
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 a triangle $ABC$, two variable points $D$ and $F$ are taken on the external angle bisector of vertex $A$ such that the product of the distances $AD$ and $AF$ is constantly equal to $AB \times AC$. What is the locus of the point $M$ determined by the intersection of the lines $DB$ and $FC$?

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