
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 \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) satisfy \(\|\mathbf{a}\| = \|\mathbf{b}\| = 1\) and \(\|\mathbf{c}\| = 3\). Also, \(\mathbf{a} \times (\mathbf{a} \times \mathbf{c}) + \mathbf{b} = \mathbf{0}\) and \(\mathbf{d} \cdot \mathbf{a} = \mathbf{d} \cdot \mathbf{b} = 0\) where \(\mathbf{d}\) is another vector in the space. 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}
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