
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
For the function $f(x)=x|x|+px+q$, four propositions are provided:
$①$ When $q=0$, $f(x)$ is an odd function.
$②$ The graph of $y=f(x)$ is symmetric with respect to the point $(0,q)$.
$③$ When $p=0$ and $q > 0$, the equation $f(x)=0$ has exactly one real root.
$④$ The equation $f(x)=0$ has at most two real roots.
The correct propositions are indicated by their numbers: __________.

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