
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 \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) be unit vectors such that the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\alpha\), and the angle between \(\mathbf{c}\) and \(\mathbf{a} \times \mathbf{b}\) is also \(\alpha\). If \(\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \frac{1}{8}\), find the smallest possible value of \(\alpha\), in degrees.

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