
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
After studying, the general conclusion of the problem "$1+2+3+\ldots +100=$?" is $1+2+3+\ldots +n=\frac{1}{2}n\left(n+1\right)$, where $n$ is a positive integer. Now, let's study a similar problem: $1\times 2+2\times 3+3\times 4+\ldots +n\left(n+1\right)=$?<br/>Observe the following three special equations:<br/>$1\times 2=\frac{1}{3}(1\times 2\times 3-0\times 1\times 2)$,<br/>$2\times 3=\frac{1}{3}(2\times 3\times 4-1\times 2\times 3)$,<br/>$3\times 4=\frac{1}{3}(3\times 4\times 5-2\times 3\times 4)$,<br/>By adding the two sides of these three equations, we get $1\times 2+2\times 3+3\times 4=\frac{1}{3}\times 3\times 4\times 5=20$.<br/>Based on the information, directly write down the calculation results of the following expressions.<br/>$(1)1\times 2+2\times 3+3\times 4+\ldots +10\times 11$;<br/>$(2)1\times 2+2\times 3+3\times 4+\ldots +n\left(n+1\right)$.

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