
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
Find all natural numbers \( n \in \mathbf{N} \) for which there exists a value of \( k \in \{1, 2, \ldots, n-1\} \) such that the equality

\[ 
2 \binom{n}{k} = \binom{n}{k-1} + \binom{n}{k+1}
\]

holds true.

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