
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 in \(\mathbb{R}^3\), and let \(E\) be the determinant of the matrix whose column vectors are \(\mathbf{u} + 2\mathbf{v}\), \(\mathbf{v} + 2\mathbf{w}\), and \(\mathbf{w} + 2\mathbf{u}\). Determine the determinant of the matrix whose column vectors are \(\mathbf{u} \times (\mathbf{v} + 2\mathbf{w})\), \((\mathbf{v} + 2\mathbf{w}) \times (\mathbf{w} + 2\mathbf{u})\), and \((\mathbf{w} + 2\mathbf{u}) \times \(\mathbf{u} + 2\mathbf{v})\), expressed in terms of \(E\).

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