
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 and then answer the questions. When simplifying and operating square roots, sometimes we encounter expressions like $\frac{3}{\sqrt{5}-\sqrt{2}}$, in fact, we can further simplify it:<br/>Method 1: $\frac{3}{\sqrt{5}-\sqrt{2}}=\frac{3(\sqrt{5}+\sqrt{2})}{(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})}=\frac{3(\sqrt{5}+\sqrt{2})}{(\sqrt{5})^{2}-(\sqrt{2})^{2}}=\sqrt{5}+\sqrt{2}$;<br/>Method 2: $\frac{3}{\sqrt{5}-\sqrt{2}}=\frac{5-2}{\sqrt{5}-\sqrt{2}}=\frac{(\sqrt{5})^{2}-(\sqrt{2})^{2}}{\sqrt{5}-\sqrt{2}}=\frac{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}{\sqrt{5}-\sqrt{2}}=\sqrt{5}+\sqrt{2}$.<br/>The steps of this simplification are called rationalizing the denominator.<br/>$(1)$ Write down the simplified results directly: $\frac{1}{\sqrt{2}+1}=$______, $\frac{1}{\sqrt{7}+\sqrt{6}}=$______;<br/>$(2)$ Please refer to the above simplification methods, simplify using two methods: $\frac{3}{\sqrt{6}-\sqrt{3}}$;<br/>$(3)$ Calculate: $\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}}+\ldots +\frac{1}{\sqrt{10}+\sqrt{9}}$.

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