
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
If the function $f(x)$ satisfies $f(x+y) = f(x) + f(y)$ for any real numbers $x, y$, and $f(2) = 0$, then which of the following statements is correct?  
① $f(x)$ is a periodic function; ② $f(x)$ is an odd function; ③ $f(x)$ is symmetric about the point $(1,0)$; ④ $f(x)$ is symmetric about the line $x=1$.

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