
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 \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) be vectors, and let \(E\) be the determinant of the matrix whose column vectors are \(2\mathbf{u}, \mathbf{v}, \mathbf{w}\). Determine the determinant of the matrix whose column vectors are \(\mathbf{u} \times \mathbf{v}, \mathbf{v} \times \mathbf{w},\) and \(\mathbf{w} \times (2\mathbf{u})\) and express it in the form of \(m \cdot E^p\). Enter the ordered pair \((m,p)\).

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