
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 the function $y=f(x)$, where $x \in \mathbb{R}$, consider the following four propositions:  
① If $f(1+2x)=f(1-2x)$, then the graph of $f(x)$ is symmetric about the line $x=1$;  
② The graphs of $f(x-2)$ and $f(2-x)$ are symmetric about the line $x=2$;  
③ If $f(x)$ is an even function, and $f(2+x)=-f(x)$, then the graph of $f(x)$ is symmetric about the line $x=2$;  
④ If $f(x)$ is an odd function, and $f(x)=f(-x-2)$, then the graph of $f(x)$ is symmetric about the line $x=1$.  
Among these propositions, the correct ones are.

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