
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
Let the function $f(x) = |x|x + bx + c$. Consider the following four statements:  
① When $b=0$ and $c>0$, the equation $f(x)=0$ has only one real root;  
② When $c=0$, $y=f(x)$ is an odd function;  
③ The graph of $y=f(x)$ is symmetric about the point $(0,c)$;  
④ The equation $f(x)=0$ has at most two distinct real roots.  
Among the above statements, the correct ones are \_\_\_\_\_\_.

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