
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 vectors $\mathbf{a}, \mathbf{b}, \mathbf{c},$ and $\mathbf{d}$ satisfy $\|\mathbf{a}\| = 1, \|\mathbf{b}\| = \|\mathbf{d}\| = 1, \|\mathbf{c}\| = 3$, and $\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} + \mathbf{d} = \mathbf{0}$. If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{c}$, find all possible values of $\theta$ in degrees.

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