
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 following four propositions:  
① A necessary but not sufficient condition for a function $f(x)$ defined on $\mathbb{R}$ to be an odd function is $f(0)=0$;  
② The graph of the function $f(a-x)$ is symmetric to the graph of the function $f(a+x)$ about the line $x=a$;  
③ If the function $y=f(x)$ has an inverse function $y=f^{-1}(x)$, then the function $y=f^{-1}(x-1)-2$ definitely has an inverse function, and its inverse function is $y=f(x+2)+1$;  
④ The range of the function $f(x)$ is definitely equal to the range of the function $f(x+1)$, but their domains are different.  
Among these, the true propositions are.

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