
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
Two points, \(M\) and \(N\), move along two perpendicular lines such that the distance \(MN\) remains constant. Let \(P\) be an arbitrary point on the segment \(MN\). What is the locus of points \(P\) as points \(M\) and \(N\) traverse the section determined by the problem (of length \(2 \cdot MN\))?

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