
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
Given skew perpendicular lines \( a \) and \( b \) and a point \( P \). What is the set of points \( M \) for which the sum of the lengths of the projections of the segment \( PM \) onto the lines \( a \) and \( b \) is constant? 

[The projection of a point \( A \) onto a line \( l \) is a point \( A' \in l \) such that \( (AA') \perp l \); if \( A \in l \), then \( A' = A \). The projection of a segment \( AB \) onto a line \( l \) is the segment \( A'B' \), where points \( A' \) and \( B' \) are the projections of points \( A \) and \( B \) onto the line \( l \).]

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