
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
Find the scalar values $k$ and $m$ such that for any vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ with $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0},$ the equation
\[ k (m\mathbf{b} \times \mathbf{a}) + m\mathbf{b} \times \mathbf{c} + m\mathbf{c} \times \mathbf{a} = \mathbf{0} \]
holds true.

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