
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 point P is a moving point on the circle $x^2+y^2=4$, and a perpendicular line PH is drawn from point P to the x-axis with H as the foot of the perpendicular. Let the midpoint of the segment PH be E, and denote the trajectory equation of point E as C, with point A being $(0,1)$.

(1) Find the trajectory equation C of the moving point E;  
(2) If a line $l$ with slope $k$ passes through point A $(0,1)$ and intersects curve C at another point B, find the maximum area of $\triangle OAB$ and the equation of line $l$ at this time;  
(3) Is there a line $l$ with direction vector $\overrightarrow{a}=(1,k)$ ($k\neq0$) that intersects curve C at two distinct points M and N, and satisfies $|\overrightarrow{AM}|=|\overrightarrow{AN}|$? If it exists, find the range of values for $k$; if not, explain why.

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