
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 $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors, and let $D$ be the determinant of the matrix $[\mathbf{a}, \mathbf{b}, \mathbf{c}]$. Consider the matrix formed by column vectors $\mathbf{a} + \mathbf{d},$ $\mathbf{b} + 2\mathbf{d},$ and $\mathbf{c} + 3\mathbf{d}$ where $\mathbf{d}$ is another vector. Calculate the determinant of this new matrix and express it in terms of $D$ and $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$.

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