
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 \(\mathcal{M}\) be the set of \(5 \times 5\) real matrices of rank 3. Given a matrix \(A\) in \(\mathcal{M}\), the set of columns of \(A\) has \(2^{5}-1=31\) nonempty subsets. Let \(k_{A}\) be the number of these subsets that are linearly independent.

Determine the maximum and minimum values of \(k_{A}\), as \(A\) varies over \(\mathcal{M}\).

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