
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
In the study of functions, we often encounter a type of function like $y=\frac{kx+f}{dx+e}$ (where $k$, $f$, $d$, $e$ are real constants and $d\neq 0$), which we call a linear fractional function. Please complete the following questions based on the given conditions.<br/>$(1)$ Let $a$ be a real number. For the function $f(x)=\frac{x+a}{x-a}$, discuss the parity of the function $y=f(x)$ based on different values of $a$, and provide the reasons;<br/>$(2)$ Let $m$ be a real number. For the function $g(x)=\frac{x-m+1}{x-2m}$, if $g(x) \lt 0$ holds as a sufficient but not necessary condition for $\frac{1}{3}＜x＜\frac{1}{2}$, find the range of values for $m$;<br/>$(3)$ Let $n$ be a real number. For the function $h(x)=4-\frac{1}{x}$, if there exists an interval $[{λ,μ}]⊆({\frac{1}{3},+∞})$ such that $\{y\left|\right.y=h\left(x\right),x\in \left[\lambda ,\mu \right]\}=\left[n\lambda ,n\mu \right]$, find the range of values for $n$.

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