
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
Given a matrix \(A\) whose column vectors are \(\mathbf{a}, \mathbf{b}, \mathbf{c}\), let the determinant \(D\) be defined as \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\). Determine the determinant of a matrix \(B\) whose column vectors are \(\mathbf{2a} + \mathbf{b}\), \(\mathbf{2b} + \mathbf{c}\), and \(\mathbf{2c} + \mathbf{a}\).

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