
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
On a plane, a segment \( \mathrm{AB} \) of length 1 is given, and an arbitrary point \( M \) is located on this segment. Squares \( AMCD \) and \( MBEF \) are constructed on the segments \( AM \) and \( MB \) as sides, respectively, lying on the same side of \( AB \). Let \( \mathrm{P} \) and \( \mathrm{Q} \) be the points of intersection of the diagonals of these squares, respectively. Find the geometric locus of the midpoints of the segments \( PQ \) as the point \( M \) moves along the entire segment \( AB \).

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