
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
There exists a scalar $k$ such that for any vectors $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ satisfying $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{d},$ the equation
\[k (\mathbf{b} \times \mathbf{a}) + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{d} \times \mathbf{a}\] holds. Find $k.$

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