
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 before answering. From $(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})=(\sqrt{5})^{2}-(\sqrt{2})^{2}=3$, 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 of each other. When calculating square roots, using rationalizing factors can sometimes eliminate 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 $\sqrt{3}+1$ is ______;  
$(2)$ Simplify the expressions by eliminating square roots in the denominators: $\frac{3}{2\sqrt{3}}=$______, $\frac{3}{3+\sqrt{6}}=$______;  
$(3)$ Calculate using this rule: $(\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\ldots +\frac{1}{\sqrt{2009}+\sqrt{2008}})(\sqrt{2009}+1)$.

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