
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
Reading material:<br/>($1$) When simplifying and operating square roots, sometimes we encounter expressions like $\frac{2}{{\sqrt{3}+1}}$, in fact, we can further simplify it: $\frac{2}{{\sqrt{3}+1}}=\frac{{2×({\sqrt{3}-1})}}{{({\sqrt{3}+1})({\sqrt{3}-1})}}=\frac{{2×({\sqrt{3}-1})}}{{{{({\sqrt{3}})}^2}-{1^2}}}=\sqrt{3}-1$.<br/> We call this process rationalizing the denominator of a fraction.<br/>($2$) If we can find two real numbers $x$ and $y$ such that $x+y=a$ and $xy=b$, then $\sqrt{a+2\sqrt{b}}=\sqrt{{{({\sqrt{x}})}^2}+{{({\sqrt{y}})}^2}+2\sqrt{x}⋅\sqrt{y}}=\sqrt{{{({\sqrt{x}+\sqrt{y}})}^2}}=\sqrt{x}+\sqrt{y}$, then we call $\sqrt{a+2\sqrt{b}}$ a "harmonic quadratic root", and the above process is called simplifying "harmonic quadratic root".<br/> For example: $\sqrt{3+2\sqrt{2}}=\sqrt{{{({\sqrt{1}})}^2}+{{({\sqrt{2}})}^2}+2\sqrt{1}⋅\sqrt{2}}=\sqrt{{{({1+\sqrt{2}})}^2}}=1+\sqrt{2}$.<br/> Based on the reading material, solve the following problems:<br/>$(1)$ Simplify: $\frac{2}{{\sqrt{3}-1}}$;<br/>$(2)$ Simplify "harmonic quadratic root".<br/>① $\sqrt{4+2\sqrt{3}}=$______;<br/>② $\sqrt{6-2\sqrt{5}}=\_\_\_\_\_\_.$<br/>$(3)$ Given $m=\frac{1}{{\sqrt{5+2\sqrt{6}}}}$, $n=\frac{1}{{\sqrt{5-2\sqrt{6}}}}$, find the value of $\frac{{m-n}}{{m+n}}$.

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