
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 know that the sum of any rational number and an irrational number is an irrational number, the product of any non-zero rational number and an irrational number is an irrational number, and the product of zero and an irrational number is zero. Therefore, if $mx+n=0$, where $m$ and $n$ are rational numbers and $x$ is an irrational number, then $m=0$ and $n=0$. Using the above knowledge to solve the following problems:<br/>
$(1)$ If $(m+1)\sqrt{3}+n-2=0$, where $m$ and $n$ are rational numbers, find the values of $m$ and $n$;<br/>
$(2)$ If $3m-n+(2m-\frac{1}{2}n+4)\sqrt{5}=2$, where $m$ and $n$ are rational numbers, find the cube root of $n-4m$;<br/>
$(3)$ If $m$ and $n$ are both rational numbers, and $(m+1)\sqrt{2}+m-17=2\sqrt{2}-n^2$, find the arithmetic square root of $|m+n|$.

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