
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 nonzero vectors, no two of which are parallel, such that
\[
(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.
\]
Let \(\theta\) be the angle between \(\mathbf{b}\) and \(\mathbf{c}\). Find \(\sin \theta\).

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