
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 propositions:  
① When $c=0$, $f(-x) = -f(x)$ always holds.  
② When $b=0$ and $c>0$, the equation $f(x) = 0$ has only one real root.  
③ The graph of the function $y=f(x)$ is symmetric about the point $(0, c)$.  
④ The equation $f(x) = 0$ has at most two real roots.  
Among these, the correct propositions are numbered as follows.

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