
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 following definition: If two real numbers $a$ and $b$ satisfy the fractional equation about $x$ $\frac{a}{x}+1=b$ with the solution $x=\frac{1}{a+b}$, then we call the real numbers $a$ and $b$ a "equation number pair" of the fractional equation about $x$, denoted as $\left[a,b\right]$. For example, $a=2$, $b=-5$ is an "equation number pair" of the fractional equation about $x$ $\frac{a}{x}+1=b$, denoted as $\left[2,-5\right]$.

$(1)$ Determine which of the number pairs $\left[3,-5\right]$ and $\left[-2,4\right]$ are "equation number pairs" of the fractional equation about $x$ $\frac{a}{x}+1=b$______; (only fill in the numbers)<br/>$(2)$ If the number pair $\left[n,3-n\right]$ is an "equation number pair" of the fractional equation about $x$ $\frac{a}{x}+1=b$, find the value of $n$;<br/>$(3)$ If the number pair $\left[m-k,k\right](m\neq -1$ and $m\neq 0$, $k\neq 1)$ is an "equation number pair" of the fractional equation about $x$ $\frac{a}{x}+1=b$, express $k$ in terms of $m$ using an algebraic expression.

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