
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 and then answer the questions: From $(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})=(\sqrt{5})^2-(\sqrt{3})^2=2$, it can be seen that when two algebraic expressions containing square roots are multiplied, the product does not contain square roots. We call these two algebraic expressions rationalizing factors. When calculating square roots, using rationalizing factors can sometimes eliminate the square roots in the denominator. For example: $\frac{1}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\sqrt{3}-\sqrt{2}$. Please complete the following questions:  
$(1)$ The rationalizing factor of $\sqrt{2}-1$ is ______.  
$(2)$ Simplify the expression by eliminating the square root in the denominator: $\frac{3}{3-\sqrt{6}}=\_\_\_\_\_\_.(Write the result directly)$  
$(3)$ Compare $\sqrt{2019}-\sqrt{2018}$ and $\sqrt{2018}-\sqrt{2017}$. (Fill in $ \gt $ or $ \lt $)  
$(4)$ Calculate the value of the following expression using the pattern you discovered: $(\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+⋯+\frac{1}{\sqrt{2024}+\sqrt{2023}})(\sqrt{2024}+1)$.

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