
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
The square \(ABCD\) rotates around its fixed center. Find the geometric locus of the midpoints of segments \(PQ\), where \(P\) is the foot of the perpendicular dropped from point \(D\) to the fixed line \(l\), and \(Q\) is the midpoint of side \(AB\).

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