
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 function $f(x)$ is an odd function defined on $\mathbb{R}$. Consider the following statements:  
① $f(0) = 0$;  
② If $f(x)$ has a minimum value of $-1$ on $(0, +\infty)$, then $f(x)$ has a maximum value of $1$ on $(-\infty, 0)$;  
③ If $f(x)$ is increasing on $[1, +\infty)$, then $f(x)$ is decreasing on $(-\infty, -1]$;  
④ If $x > 0$, $f(x) = x^2 - 2x$; then for $x < 0$, $f(x) = -x^2 - 2x$.  
Among these statements, the correct ones are.

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