
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
$(1)$ Given $m=\sqrt[{b-1}]{{a+4}}$ is the arithmetic square root of $a+4$, and $n=\sqrt[{a-2}]{{3b-1}}$ is the cube root of $3b-1$, find the cube root of $m-2n$;<br/>$(2)$ If $m=\sqrt{1-a}+\sqrt{a-1}+1$, and the arithmetic square root of $n$ is $5$, find the square root of $3n+6m$.

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