
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 that the standard equation of circle C is $(x-2)^2 + (y-4)^2 = k$ ($k > 0$), if circle C intersects the y-axis at points A and B, with point A above point B, and circle C intersects the x-axis at points E and F, with point E to the right of point F, determine the equation for the trajectory of the midpoint M of AE.

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