
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 $y=x+\frac{t}{x}$ has the following property: if the constant $t \gt 0$, then the function is monotonically decreasing on $({0,\sqrt{t}}]$ and monotonically increasing on $[{\sqrt{t},+∞})$.  
$(1)$ Given $f(x)=\frac{x}{{{x^2}-8x+4}}$, $\left(1\leqslant x\leqslant 3\right)$, using the above property, find the range of the function $f\left(x\right)$;  
$(2)$ For the functions $f\left(x\right)$ and $g\left(x\right)=x^{2}+2ax-a^{2}$ in (1), if for any $x_{1}\in \left[1,3\right]$, there exists $x_{2}\in \left[0,1\right]$ such that $g(x_{2})f(x_{1})=1$ holds, find the range of the real number $a$.

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