
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
Among the following propositions:  
① If the domain of the function $f(x)$ is $\mathbb{R}$, then $g(x) = f(x) + f(-x)$ is definitely an even function;  
② If $f(x)$ is an odd function defined on $\mathbb{R}$, and for any $x \in \mathbb{R}$, $f(x) + f(2+x) = 0$, then the graph of the function $f(x)$ is symmetric about the line $x=1$;  
③ Given $x_1$, $x_2$ are two values within the domain of the function $f(x)$, and $x_1 < x_2$, if $f(x_1) > f(x_2)$, then $f(x)$ is a decreasing function;  
④ If $f(x)$ is an odd function defined on $\mathbb{R}$, and $f(x+2)$ is also an odd function, then $f(x)$ is a periodic function with a period of 4.  
The correct proposition numbers are.

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