
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
Alice wants to color the integers between 2 and 8 (inclusive) using $k$ colors. She requires that if $m$ and $n$ are integers between 2 and 8 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ must be of different colors. Determine the smallest integer $k$ for which Alice can color the integers $2,3, \ldots, 8$ using $k$ colors.

Alice wants to color the integers between 2 and 31 (inclusive) using $k$ colors. She requires that if $m$ and $n$ are integers between 2 and 31 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ must be of different colors. Determine the smallest integer $k$ for which Alice can color the integers $2,3, \ldots, 31$ using $k$ colors.

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