
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 \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{2} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let \(\phi\) be the angle between \(\mathbf{v}\) and \(\mathbf{w}\). Determine \(\sin \phi.\)

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