
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
Consider the equation $1\cdot n + 2\cdot (n-1) + 3\cdot (n-2) + \ldots + n\cdot 1 = \frac{1}{6} n(n+1)(n+2)$. If the left side is denoted as $f(n)$, use mathematical induction to verify the equation. For the second step of the induction process, that is, when $n$ changes from $k$ to $k+1$, calculate the change in the left side of the equation $\Delta f(n) = f(k+1) - f(k)$.

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