
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
Let \(ABCD - A'B'C'D'\) be a cube. \(X\) is a moving point on segment \(AC\), and \(Y\) is a moving point on segment \(B'D'\).
1. Find the locus of the midpoints of segment \(XY\).
2. If \(Z\) is a point on segment \(XY\) such that \(ZY = 2XZ\), find the locus of point \(Z\).

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