
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 \( n \) and \( k \) be integers such that \( 1 \leq k \leq n \). Provide two proofs of the following formula: \(\binom{n+1}{k+1} = \binom{n}{k} + \binom{n-1}{k} + \ldots + \binom{k}{k}\).

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