
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
If the angle between two skew lines \(a\) and \(b\) (defined as the angle between their positive directions) is \(\theta\) (\(0 < \theta < \pi\)), and points \(A', E\) on line \(a\) and points \(A, F\) on line \(b\) are chosen such that the positive directions from \(A'\) to \(E\) and from \(A\) to \(F\) are the same as the positive directions of lines \(a\) and \(b\), respectively, and \(\angle EA'A = \alpha\), \(\angle A'AF = \beta\), and \(A'A = m\), then the distance between lines \(a\) and \(b\) is

$$
d = \frac{m}{\sin \theta} \sqrt{1 - \cos^2 \theta - \cos^2 \alpha - \cos^2 \beta - 2 \cos \alpha \cdot \cos \beta \cdot \cos \theta} .
$$

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