
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 $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be vectors, and let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$. Then the determinant of the matrix whose column vectors are $\mathbf{b} \times \mathbf{c},$ $\mathbf{c} \times \mathbf{a},$ and $\mathbf{a} \times \mathbf{b}$ is equal to
\[k \cdot D^n.\] 
Determine the ordered pair $(k,n)$.

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