
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
Reading comprehension: Read the following material and then answer the questions: When simplifying and operating quadratic radicals, we sometimes encounter expressions like: $\frac{3}{\sqrt{5}}$, $\sqrt{\frac{2}{3}}$, $\frac{2}{\sqrt{3}+1}$, $\ldots$ In fact, we can further simplify them so that the denominator of the expression does not contain radicals: for example, $\frac{3}{\sqrt{5}}=\frac{3×\sqrt{5}}{\sqrt{5}×\sqrt{5}}=\frac{3\sqrt{5}}{5}$; (1) $\sqrt{\frac{2}{3}}=\frac{\sqrt{2×3}}{\sqrt{3×3}}=\frac{\sqrt{6}}{3}$; (2) $\frac{2}{\sqrt{3}+1}=\frac{2(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)}=\frac{2(\sqrt{3}-1)}{(\sqrt{3})^{2}-1}=\sqrt{3}-1$; (3) This kind of simplification is called rationalizing the denominator. $\frac{2}{\sqrt{3}+1}$ can also be simplified using the following method: $\frac{2}{\sqrt{3}+1}=\frac{3-1}{\sqrt{3}+1}=\frac{(\sqrt{3})^{2}-1}{\sqrt{3}+1}=\frac{(\sqrt{3}+1)(\sqrt{3}-1)}{\sqrt{3}+1}=\sqrt{3}-1$. (4) Please answer the following questions: (1) Simplify: $\frac{3}{\sqrt{6}+\sqrt{3}}$. (2) Simplify: $\frac{3}{\sqrt{4}+\sqrt{1}}+\frac{3}{\sqrt{7}+\sqrt{4}}+\frac{3}{\sqrt{10}+\sqrt{7}}$. (3) Conjecture the value of $\frac{3}{\sqrt{4}+\sqrt{1}}+\frac{3}{\sqrt{7}+\sqrt{4}}+\frac{3}{\sqrt{10}+\sqrt{7}}+\ldots +\frac{3}{\sqrt{3n+1}+\sqrt{3n-2}}$ (you can write the result directly).

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