
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 the solution to the one-variable linear equation $ax=b$ as $x=b-a$ with respect to $x$, then we call this equation a "difference solution equation". For example, the solution to $3x=4.5$ is $x=4.5-3=1.5$, so the equation $3x=4.5$ is a "difference solution equation". Please answer the following questions based on the above rules:<br/>【Understanding of Definition】<br/>$(1)$ Determine: The equation $2x=4$ is ______ a difference solution equation. (Fill in "is" or "is not")<br/>$(2)$ If the one-variable linear equation $4x=m$ is a "difference solution equation", find the value of $m$;<br/>【Application of Knowledge】<br/>$(3)$ Given that the one-variable linear equation $4x=ab+a$ is a "difference solution equation", then $3\left(ab+a\right)=\_\_\_\_\_\_.$<br/>$(4)$ Given that the one-variable linear equations $4x=mn+m$ and $-2x=mn+n$ are both "difference solution equations", find the value of the algebraic expression $3\left(mn+m\right)-9\left(mn+n\right)^{2}$.

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