
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
Assuming the function $f(x) = x|x| + bx + c$, consider the following four propositions:
① If $f(x)$ is an odd function, then $c=0$.
② When $b=0$, the equation $f(x)=0$ has exactly one real root.
③ The graph of $f(x)$ is symmetric about the point $(0,c)$.
④ If $b \neq 0$, the equation $f(x)=0$ must have three real roots.
Which of the above propositions are correct? (Fill in the serial numbers)

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