
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
A line $e$ rotates around the point $M$. The segment $A B$ of variable length is positioned on the line $e$ such that the product $A M \cdot M B$ is constant. The midpoint $F$ of the segment $A B$ moves on a circle passing through $M$, and $A M < B M$. What is the locus of the points $A$ and $B$?

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