
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
Calculate the sum of the series $1 \times 2 + 2 \times 3 + \ldots + n(n+1)$. A student learned the following method: first rewrite the $k^{th}$ term as $k(k+1) = \frac{1}{3}[k(k+1)(k+2) - (k-1)k(k+1)]$. Using this, we get:
$$1 \times 2 = \frac{1}{3}(1 \times 2 \times 3 - 0 \times 1 \times 2),$$
$$2 \times 3 = \frac{1}{3}(2 \times 3 \times 4 - 1 \times 2 \times 3),$$
$\ldots$
$$n(n+1) = \frac{1}{3}[n(n+1)(n+2) - (n-1)n(n+1)].$$
By summing these up, we find that $1 \times 2 + 2 \times 3 + \ldots + n(n+1) = \frac{1}{3}n(n+1)(n+2)$.

Analogously, calculate the sum of the series $1 \times 2 \times 3 + 2 \times 3 \times 4 + \ldots + n(n+1)(n+2)$.

The result is __________.

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