
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
Given vectors $\mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w}$ where $\|\mathbf{u}\| = 2$, $\|\mathbf{v}\| = \|\mathbf{w}\| = 3$, and they satisfy the following equation:
\[\mathbf{u} \times (\mathbf{u} \times \mathbf{w}) + 2\mathbf{v} = \mathbf{0}\]
If $\phi$ is the angle between $\mathbf{u}$ and $\mathbf{w}$, find all possible values of $\phi$.

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