
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
Given $x=\frac{\sqrt{3}-1}{2}$, $y=\frac{\sqrt{3}+1}{2}$, $m=\frac{1}{x}-\frac{1}{y}$, $n=\frac{y}{x}+\frac{x}{y}$.

$(1)$ Find the values of $m$ and $n$.

$(2)$ If $\sqrt{a}-\sqrt{b}=n+2$ and $\sqrt{ab}=m$, find the value of $\sqrt{a}+\sqrt{b}$.

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