
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
Definition: We call $({\sqrt{a}+\sqrt{b}})$ and $({\sqrt{a}-\sqrt{b}})$ a pair of "dual expressions" because $({\sqrt{a}+\sqrt{b}})({\sqrt{a}-\sqrt{b}})={({\sqrt{a}})^2}-{({\sqrt{b}})^2}=a-b$. Therefore, constructing "dual expressions" and then multiplying them can effectively eliminate the "$\sqrt{}$" in $({\sqrt{a}+\sqrt{b}})$ and $({\sqrt{a}-\sqrt{b}})$, so quadratic radical division can be calculated in this way: for example, $\frac{{2+\sqrt{2}}}{{2-\sqrt{2}}}=\frac{{{{({2+\sqrt{2}})}^2}}}{{({2-\sqrt{2}})×({2+\sqrt{2}})}}=3+2\sqrt{2}$. In this way, by multiplying both the numerator and denominator by an expression to eliminate the square root in the denominator or the denominator in the square root, it is called rationalizing the denominator. Based on the above material, understand and apply the method provided in the material to answer the following questions:  
$(1)$ The relationship between the dual expressions $2+\sqrt{3}$ and $2-\sqrt{3}$ is ______  
$A$. They are additive inverses  
$B$. They are reciprocals  
$C$. Their absolute values are equal  
$D$. They are unrelated  
$(2)$ Given $x=\frac{1}{{\sqrt{5}-2}}$, $y=\frac{1}{{\sqrt{5}+2}}$, find the value of $\frac{{x-y}}{{{x^2}y+x{y^2}}$  
$(3)$ Solve the equation: $\sqrt{24-x}-\sqrt{8-x}=2$ (Hint: Use the knowledge related to "dual expressions" and let $\sqrt{24-x}+\sqrt{8-x}=t$).

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