
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 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  $\mathcal{M}$ .
*The rank of  a matrix is the dimension of the span of its columns.*

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