
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
A cube with edge length $a$ has two parallel faces $ABCD$ and $A^{\prime}B^{\prime}C^{\prime}D^{\prime}$. A segment $XY$ of length $a$ has its point $X$ on the edge $AA^{\prime}$ and point $Y$ on the plane of $ABCD$. What is the locus of the midpoints of the segments $XY$?

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