
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 $D$ be the determinant of the matrix whose column vectors are $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}.$ Find the determinant of the matrix whose column vectors are $k\mathbf{a} + \mathbf{b},$ $\mathbf{b} + k\mathbf{c},$ and $k\mathbf{c} + \mathbf{a},$ in terms of $D$ and the scalar $k.$

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