
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 $E$ be the determinant of a matrix whose column vectors are $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}.$ Find the determinant of the matrix whose column vectors are $2\mathbf{u} + \mathbf{v},$ $\mathbf{v} + 2\mathbf{w},$ and $2\mathbf{w} + \mathbf{u},$ in terms of $E.$

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