
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 $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ be vectors, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{a}, \mathbf{b}, \mathbf{c}$. Define a new matrix whose column vectors are $\mathbf{a} \times \mathbf{b}, \mathbf{b} \times \mathbf{c}, (\mathbf{c} \times \mathbf{a}) \times \mathbf{d}$. Find the determinant $E'$ of this new matrix in terms of $E$ and determine the factor $k$ such that $E' = k \cdot E^m$ for some integer $m$. Enter the ordered pair $(k,m)$.

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