
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 odd function $f(x)$ is defined on $(-\infty, 0) \cup (0, +\infty)$, and its range is $\mathbb{R}$. It holds that $f(x) > 0$ if and only if $x > 1$.  
Regarding $f(x)$, the following statements are made:  
1. $f(-1) = 0$;  
2. The equation $f(x) = 0$ has infinitely many solutions;  
3. $f(x)$ has a minimum value but no maximum value;  
4. The graph of $f(x)$ is symmetric about the origin, and $f(x)$ is a periodic function.  
Among these statements, the correct ones are.

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