
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 in $\mathbb{R}^3$, and let $D$ be the determinant of the matrix whose columns are $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$. Let $\alpha$ be a non-zero scalar. Then, the determinant of the matrix whose columns are $\alpha \mathbf{a} \times \mathbf{b},$ $\mathbf{b} \times \mathbf{c},$ and $\mathbf{c} \times \mathbf{a}$ is equal to
\[ k \cdot (\alpha D)^n. \]
Find the ordered pair $(k,n)$.

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