
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
Carefully observe the following equations:<br/>The $1$st one: $2^{2}-1=1\times 3$;<br/>The $2$nd one: $3^{2}-1=2\times 4$;<br/>The $3$rd one: $4^{2}-1=3\times 5$;<br/>The $4$th one: $5^{2}-1=4\times 6$;<br/>The $5$th one: $6^{2}-1=5\times 7$;<br/>$\ldots $<br/>These equations reflect a certain operation rule among natural numbers. Answer the following questions as required:<br/>$(1)$ Please write down the $6$th equation: ______;<br/>$(2)$ Let $n(n\geqslant 1)$ represent a natural number, the $n$th equation can be represented as ______;<br/>$(3)$ Using the above conclusion, calculate: $\frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+\cdots+\frac{1}{2024^2-1}$.

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