
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
Given a cube \(A B C D -A^{\prime} B^{\prime} C^{\prime} D^{\prime}\). Point \(X\) moves at a constant speed along the perimeter of square \(A B C D\) in the direction \(A \to B \to C \to D \to A\), and point \(Y\) moves at the same constant speed along the perimeter of square \(B^{\prime} C^{\prime} C B B^{\prime}\) in the direction \(B^{\prime} \to C^{\prime} \to C \to B \to B^{\prime}\). Both points \(X\) and \(Y\) start their motion simultaneously from points \(A\) and \(B^{\prime}\) respectively. Find the trajectory of the midpoint of the line segment \(X Y\).

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