
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  $ABC$  be an equilateral triangle. For a point  $M$  inside  $\vartriangle ABC$ , let  $D,E,F$  be the feet of the perpendiculars from  $M$  onto  $BC,CA,AB$ , respectively. Find the locus of all such points  $M$  for which  $\angle FDE$  is a right angle.



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