
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
In space, two different points \( A \) and \( B \) are used to draw two skew lines \( m \) and \( n \) perpendicular to the line \( AB \). On the line \( m \), a point \( C \) different from \( A \) is selected, and on the line \( n \), a point \( D \) different from \( B \) is selected. Let the length of \( AB \) be \( a \), the length of \( CD \) be \( b \), and the angle between \( m \) and \( n \) be \( \theta \). Determine the radius of the sphere passing through the four points \( A \), \( B \), \( C \), and \( D \).

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