
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 \(\mathbf{b}\) and \(\mathbf{c}\) be vectors such that \(\|\mathbf{b}\|=2\), \(\|\mathbf{c}\|=3\), and the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(\frac{\pi}{3}\). A vector \(\mathbf{a}\) is orthogonal to both \(\mathbf{b}\) and \(\mathbf{c}\) and is expressed as \( \mathbf{a} = k (\mathbf{b} \times \mathbf{c}) \) for some constant \( k \). Find the possible values of \( k \) such that \(\mathbf{a}\) is a unit vector.

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