
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

Simplify the following sum:

$$
\begin{gathered}
S=1 \cdot 2 \cdot \ldots \cdot k+2 \cdot 3 \cdot \ldots(k+1)+3 \cdot 4 \cdot \ldots \cdot(k+2)+\ldots \\
+n \cdot(n+1) \cdot \ldots \cdot(n+k-1)
\end{gathered}
$$

(The sum consists of $n$ terms, each with a product of $k$ numbers.)

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