
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
We define: If the sum of two real numbers is equal to the product of these two real numbers, then these two real numbers are called "sum-product equal number pairs", that is, if $a+b=ab$, then $a$ and $b$ are called "sum-product equal number pairs", denoted as $\left(a,b\right)$. For example: $\frac{3}{2}+3=\frac{3}{2}\times 3$, $\frac{1}{2}+(-1)=\frac{1}{2}\times (-1)$, then the number pairs $(\frac{3}{2},3)$, $(\frac{1}{2},-1)$ are called "sum-product equal number pairs".<br/>$(1)$ Determine whether $\left(-2,4\right)$ and $(\sqrt{2}+2,\sqrt{2})$ are "sum-product equal number pairs", and explain the reasons;<br/>$(2)$ If $\left(m,n\right)$ (where $m,n\neq 1$) is a "sum-product equal number pair", then $m$______(expressed as an algebraic expression containing $n$).

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