
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
The vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) are defined such that \(\|\mathbf{u}\| = 2\), \(\|\mathbf{v}\| = 1\), and \(\|\mathbf{w}\| = 4,\) and
\[\mathbf{u} \times (\mathbf{u} \times \mathbf{w}) + 2\mathbf{v} = \mathbf{0}.\]
Determine all possible values of \(\phi\), the angle between \(\mathbf{u}\) and \(\mathbf{w}\).

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