
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
Let \( k \) and \( n \) be positive integers and let
\[ 
S = \left\{ \left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1} + \cdots + a_{k} = k \right \}.
\]
Determine, with proof, the value of
\[ 
\sum_{\left(a_{1}, \ldots, a_{k}\right) \in S} \binom{n}{a_{1}} \binom{a_{1}}{a_{2}} \cdots \binom{a_{k-1}}{a_{k}}
\]
in terms of \( k \) and \( n \), where the sum is over all \( k \)-tuples \(\left(a_{1}, \ldots, a_{k}\right)\) in \( S \).

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