
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
Given the function $f(x) = x|x| + px + q$ ($x \in \mathbb{R}$), consider the following four statements: ① A necessary and sufficient condition for $f(x)$ to be an odd function is $q=0$; ② The graph of $f(x)$ is symmetric about the point $(0, q)$; ③ When $p=0$, the solution set of the equation $f(x)=0$ is always non-empty; ④ The number of solutions to the equation $f(x)=0$ is always no more than two.  
Among these, the correct statement(s) is/are.

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