
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
Given the sequence $a\_n=n(n+1)$, using the formula $n(n+1)=\frac{n(n+1)(n+2)-(n-1)n(n+1)}{3}$, derive the sum of the first $n$ terms of the sequence, $S\_n=\frac{n(n+1)(n+2)}{3}$. Now, let $b\_n=n(n+1)(n+2)$. By analogy with the previous method, find the sum of the first $n$ terms of the sequence $T\_n=$ \_\_\_\_\_\_.

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