
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
Read the following material first, then answer the following questions.  
$\because ({\sqrt{a}+\sqrt{b}})({\sqrt{a}-\sqrt{b}})=a-b$,  
$\therefore a-b=({\sqrt{a}+\sqrt{b}})({\sqrt{a}-\sqrt{b}})$.  
For example: $\because ({\sqrt{15}+\sqrt{14}})({\sqrt{15}-\sqrt{14}})=1$,  
$\therefore \frac{1}{{\sqrt{15}-\sqrt{14}}}=\sqrt{15}+\sqrt{14}$.  
This kind of transformation is called rationalizing the denominator.  
Use the above method to calculate the following expressions:  
$(1)({\frac{1}{{\sqrt{2}+1}}+\frac{1}{{\sqrt{3}+\sqrt{2}}}+\frac{1}{{\sqrt{4}+\sqrt{3}}}+?+\frac{1}{{\sqrt{2022}+\sqrt{2021}}}+\frac{1}{{\sqrt{2023}+\sqrt{2022}}}})×({\sqrt{2023}+1})$.  
$(2)$ Calculate $\frac{{12}}{{5+\sqrt{13}}}+\frac{5}{{\sqrt{13}-2\sqrt{2}}}-\frac{4}{{2\sqrt{2}+3}}$.

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