
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 $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be unit vectors such that the angle between $\mathbf{u}$ and $\mathbf{v}$ is $\beta,$ and the angle between $\mathbf{w}$ and $\mathbf{u} \times \mathbf{v}$ is $\frac{\beta}{2}.$ If $\mathbf{v} \cdot (\mathbf{w} \times \mathbf{u}) = \frac{\sqrt{3}}{8},$ find the smallest possible value of $\beta,$ in degrees.

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