
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 there be a line \( YY^{\prime} \) and a point \( A \) outside of it. We connect \( A \) with an arbitrary point \( B \) on the line \( YY^{\prime} \), and then draw a rectangle \( ABMN \) such that one side is \( AB \) and the diagonal \( BN \) is perpendicular to \( YY^{\prime} \). As \( B \) moves along the line \( YY^{\prime} \), determine:

(1) What is the locus of the point \( O \) where the diagonals \( BN \) and \( AM \) intersect?

(2) What is the locus of the point \( M \)?

(3) What is the locus of the point \( N \)?

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