
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 the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$: 

1. Let $X$ be an arbitrary point on the diagonal $AC$ of the cube, and $Y$ be an arbitrary point on the diagonal $B' D'$.
    a) What is the locus of all midpoints of the segment $XY$?
    b) Consider all points $Z$ on the segment $XY$ such that $ZY = 2XZ$. Find the locus of all such points $Z$.

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