
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
Vectors $\mathbf{d}, \mathbf{e},$ and $\mathbf{f}$ satisfy $\|\mathbf{d}\| = 2, \|\mathbf{e}\| = 1,$ and $\|\mathbf{f}\| = 3.$ The vector equation given is:
\[
\mathbf{d} \times (\mathbf{d} \times \mathbf{f}) + 2\mathbf{e} = \mathbf{0}.
\]
Find all possible values of $\theta$, where $\theta$ is the angle between $\mathbf{d}$ and $\mathbf{f}$.

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