
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 \( \mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = 0 \), and the angle between \( \mathbf{b} \) and \( \mathbf{c} \) is \( \frac{\pi}{3} \). Then
\[ \mathbf{a} = k (\mathbf{b} \times \mathbf{c}) \]
for some constant \( k \). Find all possible values of \( k \).

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