
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 $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be unit vectors such that $\mathbf{a} \cdot \mathbf{b} = 0$ and $\mathbf{a} \cdot \mathbf{c} = 0$. The angle between $\mathbf{b}$ and $\mathbf{c}$ is $\frac{\pi}{3}$. Then,
\[\mathbf{a} = k (\mathbf{b} \times \mathbf{c})\]
for some constant $k$. Enter all the possible values of $k$, separated by commas.

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