
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
 $AB$  and  $CD$  are two fixed parallel chords of the circle  $S$ .  $M$  is a variable point on the circle.  $Q$  is the intersection of the lines  $MD$  and  $AB$ .  $X$  is the circumcenter of the triangle  $MCQ$ . 
Find the locus of  $X$ . 
What happens to  $X$  as  $M$  tends to 
(1)  $D$ , 
(2)  $C$ ? 
Find a point  $E$  outside the plane of  $S$  such that the circumcenter of the tetrahedron  $MCQE$  has the same locus as  $X$ .


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