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

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