
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 distinct points $A$ and $B$ respectively, perpendicular to the line $AB$. On line $m$, a point $C$ distinct from $A$ is taken, and on line $n$, a point $D$ distinct from $B$ is taken. Let the length of $AB$ be $a$, the length of $CD$ be $b$, and the angle between $m$ and $n$ be $\theta$. Then the radius of the sphere passing through points $A$, $B$, $C$, and $D$ is $\qquad$.

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