
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
The great mathematician Gauss once studied a problem like this when he was in school: $1+2+3+\ldots +n=$? After research, the conclusion to this problem is $1+2+3+\ldots +n=\frac{1}{2}n(n+1),$ (where $n$ is a positive integer). Now let's study a similar problem: $1\times 2+2\times 3+\ldots +n(n+1)=$? Observing the following three special equations and filling in the blanks: $1\times 2=\frac{1}{3}(1\times 2\times 3-0\times 1\times 2),$ $2\times 3=\frac{1}{3}(2\times 3\times 4-1\times 2\times 3),$ $3\times 4=\frac{1}{3}(3\times 4\times 5-2\times 3\times 4),$ ① $4\times 5=\frac{1}{3}\times $______, $\ldots$ By adding both sides of the first two equations, we get $1\times 2+2\times 3=\frac{1}{3}\times 2\times 3\times 4$. By adding both sides of the three equations, we get $1\times 2+2\times 3+3\times 4=\frac{1}{3}\times 3\times 4\times 5$. Based on the above information, complete the blanks: ② Calculate $1\times 2+2\times 3+3\times 4+4\times 5=\frac{1}{3}\times $______; ③ Calculate: $1\times 2+2\times 3+\ldots +10\times 11=$______; ④ Calculate: $1\times 2+2\times 3+3\times 4\ldots +n(n+1)=$______; ⑤ Based on the above information, try to calculate: $1\times 2\times 3+2\times 3\times 4+3\times 4\times 5\ldots +10\times 11\times 12$; Conjecture: $1\times 2\times 3+2\times 3\times 4+3\times 4\times 5\ldots +n(n+1)(n+2)=\_\_\_\_\_\_$.

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