
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
The domain of the function $f(x)$ is $\mathbb{R}$. If both $f(x+1)$ and $f(x-1)$ are odd functions, then consider the following statements about $f(x)$:  
① $f(x)$ is an even function.  
② $f(x)$ is an odd function.  
③ $f(x) = f(x+2)$.  
④ $f(x+3)$ is an odd function.  
Among these, the correct statement(s) is (are) ____.

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