
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 \( n \geq 3 \) and \( k \geq 1 \). Determine the smallest integer \( \ell \) with the following property:

For every convex polygon \( \mathcal{P} \) with \( n \) vertices and \( A \) a set of \( k \) points inside \( \mathcal{P} \), there exists an \( \ell \)-gon whose vertices are vertices of \( \mathcal{P} \) and that contains all the points of \( A \).

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