
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
Xiao Mei, based on her experience in learning "Numbers and Expressions," wants to explore the rules of operation of the following quadratic radicals through the method of "from specific to general." Here is her exploration process, please complete it:<br/>The first equation: $\sqrt{1+\frac{1}{3}}=\sqrt{\frac{3+1}{3}}=\sqrt{4×\frac{1}{3}}=2\sqrt{\frac{1}{3}}$,<br/>The second equation: $\sqrt{2+\frac{1}{4}}=\sqrt{\frac{8+1}{4}}=\sqrt{9×\frac{1}{4}}=3\sqrt{\frac{1}{4}}$,<br/>The third equation: $\sqrt{3+\frac{1}{5}}=4\sqrt{\frac{1}{5}}$,<br/>$(1)$ Specific operation, following the third equation, write out the fifth equation:______.<br/>$(2)$ Observation, induction, make a conjecture, and verify the conjecture.<br/>If $n$ is a positive integer, express the above operation rule with $n$ in the formula:______.<br/>Verify the conjecture:<br/>$(3)$ Apply the operation rule to simplify: $\sqrt{2022+\frac{1}{2024}}×\sqrt{4048}$.

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