
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 $f(x)=(\frac{1}{3})^{x}$, its inverse function is $y=g(x)$.  
$(1)$ If the domain of $g(mx^{2}+2x+1)$ is $\mathbb{R}$, find the range of the real number $m$;  
$(2)$ When $x\in[-1,1]$, find the minimum value $h(a)$ of the function $y=[f(x)]^{2}-2af(x)+3$;  
$(3)$ Determine whether there exist real numbers $m > n > 3$ such that the domain of the function $y=h(x)$ is $[n,m]$ and its range is $[n^{2},m^{2}]$. If such numbers exist, find the values of $m$ and $n$; if not, explain why.

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