
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
When simplifying square roots, we sometimes encounter expressions like $\frac{1}{{\sqrt{2}+1}},\frac{1}{{\sqrt{3}-\sqrt{2}}}$, for which we can further simplify by rationalizing the denominator. For example: $\frac{1}{{\sqrt{2}+1}}=\frac{{1×({\sqrt{2}-1})}}{{({\sqrt{2}+1})({\sqrt{2}-1})}}=\frac{{\sqrt{2}-1}}{{2-1}}=\sqrt{2}-1$ and $\frac{1}{{\sqrt{3}-\sqrt{2}}}=\frac{{1×({\sqrt{3}+\sqrt{2}})}}{{({\sqrt{3}-\sqrt{2}})({\sqrt{3}+\sqrt{2}})}}=\frac{{\sqrt{3}+\sqrt{2}}}{{3-2}}=\sqrt{3}+\sqrt{2}$. 

$(1)$ Simplify $\frac{2}{{\sqrt{5}+\sqrt{3}}}$ using the above method; 

$(2)$ Calculate: $({\frac{1}{{\sqrt{2}+1}}+\frac{1}{{\sqrt{3}+\sqrt{2}}}+\frac{1}{{\sqrt{4}+\sqrt{3}}}+\ldots+\frac{1}{{\sqrt{2023}+\sqrt{2022}}}})×({\sqrt{2023}+1})$.

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