
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
Observe the following equations:<br/>The $1$st equation: $a_{1}=\frac{1}{1×2}=1-\frac{1}{2}$;<br/>The $2$nd equation: $a_{2}=\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$;<br/>The $3$rd equation: $a_{3}=\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$;<br/>The $4$th equation: $a_{4}=\frac{1}{4×5}=\frac{1}{4}-\frac{1}{5}$;<br/>$\ldots $<br/>Please answer the following questions:<br/>$(1)$ According to the pattern of the above equations, write down the $5$th equation: $a_{5}=\_\_\_\_\_\_=\_\_\_\_\_\_$;<br/>$(2)$ Express the $n$th equation using an expression containing $n$;<br/>$(3)$ Calculate: $\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+⋯+\frac{1}{2023×2024}$.

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