
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:<br/>$1×2=\frac{1}{3}×(1×2×3-0×1×2)$;<br/>$2×3=\frac{1}{3}×(2×3×4-1×2×3)$;<br/>$3×4=\frac{1}{3}×(3×4×5-2×3×4)$,<br/>After reading the above material, please complete the following questions:<br/>$(1)$ According to the above material, the fourth equation is: $4\times 5=\_\_\_\_\_\_,$ the $n$th equation is: $n\left(n+1\right)=$______;<br/>$(2)$ Calculate: $1\times 2+2\times 3+3\times 4+\ldots +n\left(n+1\right)$, (express using a formula containing $n$);<br/>$(3)$ Calculate: $1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+\ldots +17\times 18\times 19$.

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