
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
Determine the scalar $k$ such that for any vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ satisfying $\mathbf{a} + 2\mathbf{b} + \mathbf{c} = \mathbf{0}$, the equation
\[k (\mathbf{b} \times \mathbf{a}) + 2\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}\]
always holds.

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