
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 $f(x)=\frac{a(x-b)}{(x-b)^{2}+c}(a\neq 0,b\in R,c > 0)$, $g(x)=m[f(x)]^{2}-n(mn > 0)$, provide the correct propositions among the following:

1. When $b=0$, the function $f(x)$ is monotonically increasing on $(0, \sqrt {c})$ and monotonically decreasing on $(\sqrt {c},+\infty)$;
2. The graph of the function $f(x)$ is symmetric about a point on the $x$-axis;
3. There exist real numbers $p$ and $q$ such that $p\leqslant f(x)\leqslant q$ holds for any real number $x$;
4. The solution set of the equation $g(x)=0$ regarding $x$ could be $\{-3,-1,0,1\}$.

The correct proposition numbers are _______.

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