
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 $|2a+b|$ and $\sqrt{3b+12}$ are opposite in sign.<br/>① Find the square root of $2a-3b$;<br/>② Solve the equation in terms of $x$: $ax^{2}+4b-2=0$.<br/>$(2)$ Given that the square root of a positive real number $y$ is $m$ and $m+b$.<br/>① When $b=8$, find $m$.<br/>② If $m^{2}y+\left(m+b\right)^{2}y=50$, find the value of $y$.<br/>③ Under the condition in ②, if $k$ is the decimal part of $-\sqrt{y}+1$, find the value of ${({k+\sqrt{5}})^2}$. (Note: The decimal part of a number refers to the number minus the largest integer not exceeding the number)

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