
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 $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})={(\sqrt{a})}^{2}-{(\sqrt{b})}^{2}=a-b$, which effectively eliminates the square roots. Therefore, some problems can be solved by constructing "dual expressions". For example, given $\sqrt{18-x}-\sqrt{11-x}=1$, find the value of $\sqrt{18-x}+\sqrt{11-x}$ as follows: Because $(\sqrt{18-x}-\sqrt{11-x})×(\sqrt{18-x}+\sqrt{11-x})={(\sqrt{18-x})}^{2}-{(\sqrt{11-x})}^{2}=18-x-11+x=7$, so $\sqrt{18-x}+\sqrt{11-x}=7$. 

$(1)$ Given: $\sqrt{20-x}+\sqrt{4-x}=8$, find: ① $\sqrt{20-x}-\sqrt{4-x}=$______; ② Combine the given condition and the result of question ① to solve the equation: $\sqrt{20-x}+\sqrt{4-x}=8$; 

$(2)$ The range of values for $x$ in the algebraic expression $\sqrt{10-x}+\sqrt{x-2}$ is ______, the maximum value is ______, and the minimum value is ______; 

$(3)$ Calculate: $\frac{1}{3\sqrt{1}+\sqrt{3}}+\frac{1}{5\sqrt{3}+3\sqrt{5}}+\frac{1}{7\sqrt{5}+5\sqrt{7}}+\ldots+\frac{1}{2023\sqrt{2021}+2021\sqrt{2023}}$.

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