
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 skew lines $m$ and $n$ are drawn through two different points $A$ and $B$, respectively, and both are perpendicular to the line $AB$. Points $C$ and $D$ are taken on lines $m$ and $n$ respectively, with $C$ not at $A$ and $D$ not at $B$. Given that the length of $AB$ is $a$, the length of $CD$ is $b$, and the angle between $m$ and $n$ is $\theta$, find 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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