proposition numbers)

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