
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 $f(x)=|x|x+bx+c$, determine the correct proposition(s) from the following options and fill in the blank with their corresponding numbers.

$①$ When $b > 0$, the function $f(x)$ is monotonically increasing on $R$;

$②$ When $b < 0$, the function $f(x)$ has a minimum value on $R$;

$③$ The graph of the function $f(x)$ is symmetric about the point $(0,c)$;

$④$ The equation $f(x)=0$ may have three real roots.

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